4:- {”A' fireH 'v v :Iva'm'pj‘ .,,. .- Nn‘r ...... ..,A.A.,.A,,_ A. ,.,. _ -A- .- -: «‘04 .J. p". ' ' w-uw. Al 1..., «"1 ’I I" r;-.-";.,...r ’- rm l- sw. Let us examine the underlying economic meaning of this relation. Under full employment conditions an increase in investment expenditure must bring about an increase in both the ratio of investment to output (I/Y) and also an increase in the ratio of saving to output (S/Y). This is a necessary condition if a new equilibrium is to be obtained. If the saving-output ratio did not rise, the result would be a continuous upward movement of the general level of prices. With a higher absolute level of investment, the continued equilibrium can only be achieved either by a change in the propensity 13 to save itself, or by a shift in the distribution of income from the class with the lower propensity to save to the class with the higher propensity to save. But Kaldor rules out the first possibility by his assumption that both sw and 3p are constant. Here we can see how important the assumption of differential saving propensities between wage earner and profit earner plays in this theory. This is the essential feature of this macro distribution theory. Whenever there is a rise in the investment-output ratio, the only way the economy can stay in equilibrium is to increase the saving-output ratio for the whole economy by the redistribution of income in favor of the profit earner. The mechanism to redistribute income for new equilibrium is essentially that of the price level. The increase in investment expenditure under full employment conditions leads initially to a general rise in prices. But in the absence of the economic force to equalize marginal productivity the real wage rate, no mechanism exists to insure that money wages rise at the same rate as prices (Sen, 1963; Kaldor, 1956). The failure of money wages to keep pace with the rise in the general price level will thus reduce the real income of wage earners. This inflation-induced shift in the distribution of income in favor of profits will raise the overall level of real saving in the economy. This process will continue until the saving-output ratio is once again in equilibrium with the investment-output ratio. But the critical assumption of this theory is that the propensity to save of the profit earner is greater than that of the wage earner. Without this assumption, any change in the distribution of income will not l4 affect the saving—output ratio, and thus the system would be unstable. One of the logical slips in Kaldor's theory was pointed out and corrected by Pasinetti (Pasinetti, 1962). He pointed out that when any individual saves a part of his income, he must also be allowed to own it, otherwise he would not save at all. Thus it is clear that some part of total profits must accrue to workers as a result of their past savings. Pasinetti reformulated the original Kaldor model so as to reflect this observation. But his basic system of relations is rather similar to that of Kaldor. The neo-classical distribution theory is the generalization of Ricardo's marginal principle distribution theory, so as to make this principle hold true for any factor. Under competitive conditions, any factor variable in supply will obtain a remuneration which must correspond to its marginal product. Thus this theory requires the assumption that the production function be homogeneous of the first degree for all factors. All factor prices and employment are determined by market forces. Given quantities of each factor employed, the total product is determined by the production function. Total output is distributed to the factors in the production process. The shares of the factors are determined by their relative quantities and their relative prices; the price ratios, in turn, are equal to the respective marginal rate of substitution between the factors. With a given technology, the factor shares may change as a result of changes in their relative quantities, changesin their relative prices (or marginal rates of substitution), and the relationships ‘15 between these two kinds of change. This leads to the important concept of elasticity of substitution: the ratio of proportional change of the factor ratio to the proportional change of the marginal rate of sub- stitution, or to the proportional change of the factor price ratio under competitive conditions at a given isoquant. For example, unit elasticity of substitution between capital and labor in two factor production implies constancy of the relative income shares. For another example, if the elasticity of substitution is greater than one, say two, a change of one percent in the factor price ratio will be followed by a two percent change in the ratio of factor employment. Therefore, it will result in a one percent change in the relative factor share ratio. In other words, the competitively imputed share in output of the more rapidly growing factor rises. The opposite is the case, if the elasticity of substitution is less than one. The second possibility is that in the course of time the tech- nology changes, i.e., the whole production function shifts, or the quality of factors changes. The direction and magnitude of the changes of the marginal rate of substitution at given factor ratios will depend on the characteristics of technical change. This is Hick's criterion of classifying technical progress. At given factor ratios, technical progress is capital-using, neutral, or labor-using, according as the marginal rate of technical substitution of capital for labor decreases, remains unchanged, or increases. At a given input price ratio, capital- using technical change will provide an incentive to substitute capital for labor, i.e., to increase the capital-labor ratio. 16 From the above discussion, we can easily derive two important neo- classical propositions about the relative income shares. The first holds that a factor-saving technical progress, other things constant, reduces the relative share of the income, or the output elasticity of that factor (Hicks, 1932, p. 122). The second maintains that if one factor increases in supply more rapidly than another, and if the elasticity of substitution is less than unity, then the relative share of the first factor decreases (Hicks, 1932, p. 115). This study uses the neoclassical framework of income distribution. By specifying the form of production function and the representation of technical change in the production function, the above relations of the marginal productivity theory of distribution will be discussed more specifically in Chapter III. CHAPTER II ANALYTICAL FRAMEWORK OF THE STUDY Primary distribution of income is basically the result of produc— tion and consumption decisions. Individual incomes are aggregate rewards for productive services at a given distribution of resource ownership. Given their production functions, production units attempt to employ factor services and produce products in such a way as to maximize their flows of profit. These decisions determine demand functions for factor services and supply functions for products in terms of factor and pro- duct prices as parameters. On the other hand, consumption units attempt to offer factor services and purchase products in such a way as to attain maximum utility flows. These decisions determine demand func- tions for products and supply functions for factor services in terms of factor and product prices as parameters. Given the demand and supply functions determined by the production and consumption units, the market adjusts toward a set of prices for factor services and products that clear all product and factor markets. In the process of the above relations of economic forces, output is produced and distributed to the resource owners. Each factor receives its price which is determined in the market based on its marginal productivity. This chapter will review some relationships of production parameters to the distribution. In the first section we will discuss some distributional aspects of production functions, particularly the CES (Constant Elasticity of Substitution) function which will be used in this study. The second 17 18 section will discuss some alternative formulations of technical change which are widely used in empirical work, and a factor-augmenting assumption of technical change will be introduced in our production relation. The final section will derive distributional relationships with our specified production function and technical change under the marginal productivity theory of wages. Aggregate Production Function A production function is an attempt to describe the physical facts of a given technology, showing the relation by which the services of productive factors are transformed into output. The conceptual basis for believing in the existence of a simple and stable relation- ship between measures of aggregate inputs and a measure of aggregate output is not very sound. But an aggregate production function is a very convenient concept in many economic areas, and it has served as a basic framework for many empirical studies. The simplest form of a production function widely used in neo— classical macrodistribution analysis is the Cobb-Douglas function. The constant exponent parameters of this function are direct criteria of distribution between factors. Thus the relative share of labor, or the elasticity of production with regard to labor, is independent of the capital intensity or capital accumulation relative to labor input. This relationship can be seen as follows. From the CD function, Y = a La KB, we can derive the marginal productivities of labor and capital as MPL = a(Y/L) and MPK = B(Y/K), where Y is output, L and K are labor and capital inputs respectively, and a, a and B are parameters. 19 Assuming competitive markets, a(Y/L) = W/P and 8(Y/K) = R/P, or a = (L'W)/(Y°P) and B = (R'K)/(Y-P), where W and R are money wage rate and per unit return to capital, respectively and P is price of output. Thus in equilibrium, the coefficients a and B will directly measure the share of total receipts paid to labor and capital. Another critical feature of the CD function is the extent to which it permits the substitutability between factors. We can see this by examining the marginal rate of substitution of labor for capital as MRSLK = MPL/MPK = (I'K)/(B'L). From this one may easily see that the elasticity of substitution, which is defined as the pro- portionate change in the ratio of capital to labor divided by the pro— portionate change in the ratio of marginal productivities, is unitary in the CD function.* Thus for any level of output and inputs of the CD function, one input can always be substituted for another input at a fixed elasticity of unity. But many production processes may have very low elasticities of substitution, or one may prefer a function with fixed requirements of each input in order to produce a unit of output, which is called the Leontief production function. Or at the other extreme, it may be possible to imagine processes where the extent of substitution is very high indeed. The isoquants may be nearly flat. Unitary elasticity of substitution is required for constancy of relative shares, since it *From the expression of MRS ln(K/L) = ln(B/a) + ln(MPL/MPK). LK’ dln(K/L) _ 1 dln(MPL/MZPK) ‘ ° Thus by definition, 0 = 20 implies that the relative quantities of inputs vary by the same pro— portions as their relative prices. Thus the CD function is not able to explain the movements of relative factor shares. The CES production function does not assume that the elasticity of substitution is unitary, but does assume that it is constant. There— fore, the CES function can allow for, and suggest the direction of, changes in relative shares. The rest of this section will review the basic form and some properties of the CES production function which will be used in this study. The general two-factor CES production function can be written as: - _ y. Y = y[6K p + (1-6)L 9]" o (2-1) where y, 6, p and v are respectively the parameters of efficiency, capital intensity or distribution, substitution, and degree of returns to scale. This CES function was derived from an empirical relationship between average productivity of labor and the real wage rate (logarithmic relation) and from the assumptions of competitive equilibrium (Arrow, et a1., 1961). The interpretation of the efficiency parameters, y, can be easily seen. Clearly this varies according to the units in which output and input are measured. But if we use the same units for measuring inputs and output, and we compare production.functions with different y's, the one with the higher 7 will have the more efficient production relation. It will also be easy to observe the meaning of the scale parameter v. The increase in both inputs by a factor A will give rise to an expansion of output of Av, and so when v > 1 there will be increasing returns to scale, and when v < 1 there will be decreasing returns. 21 If there are constant returns to scale (v=l), then the marginal product relations can be written as: _1_ MPL = (1-5)y‘° (Y/L)O (2-2) 1 MPK = 67-p(Y/K)U (2-3) where o = (1+p)-1. o is the elasticity of substitution between capital and labor, as will be shown below. The Cobb—Douglas (CD) function is the special case with p=0, in which case the exponents are 5 and (1-6) respectively; this accounts for calling 6 the distribution parameter. The CES production function has all the properties of the neo- classical production function and includes the Cobb-Douglas and Leontif production functions as special cases. From (2-2) and (2-3) it is obvious that the CES function satisfies the condition that marginal product of each input, MPL and MPK, are positive. It also can be shown that 3MPL/3L and 3MPK/8K are negative for reasonable values of the scale parameter v, including v=1. For sufficiently large values of v, these partial derivatives become positive, but such large values of v are not likely (Brown, 1966). Also if p is positive, the CES function does reach a finite maximum as one factor increases while the other is held constant. Hence limits do exist when 0, the elasticity of substitution, is less than unity. However, when the elasticity of sub- titution is greater than unity, the function does not have a limit. We have mentioned above that when p=0, i.e., the elasticity of substitution is unity, the CES reduces to a CD function. The relation between the two functions can be shown by means of a Taylor series 22 approximation of the CES function; this approximation may also be used in one method of estimating the parameters of the CES function (see Chapter V). From the CES function (2-1), the following can be defined: Z(p) = 5K7“ + (1-6)°L-p, and ¢(p) = 1n 2(0)- The term ¢(p) can now be expanded around the value p=0 in a Taylor Series approximation. Then, disregarding the terms of third and higher orders, the expansion becomes 2 ¢(0) + p¢'(0) +-% ¢"(0) + . ¢(o) (1n K + 1n L)2 + . . 2 = -p[61n K + (1—6)1n L] + E_§él:§l_ Taking logs of the CES function (2-1), we get 1n Y = ln Y - g-ln Z(p), i.e., 1n Y = 1n Y + v6 In K + v(l-6) In L - 2Bg-‘é-l-tél-(ln K - 1n L)2 <2-4) This form is linear in the unknown parameters and allows direct estimation of parameters; this estimation method was proposed by Kmenta (Kmenta, 1967). Setting p=0 in equation (2—4), we get 1n Y = 1n y + v6 In K + v(l-6) 1n L. (2-5) Thus when p=0 the CES function is a CD function. The marginal rate of substitution of labor for capital can be derived by taking the ratio of the marginal product of labor to the marginal product of capital. From (2—2) and (2—3), the marginal rate of substitution of labor for capital (MRSLK) can be expressed as: 1 MRS =—=——--—-—-—(—) (2—6) This equation suggests some important relations. A small value of 6 implies that the production process has labor intensive characteristics. For a small value of 6, the marginal product of labor is high relative to that of capital for a given capital—labor ratio. Thus a unit reduc- tion in the labor input has to be compensated for by a larger increase in the rate of capital than if the process were less labor intensive (larger value of 6). In this sense, 6 is a measure of capital intensity of the technology. Another aspect to note in the equation (2-6) is how the elasticity of substitution, 0, affects the marginal rate of substitution. At a given factor ratio, a high value of 0 means that capital can be easily substituted for labor. In other words, if we reduce the rate of capital input by one unit we have to increase the rate of labor input by a greater amount when the factors are not easily substituted for each other, other things being constant. A high value of 0 means greater similarity between inputs, and vice versa. Equation (2—6) also implies that when the value of o is low, diminishing returns to labor set in more rapidly than when the value of o is at a higher level. To show that o is the elasticity of substitution, let us take the equation (2-6) in logarithms. We get ln(MPL/MPK) = 1%? + 2]? 1n Q). (24) Taking the derivative with respect to we get I, d(MPL/MPK) ___ L d(K/L) MPL/MPK o K/L 24 The solution for o is = d(K/L) / d(MPL/MPK) K/L MPL/MPK (2'8) which is the definition of the elasticity of substitution. When there exists equilibrium in the market, we can write Equation (2-6) as 1-6 K'%' w ‘—3— (L) = f. (2-9) where r is the real rental of a unit of capital and w is the real wage rate. Multiplying-i-in both sides of Equation (2-9), we get = —— <-—) (2-10) The expression (w-L)/(r-K) is the relative share of labor to capital. Clearly, as the value of 0 goes to unity, the relative share of labor to capital approaches to the value of (1-6)/6. Hence, in a Cobb-Douglas production function, the ratio of relative factor shares can be represented by the ratio of the exponents, or production elasticities of the factors. But with the CES production function, 6 is not sufficient to determine the distribution of income between factors. We also require a knowledge of the substitution parameter, 0. In the CD function, we found that the distribution of income between factors depends on only the coefficients of the production function, and thus the distribution did not vary with the factor ratio. With the CES production function, however, we see that the distribution is a function of the factor ratio. 25 The greater the deviation of o from zero, the greater the effect of the factor ratio on the distribution of income. So far, our production function has been related to physical output and physical inputs, in which the relation was assumed stable over time. But many empirical works show that there is an important factor which changes the production relation and thus changes output or the productivities of physical inputs at given factor inputs. This intangible factor is named simply technical change. Sometimes it is interpreted as an aggregate effect of missing inputs in the production relation, such as investment in research and extension, education and other social investments which change the environment on which the production relations are based. There are numerous empirical studies which have attempted to incorporate these various input variables in production relations. The next section will discuss some possible alternative hypotheses which represent the variable, so-called technical change, in a production function framework. Representation of Technical Change In empirical work, technical progress may be formulated as "embodied" or "disembodied." "Disembodied" technical progress applies equally to all resources in current use. But in "embodiment" assumption, technical progress is embodied only in the new capital equipment of improved design or in new labor of enhanced skill as opposed to existing machines or to labor trained at earlier times. Thus the investment in new equipment or new skills is the essential carrier of new technology. 26 A general way to represent technical change is to write Y = F(L, K; T), where Y is output, L and K are inputs measured by physical unit, and T is a parameter or a vector of parameters, each value of which corresponds to a different level of technology. If we conceive of technical change altering the production function, then the parameters of that function must change with different levels of T, e.g., in our CES case, 6, p, y and v. Any one of the parameters can be thought of as a function of T, or simply a smooth or discrete function of time. The discrete concept of technical change was used to measure technical change by Brown who refers to technical epochs (Brown, 1966). One very restrictive but convenient specification of technical change is that the effects of technical progress are neutral or uniform in the sense that marginal rates of substitution do not change with different levels of technology at a given capital-labor ratio. In that case, we could write the shifting production function as Y = a(t) F(K,L). The production function shifts over time simply by a uniform upward dis- placement of the whole function without disturbing the balance between capital and labor in current production. It implies that the distribu- tion of income remains unchanged at the same capital-labor ratio in a competitive economy. This formulation was used by Solow (Solow, 1957) to estimate the rate of technical change after testing neutrality. This formulation implies that all technical progress is a way of improving the organization and operation of inputs without reference to the nature of inputs themselves. Technical change is disembodied. 27 The isoquant contours of the production function shift inward toward the origin as time passes. ”It floats down from the outside." (Solow, 1959). Another assumption to represent technical progress is that technical advance takes the form of making labor and capital input more productive. Formally, we could write Y = F(a(t)K, b(t)L) where a(t) and b(t) are factor augmentation rates (Solow, 1967). Thus a(t)K and b(t)L are interpreted as inputs of K and L in efficiency units. Under this assumption, we can write the production function as stable by adjusting inputs in efficiency units. But if we take input variables as natural units, the production function will shift over time. Technical change is said to be purely labor augmenting if A(t) = 0 and b(t) > O,* whereas it was purely capital augmenting if b(t) = 0 and a(t) > 0. It is equally capital and labor augmenting if a(t) = b(t) > 0. These definitions of technical change are related to the various classifications of technical change, which are based on its effect on the relative factor shares. The theory of marginal productivity is used to analyze the effects of technical change on factor income. For this reason, the technical change has been classified according to its neutrality in terms of its effect on the relative shares of labor and capital: neutral, capital— saving, or labor-saving technology. *The dot over a variable indicates the proportional rate of change (or growth rate) of the variable, for example, ° = d 1n X _ d§_ 1 X dt “x dt' 28 Hicks classified the factor saving bias of technical change at a constant capital-labor ratio (Hicks, 1932). His definition of neutrality means that the technical change does not affect the marginal rate of substitution of capital for labor at a given capital-labor ratio. Thus in competitive conditions, it implies that the distribution of income remains unchanged at the same capital—labor ratio. This definition of neutrality is equivalent to the above definition of the equally capital and labor augmenting technical change. Technical change is said to be labor-saving (or capital—using) if the relative share of labor is decreasing at a constant capital-labor ratio. Technical change is said to be capital-saving (or labor-using) if the relative share of labor is increasing at a constant capital-labor ratio. But Harrod measures the bias of technical change along a constant capital-output ratio, and Solow measures it along a constant labor— output ratio. By these classifications, Harrod's neutrality is equivalent to pure labor augmenting, and Solow's neutrality is equivalent to pure capital augmenting technical change (Uzawa, 1961; Allen, 1968). Thus factor augmenting specification of technical change is a generalization of various definitions of neutrality. It seems usual to think of technical change as occurring through time, perhaps smoothly, particularly at the macro level, as knowledge accumulates. Technical progress proceeds at a proportional rate %-%%3 generally varying over time. If it is at a constant proportional rate m, then %-%%-= m, with a=l at t=0. Hence a(t) = em". This assump- tion has been widely used in empirical work. It may be doubtful that inventive and innovative processes within a single firm can be 29 represented by smooth exponential time trends. However, when many discrete and almost random influences are aggregated, it seems reasonable to suppose that aggregate technical change can be represented by a smooth time trend. With the above specification of factor augmenting technical change, the CES production function may be written as 1 Y = ytsx>"’ + <1—s>L>“’1 p (2-11) where a(t) and b(t) represent efficiency indexes of the conventional inputs of capital and labor. L and K represent conventional measures of the physical flow of labor and capital inputs. The changes in a(t) and b(t) through time are interpreted as capital-augmenting and labor-augmenting technical changes, although this says nothing about the sources of such efficiency growth. Note that the neutral component of technical progress is also embodied in a(t) and b(t) if we assume that the neutral efficiency parameter y is constant over time. The increase in a(t) or b(t) has the same effect on output as an equiproportional increase in inputs. Therefore factor augmentation restricts technical change so that it cannot alter the form or the parameters of the production function. It enters by changing the quantity of the effective factor unit. The course of technical progress is often described by an index of the rate of progress (R) and an index of its bias (B). The rate of technical progress, R, is defined as 30 Considering F as homogeneous of degree one in the two inputs, the rate of change can be rewritten as* KF + LF . . _ Kt Lt _ , _ , R ' KFK + LFL SL FL + (1 SL) F K (2-12) where FKt and FLt are the changes over time of the marginal products of capital and labor respectively, and S is the wage income share. Thus L the rate of technical progress is the share-weighted sum of the pro— portional changes in the marginal products of each factor. Using the Hicksian manner, the bias of technical progress B is defined as the proportional change in the marginal rate of substitution of labor for capital, i.e., - 5.1.. = ' _ _ B — dt 1n (FR/FL) FK FL (2 13) Note that FK and FL are functions of time. It shows what happens to the marginal rate of substitution between L and K for a fixed capital-labor ratio as the level of technology t changes. With the production function (2-11) the bias of technical progress (B) can be expressed in terms of a and b by using the definition of bias.** B = (1713- ) (51-13) (2—14) From the above expression, it is apparent that the concept of factor augmenting technical progress is related to the Hicksian concept *MPL and FL both refer to the marginal product of labor. We shall use whatever symbol is most convenient in each context. A similar remark applies to the symbols MPK and FK for the marginal product of capital. 1**From production function (2-11), FK = -pa and FL = -p6, where p = __ - 10 o 31 of neutral, labor-using and capital-using technical progress. The equality of the rates of growth of labor and capital efficiency, or unitary elasticity of substitution, is exactly equivalent to neutrality in Hick's sense. Technical progress is labor-using if the elasticity of substitution is less than unity and the rate of capital augmentation exceeds the rate of labor augmentation, or if the elasticity of sub— stitution is greater than unity and the rate of labor augmentation exceeds the rate of capital augmentation. Similar statements apply to capital-using technical progress. The form of factor augmentation needs to be specified when stat— istical estimation is attempted. One common specification is to assume that factor augmentation occurs at a constant exponential rate (David, 1965; Ferguson, 1969). But Lianos specified a(t) and b(t) as a(t) =aota and b(t) = botB (Lianos, 1971). This formulation implies that the rate of factor augmentation is declining since a = atul and b = Bt-l. Fishelson explained that this formulation is less restrictive than constant exponential specification (Fishelson, 1974). Marginal Productivity and Distribution Theory Based on the specified formulation of the production function and technical change, this section will derive the relationships under— lying the sources of change in the relative wage income share, which we will apply to explain its behavior. For doing this we need to review the general economic mechanism of distribution of output to the resource owners under the marginal productivity theory. The distribution mechanism in marginal productivity theory is basically the micro-economic problem of the determination of the employment and 32 the prices of the factors of production. Thus the distribution theory has come to mean the process by which factor prices are determined through the interplay of the producer's demand for the factors and the supply conditions for these factors. We assume there exists an aggregate production function with smooth factor substitutability and marginal productivities as Y = F(K, L) (2-15) where F is homOgeneous of first degree in the homogeneous inputs, capital (K) and labor (L). By its homogeneity property, the production function may be written in a per capita form as y = f(k) (2-16) k > 0 and fkk < o. The per capita form is specific to the linear and homogeneous where y =-%-and k =-%; By assumption, f function, i.e., to the case of constant returns. It is also assumed that production is based on profit maximization under perfect competi— tition. The necessary conditions for profit maximization imply that production should be pushed to the point where the marginal products are equal to the corresponding factor prices. From Equation (2-16) these conditions may be written as MPL f(k) - kfk(k) = w (2-17) MPK fk(k) = r (2-18) where w and r denote the real wage rate and the rate of return on capital. These equations are the derived input demand equations. The above conditions provide for the distribution of output to the factors. From the equations (2—17) and (2—18),we have 33 rk + w = kfk(k) + f(k) - kfk(k) = Y Hence the output divides in per capita terms as y = rk + w. Multiplying by L on both sides, it can be seen that total output is distributed as Y = rK + wL. Thus with constant returns to scale under the competitive equilibrium, the total product is just sufficient to pay each input its marginal product. Under the optimum position, the above relationships of determina- tion of factor prices and distribution of output can be seen more clearly in Figure 1. Given the per capita production function, y = f(k), which describes technical conditions of production, the rate of return on capital (r), wage rate (w) and output per capita (y) are determined by the selection of factor ratio (k). The rate of return on capital is the slope of the tangent at the point P, and the wage rate is the intercept OW which is equal to f(k) — kfk(k). Thus the share of wage income is also determined by-%%, and g% is the share of capital income. For a given stable production function, Figure 1 shows the inter— relationships among these variables: wage rate, returns to capital, capital-labor ratio, and output per unit of labor input. Any one of those variables is determined; the rest of the variables are also uniquely determined. Thus, assuming no technical change in the pro- duction, the distribution of output is determined purely by the shape of production function and the capital—labor ratio. It was also shown that, from Figure l, the wage rate (w) is the distance OW, and the rate of return to capital (r) is the slope of the tangent line wr in the competitive equilibrium. The slope of the tangent line can be expressed as the ratio of CW to 0B. But note that 34 Y1 N-..._ w l P / ; B o M k Figure l. The Distribution of Output 35 the distance OW is the wage rate. Thus, r =-%§, or OB ='¥u From this we can easily see the effect of the capital-labor ratio on the distance OB, which is the ratio of the wage rate and the rate of return to capital. The higher capital—labor ratio is associated with the higher value of ¥3 Thus the capital-labor ratio has a positive relation with g-and a negative relation with a; That is, k = f(g). The elasticity of this curve, as a usual definition, is the elasticity of substitution which we defined as the responsiveness of the capital— labor ratio (k) to the prices of capital and labor. We can also see the relationship between the elasticity of substitution and the ratio of relative shares by writing the ratio of relative shares as fig = (K/L)/(w/r)- From Figure 1 it also can be seen that higher capital—labor ratios are associated with lower values of r and with higher values of w. This implies an inverse relationship between the wage rate and the rate of return to capital, which is called "Factor price frontier" or "wage frontier" (Samuelson, 1962). It is important to note that the slope of the w—r curve (3%) at any point is equal to the capital- labor ratio (k). From Equation (2—17) %¥-= -kf > O and from kk dr- 1911:5111 51.5,” Equation (2-18) dk fkk < 0. Hence (dw)/(dr) dr dk dk k < 0. It is therefore clear that the elasticity of the wage frontier (- EA!) must equal E-k =-E§, i.e., the ratio of relative shares of w dr w wL capital and labor. So far we have examined the marginal productivity theory of distribution with a stable production function and the assumption of competitive market. The theory tells us that the distribution 36 of output is changed by the changes in the factor ratio for a given stable production function. This theory also implies that if the economy gorws in the steady state equilibrium path, or at constant capital-labor ratio and constant output-labor ratio, then the distribution of output remains constant. But when the economy is out of a long-run steady—state equi— librium growth path, there always exists the possibility to change the distribution of output as the capital-labor ratio changes, depend- ing on the nature of the production technology. There is also the possibility that the production technology is changed and biased, which may shift the production function and thus lead to a change in distribution. Under the assumptions we made in the above, the relation- ship of the relative wage income share can be derived in terms of changes in capital-labor ratio and technical progress. The share of labor (SL) is defined simply as S = (wL)/Y. L Assuming the competitive equilibrium conditions, then the real wage rate is equal to the marginal product of labor, and thus the relative share of labor SL can be written as LF _ L ___ _ sL — Y MPL/APL (2 19) Equation (2-19) says that the wage income share is equal to the ratio of the marginal product of labor to the average product of labor. It also shows how important the law of diminishing returns is to the distribution. Equation (2-19) may also be written as r' 0'0) If" *4 \ HIV-4 37 This is the definition of the elasticity of production with regard to labor. It says that the relative factor share of income is equal to the elasticity of production with regard to each factor. From the concepts introduced above, the important relationship of the rate of change in wage income share can be derived by the following procedure. This derivation procedure is based on Ferguson (Ferguson, 1968). First, the pr0portional time changes in the marginal products of labor and capital can be solved from Equations (2-12) and (2-13) as functions of the rate and bias of technical progress:* FKt ———-= R + s B (2—20) F L K F -—EE = R — (1—3 )8 (2‘21) FL L Next, the rates of growth of the marginal products and of output may be expressed as functions of the rate and bias of technical progress, the elasticity of substitution, and the rates of change in the capital and labor:** *In what follows, when a subscript is attached to a functional notation, it indicates partial differentiation. Thus, for example, Ft = 3F/3t. A superior dot denotes the proportional rate of change (or growth rate) of the variable,F for example, - d 1n F 1 dF _8F F = dt = F dt' Note thatd F# Ft =8t° For example, dF dK dL dt Ft + FK_ dt + FL dt' **From the assumption of linear homogeneity, —LFLL = KFLK, - KFkk = LFKL’ o = FKFL . See Allen (1938, pp. 340-343). Note that dFK = F FKL dt + F -95 + F dL KKdt KLEE' 38 . _ i ._. ' _ FK — R + SLB - O SL(K L) (2 22) . _ l _ ._. _ FL — R - (l-SL)B + O(1 SL)(K L) (2 23) R = R + (1-sL)(Rri) + t . (2-24) Finally, from the definition of the wage income share, the rate of change in the labor share can be expressed as O O + O - 0 SL + FL L F Substituting Equations (2-23) and (2-24) into above equation, we may get our main expression for the rate of change in labor's relative share. . 1 . . SL = -(1-SL)[B + (l-E;)(KrL)] (2-25) Equation (2—25) shows that the rate of change of the wage income share depends on the bias of technical change (B), on the value of the elasticity of substitution, 0, and on the direction of change in the Thus . ° Kt LFL FFKL ° ° FKt SL ' ° FK=E—'T ET‘K‘L>=E—'o—- K K L K Substituting Equation (2-20) into the above equation, one may get the Equation (2-22). Equation (2-23) can be obtained in the same way. dF _ dK £1.13 To obtain Equation (2—24), HOte that dt ‘ Ft + Fdet + FL dt . F KF LF . Thus, F-FE +-—E§ K +'—§L L - Substituting Equations (2~12) and (2-19) into the above equation, one may get the Equation (2-24). 39 capital-labor ratio. Of course, if the technical change is neutral, that is, B = 0, then the rate of change in the wage income share depends only on the value of the elasticity of substitution and the growth rates of capital and labor. Thus Equation (2-25) suggests that there are many ways that the relative factor share can be changed over time. For example, if the elasticity of substitution is unity, the relative share of labor will increase, remain constant, or decrease accordingly as the bias of technical change is negative, zero, or positive. If the elasticity of substitution is not unitary, there are many ways the relative share can be changed, depending on the direction and magnitude of technical bias, the elasticity of substitution, and the growth rates of capital and labor. For another example, the constancy of a relative share may be explained in two ways. The first case is where the technical progress is neutral and the elasticity of substitution is unity. For the second case, suppose that the technical progress is capital—using and that the elasticity of substitution is less than unity and the growth rate of capital-labor ratio is positive. Then the relative share will remain constant if the decrease in relative demand for labor attributable to capital-using technical progress is exactly offset by the decrease in the relative supply of labor attributable to capital deepening and inelastic substitutability. Alternatively, if the elasticity of substitution exceeds unity, shares will remain constant if the increase in the relative demand for labor attributable 40 to labor-using technical progress is precisely offset by the increase in the relative supply of labor. Unless these conditions are exactly satisfied, the relative shares will change over time. Substituting the expression for the bias of technical change (B) from the Equation (2-14) into Equation (2—25), the rate of change of wage income share can be written as . _ 2:]; . o _ . . - SL — -(l-SL)( o )[(a + K) (b + L)] (2 26) From Equation (2—26) it appears that the behavior of the relative share of labor, for a given value of 0, depends not just on the capital- labor ratio in a conventional unit but also on the changes in the productivity of the two factors. The term in brackets on the right- hand side of Equation (2-26) may be viewed as the adjusted rate of change in the capital-labor ratio, that is, physical units of capital and labor converted to effective units by using the differential rates of factor augmentation. We specified the form of factor augmentation as a(t) = aota and B -1 b(t) = bot , which was discussed in a previous section. Thus a = at and b = Bt_l. Now the rate of change of wage income share can be expressed as 3L = -<1-sL)(9—;—1->[0, the direction of the rate of change of the labor share depends on the elasticity of substitution, the growth rates of capital and labor, and the difference between a and B. This is the analytical relation underlying the relative wage income share which we will use to explain the behavior of actual wage income shares. CHAPTER III ESTIMATION OF WAGE INCOME SHARES AND SOME PRELIMINARY ANALYSIS OF THEIR CHANGES The main task of this chapter is to estimate the actual movements of the wage income share for both the agricultural and manufacturing sectors. It is the aggregate variable in which the four variables --employment, wage rate, output, and price--are directly involved. The economic forces or relationships which determine the variables were explained in the previous chapter. Using the derived relationships, the behavior of the actual wage income share will be analyzed in sub— sequent chapters. The first section will describe the sources and specifications of the data to be used for the estimations. The second section will estimate the wage income shares and observe their movements over time and between the sectors. The last section will examine the behavior of the variables which are involved directly in the movements of the wage income shares. Specifications of Variables and Sources of the Data The Measurement of Aggrggate Output and Price The output data used were the Bank of Korea (BOK) time series, which are published from 1953 to 1974 for both the sectors in "National Income in Korea." For the agricultural sector, the output was measured by the value of gross output and also the value added. Both series 41 42 were used for the estimation of the wage share. According to the descriptions of the data, the value of gross output was calculated by multiplying the annual average price of each product group by the physical quantity of each product group and summing. The basic data were provided by estimation of the Ministry of Agriculture and Fisheries (MAF). The annual average prices were calculated by weighting monthly prices by the monthly marketing volumes of the year. The basic price data have been collected every month since 1956 in the 56 selected rural areas by the National Agricultural Cooperative Federation (NACF). The output measured production in the year in which it was produced on the farm, even though some of the production may have been marketed or self-consumed in subsequent years. The intermediate products, which were produced and consumed in the production process on the same farm, were not included in the measure of gross output, but they were included to the extent that they enter into marketings. Thus the series may overestimate the gross agricultural output since it includes some double counting such as interfarm sales of such intermediate products as seed and feed. But due to the lack of reliable information, no attempts were made to measure aggregate agricultural output net of intermediate products produced and consumed within the agricultural sector, which might have provided a better definition of gross value of output. The value added series were defined as the differences between the value of gross output and purchases of intermediate products con- sumed in the production process. These include feed, seed, fertilizer, insecticides, and other items charged to current expenses. Their 43 exclusion yields the net value added to the national products by the primary agricultural factors of production, such as land, reproduceable capital, and labor. But considering the definition of gross output it is net to the individual farm rather than net to the agricultural sector of the economy. For the manufacturing sector, the output was measured by the value added. The basic sources of the gross physical output data were the Economic Planning Board (EPB) and various government and private agencies. The price data used for the aggregation of output were the wholesale prices for that portion used in domestic consumption and FOB prices for the portion exported. The major source of price data was BOK's "Wholesale Price Survey" in which data are collected in ten-day intervals for 15 selected areas. The aggregate real output was measured at 1970 constant prices for both sectors, which are reported in the publication cited above. This publication is the most complete source of output measures of sectoral aggregates. Ban's study (Ban, 1974) also prepared a time series of the aggregate output of the agricultural sector, which was measured at 1965 constant price from 1955 to 1971. But the sources of the basic data were identical, thus the two series were not significantly different. The Specifications and Measurements of Input Variables In empirical work in production economics, there are various difficulties in specifying input variables and their measurement, such as grouping of input categories, units of measurement, choice of 44 weighting system to be used for the aggregation of various inputs, deri- vation of a flow concept of service input from stock measurement, adjustment of input measure of capacity concept to actually utilized input, and so on. Many of the difficulties underlying the measurement of real factor input, particularly in capital input, have not been fully solved at the operational level. Thus various inferences or approximations are used in empirical work. In many cases these approximation techniques are largely dictated by the availability of data. we need to explain in detail the sources of data and discuss some of the problems in the specification and measurement of our input variables. The source of input data for the agricultural sector is Ban's times series, which are the most complete estimates of national aggregates of input variables. His input series were estimated basically from average per farm household data, which have been collected from 1,200 randomly selected sample farms by MAF. The average input data per farm household have been reported in "Report on the Results of Farm Household Economy Survey and Production Cost Survey of Agri- cultural Products," published annually by MAF from 1962. Before 1962, Ban used the NACF average per farm household input data, which is also based on farm surveys. For the manufacturing sector, the major sources of the data were BOK's sample survey on "Business Management" and "Monthly Earnings and Manedays of Regular Employees in Manufacturing Industry." The data have been reported since 1957 in the BOK's annual publication "Economic Statistics Yearbook." 45 The input data for the two sectors, derived from the above sources, are basically of a private accounting nature based on firm or farm household surveys. Conceptually all input and output data that enter into aggregate production relationships should be based on social accounts (Jorgenson and Griliches, 1967; Griliches, 1964). But as in most empirical work, the measurement of input and output variables are subject to the limitations of social accounting depending on the availability of data. Various public investments, such as research, extension, educa— tion, transportation and other development investments are excluded in the measurement of our aggregate inputs. All prices of inputs and outputs also reflect only private benefits and costs. The productive contributions of the excluded social inputs are costless from the point of view of private decision makers. Since no allowance was made, in the estimation procedures, for the effects of social inputs on productivity, it is likely that these effects are captured by the estimates of technical change. More specific forms and explanations of technical change were discussed in the previous chapter. Many different classifications of input variables have been used in empirical work depending mainly on the purpose of the study. Following one conventional method, this study classifies all the inputs into two groups: labor and capital. But theoretically some conditions are required for different inputs to be aggregated as a group. The necessary and sufficient conditions are stated as: (a) that the rate of substitution between inputs of different types be independent of the quantities of other inputs used with them, and 46 (b) that the marginal rate of substitution between different types of input must be constant, i.e., two types of input are perfect sub- stitutes. It will be also possible to aggregate perfectly complementary inputs with afixed ratio. However, these conditions are quite stringent to be satisfied in the real world. But as mentioned, inputs were calssified simply into two groups for the purpose of this study.* We need to specify more detailed definitions and measurements of the inputs. There were some differences in the definitions of the variables between the sectors due to the availability of data and some conceptual differences. Measures of Labor Input and wages The measurement of labor input is relatively easy compared to capital measurement. But the aggregation of different qualities of labor should be in terms of a standard unit. The marginal productivities of different kinds of labor is probably the best indicator of their quality differences. Since in competitive equilibrium, marginal products are proportional to wage rates, this suggests that labor should be weighted according to its hourly remuneration. *Conceptually, it is possible to regard the aggregate quantity index of capital inputs in the following way. One may think that the process of production has two stages such that capital, K, is a manu- factured output produced by all the individual capital goods (the capital index function), and then this K is combined with other inputs to produce the final output (production function). That is, Y = F(L, K1, K2) = H(L, K), K = G(Kl, K2). This means that the index of capital quantity is the output of a production process which uses various capital goods to produce capital in general. For more discussion, see Solow (Solow, 1956). 47 For the agricultural sector, however, the original survey data classified labor only by sex and age rather than by the wage rates actually paid. This may be because the major portion of agricultural labor is family labor, which is not paid or valued at each point of input. For this reason, the labor input was aggregated by the MAF "labor ability weighting system"* which actually considered only the age and sex factors. Using this weighting system, the labor input was aggregated as that of a 20-54 year aged man equivalent day unit of 8 hours of actual work. This measure consists of labor input actually used for all farmwork, and it includes farmers and unpaid family workers as well as hired labor. However the "ability weighting index" may not accurately reflect the quality of different labor in the sense that labor ability may be different by kinds of job or perhaps individual skills. Ban's study also pointed out that there are some differences in the labor input estimate between the MAF "ability weighting index" and wage weighting aggregation. But due to the availability of the data, the study aggregated labor inputs by using the MAF weighting index without any adjustments. Rural wage data was collected from a survey which has been con- ducted by NACF from 1956. NACF collected rural prices and wage data *The weighting index standardized labor ability as, for example, 1.0 unit for a 20-54 years aged man, .8 unit for the same aged female, .8 unit for a 54-59 years aged man, .6 unit for the same aged female and so forth. 48 every month from the 56 nationwide selected rural market areas. From the data they estimate an annual average wage rate per man-equivalent day unit by weighting monthly labor inputs. The annual average wage data have been reported since 1959 in the "Agricultural Cooperative Yearbook." Before that, wage data were derived from Ban's study. For the manufacturing sector, the quantity of labor input was calculated from the data on the number of employees and average work days per month, which have been reported since 1957 in the "Economic Statistics Year Book" published by BOK. The measure of labor was not corrected by any quality factors. It was simply aggregated by physical unit of work day. Age and sex may not be important factors for the ability of labor for manufacturing production. Certain specialized skills or experience may be more important ability correction factors, but no such relevant data for this correction was available. The annual wage series were derived from the average work days per month and the average monthly pay per employee, which are reported in the publication cited above. According to the descriptions of the data, any fringe benefits were not included. Thus the wage series derived may not accurately reflect the production decision price, and also may underestimate the wage income share. From the deriving procedure of wage data, it is also clear that some errors in the quantity measure of labor input may lead to some errors in the wage rate in the opposite direction. As has been seen, there are some possible deficiencies in our data, but any reliable information for making an adjustment was not available. 49 Measure of Capital Quantity and Its Prices There are more difficulties in the estimation of the quantity and price of capital inputs which are actually used in a given production period in which output was measured such as the well known index number problem in aggregating various heterogeneous capital goods, the conversion problem from stock to flow services which are actually associated with given production, and so on. MOre theoretically, it is often argued that it is impossible to conceive a quantity of "capital in general," the value of which is independent of the rate of interest (Robinson, 1954). The argument may depend on the concept of capital, in other words, whether capital is defined as physical goods itself, such as labor, or some abstract productive power, which may be defined as the discounted value of the future stream of revenue expected from capital goods. But in the latter definition, we need the rate of interest as given to measure the value quantity of capital goods, whereas our main purpose in analyzing the production function is to show how wages and the rate of interest are, in part, determined by the technical conditions and the factor ratio. Several different aggregation techniques have been suggested for the measurement of the capital input: aggregation in terms of other factors, such as labor time used to produce the capital goods, aggrega— tion by weighting relative expenditure shares, and aggregation by certain functional forms, which can produce "capital-in-general" from different capital goods. But none of these methods avoids the 50 complexities of the problem, and all suffer from some limitation.* As in many empirical studies, capital was defined as tangible physical productive goods, and it was aggregated in value terms. That is, capital was measured as a value quantity, which is different from a physical quantity by which labor was measured. The value aggregate was converted to real capital goods by deflating to a 1970 base price index. Under the perfect expectation and equilibrium conditions, a price weighted aggregate will measure the physical complex of capital goods in terms of its estimated ability to contribute to production over their life time. However, in view of the divergent trends in relative prices, the choice of the price-weight base year will affect capital aggregation. Of course this problem will apply equally to output aggregation. Accordingly, in principle, considerable care should be taken in the choice of the price-weight base year. However, because of time con- straints, relatively little effort was expended in this study on the selection of the base year. But the choice of 1970 as base year may be at least partly justified in the sense that the 1970 relative price structure of capital goods did not appear to be abnormal. Using the general guidelines explained above, the capital input for agricultural production was measured by the flow service concept at 1970 constant prices. Capital consisted of the depreciation charge on durable capital goods, irrigation fees, and intermediate inputs. The price of capital was derived by dividing total current expenditure *Some detailed discussion of the problem can be seen in Harcourt (Harcourt, 1972) and Kendrick (Kendrick, 1961). 51 on capital input by the quantity of capital input measured. The total current expenditure was measured by the current cost of depreciation, irrigation charges, intermediate costs and interest at 15 percent on the stock value of durable capital goods. But for manufacturing production, the capital input was measured at 1970 constant prices of all tangible durable capital goods. For the years 1963 to 1974, there are data published on the ratio of value added to tangible, durable productive assets (i.e., Vt/Kt) and also data on Vt' These data series are reported in the BOK publication. From these we calculated the implied values of Kt. For the years 1955 to 1974, there are data published on the "marginal durable capital-output ratio" (i.e., AKt/AVt). From these ratio and data on AVt, we calculated AKt’ and hence (working backward from the value of Kt in year 1963) the values of Kt in preceeding years. As seen, the measure of capital input for the manufacturing sector was based on the stock value concept. This was due to the availability of reliable data. But what we want to measure is the annual flow of capital services which is actually associated with current production. However we may be content with the measure of stock value under the assumption that the flow of services is proportional to the stock of capital. But such measure of capital input may not lead to a good approxima- tion when the average life and age of capital goods are changing. If the average life of capital goods increases, our estimate will be biased upward. And if average age is older, our estimate can be biased downward. This can be seen by pointing out that the value of the stock 52 of capital at any point in time is the current valuation of current and all future services expected from the stock, whereas what we are interested in for current production is the value of current services from this stock. The price of capital was measured by dividing total return to capital by the quantity of capital input. The total return to capital was derived by the residual concept. That is, the price of capital r = (V-wL)/K. This procedure may lead to a good approximation if the economy is close to equilibrium. Conceptually we can impute the capital price more accurately from the data on depreciation charge, interest rate, and capital gain or loss. But no such attempt could be made due to the unavailability of data. Estimation of wage Income Shares in the Agricultural and Manufacturing Sectors This section presents estimates of the wage income shares in both the agricultural and manufacturing sectors for the last two decades. The sources and specifications of the data to be employed for the estimates are described in the previous section. The behavior of their movements will be explained in the subsequent chapters. Estimation of the wage income shares is based on the simple definition SL = (W’L)/(Y°P), where W is money wage rate, L is the measure of labor input actually used, Y is total output or value added at constant price, and P is output price. As mentioned in the previous section, for the agricultural sector gross income data are used for the estimate. But for the manufacturing sector the estimate is based on value added data. 53 Both estimates, one based on gross income and one based on value added, will exhibit the same behavior of movement if the intermediate inputs have a fixed proportion to the gross income. But it is often argued that for agricultural production, the intermediate inputs are substitutable for other inputs. Thus for agricultural production we specified the production relation as between gross output and inputs, which_include intermediate inputs, rather than subtracting the intermediate inputs from both sides of the production relation. We will discuss this problem more in the next chapter. The Agricultural Sector, 1955—1974 The estimate of the wage income share for the agricultural sector is presented in Table 1. For the estimation, the labor input data included hired workers as well as farmers and unpaid family laborers. And the wage rate used for the estimates was the rate paid to hired labor. But for the period, the pr0portion of hired labor to total labor input was only about 15 to 20 percent, with a slightly declining trend over time. Thus some may argue that applying the market wage rate to the unpaid family workers may not be appropriate for the estimation in the agricultural sector. It seems true that the return to farmer and family labor is determined as residuals rather than as the market wage rates. On this ground, the alternative estimate based on the residual concept was also obtained and is presented in column 7 of Table 1. Under the equilibrium assumption, the above two alternative estimates, one based on hired wage rate and the other based on the g 54 ouonm any ma mgm can .vonvw on~o> can mo madam onu ma .mvnum m.amm Eoum woumasoamo .mmmaummma pom ocean Hausa: .mo¢z aouu .onmauamma pom .uH:: amp uaoam>aovouams you no nauseous was many «was any .«nnunma Mom .sesum m.=mm scum .Hssa-mmma pom «Am .usooau mmouw osu mo ouwnm own? osu mu .skaa :.mosuaaumum .oousom oEMw onu aoum was «now vogue ugh .:mouox cw oaooaH Hmcoauoz: .Mom Eoum .mooaum ucwumaoo chad a“ wousmmoe mma unauoo mmouu .usmuoo mmouw can mo unmouoa mm was manna vane onu vasommm 03 onus: uaouooo Hmsvumou osu ho woumawumo oaoocw mmouw 6:» mo HA me n N .mnoa :.muoavoum Hmuauaaoauw< mo ho>uam umoo :oHuosvoum was zo>usm maoaoom vaozonoow Show mo muaamom can no uuoaum: .m¢z onu scum wouoanoauo .uaaa unoam>wavonawa he vouomaus was yams“ noowq .msa-~sa .aa H neon. eons. smam. asa.a m.mon RGNH.N suw.okm «a amen. «use. Hana. one m.emm newm.a so~.osw ma swam. some. Roam. mom o.asm Hmms.a mam.sow «a samm. swam. mwos. was e.wem soH~.a ems.aam as mean. «mun. mods. mam m.~om oooo.a swm.mms on mamm. swan. owes. mes m.smm mmmw. Nom.oom mo susm. sown. mans. awn N.~oo ends. mmo.moa we Nmmm. anon. case. son R.Hme Rose. mam.mao no amen. «Ram. NNss. 0mm «.mmo Ramm. sma.s~s so amen. mean. some. HNN o.sme cows. 000.com no «Hos. Name. uses. ass H.~so omen. «mo.ome as mean. sown. ease. «as m.omo swnm. moo.~em me Noam. «New. same. was m.oae swam. on~.m~n as ssmm. ms~e. moan. sea R.smo mmmm. one.smm as seen. Name. sewn. es w.m~s ooou. mmm.mse om mssm. same. «can. mm H.Hhm Hmma. sko.mom an Boos. muse. «new. «a o.o¢m soma. Nam.eom mm seas. mace. soon. «a s.~am «moa. use.ese an emus. mace. swam. mm n.awe mama. nso.eme on ease. choc. oon. as ~.om¢ oNsH. was.aee mmaa Ase Ase Ame Ase Ame Ame adv away now New Haw Asmu\aozv flame an: .Hazv A.Huosmav Aces .aazv Age ago Amy Awe Aw% chasm owns momma muses“ wooed ooaua usauso unauao macho «Boa - mass .uouou .uouoom Housuanofiuwd oau ca ouonm oaoocH owns mo mouwaaumm .H manna. 55 residual concept, will be approximately the same. The first estimate basically assumes that all family workers, including the operator, received the same returns on their labor as hired farm workers. Thus if there is some lagged response of wage rates to changes in marginal value produc- tivity, this estimate will tend to underestimate the returns to labor during periods when farm prices are rising and to overestimate during periods when farm prices are declining. The second estimate assigns to labor the residual share remaining after the computed share for capital input has been subtracted from total output. Thus this estimate may result in an opposite bias of wage share estimate during periods when farm prices are rising or declining. For the simple comparison between sectors, the wage income share, based on the value added data, were also estimated for agriculture. The estimate is presented in column 6 of Table 1. The value added series used for the estimate are BOK data, which were calculated by subtracting the value of the intermediate inputs purchased from outside of agriculture from the value of gross output. But the value of interfarm transactions of intermediate goods was not considered in the estimation of value added. Thus this series may overestimate total agricultural value added since it includes some double counting, to the extent of inter- farm transactions of feed and seed. As a result the estimate of the wage income share based on this value added series may be biased down- ward. The three alternative estimates are consistent in the general trends. But there are some inconsistencies in the short-term fluctuations between the residual based estimate (SL3) and the hired wage based 56 estimate (S For the period 1958-60, S showed an increasing trend Ll)° while SL3 had a slightly decreasing trend. The main reasons for this inconsistency can be seen in that the output decreased in both 1959 L1 and 1960, and the output price also decreased in 1958 due to relatively large imports of surplus food. For the period 1968-71, SL1 had a de- creasing trend but S had an increasing trend. The inconsistency L3 for this period is mainly attributable to a significant decrease in labor input and.the higher price of the output (see Table 1). Assuming an equilibrium, the two estimates SL1 and SL3 should be nearly equal. Thus the difference between SL1 and SL3 can be considered the degree of disequilibrium. From Table l, we can observe the dif- ference has been significantly decreasing over time. It can be cal- culated that the average difference between the two estimates was about 10 percent for the first decade and about 5 percent for the second decade. Considering that about 20 percent of the total labor input was hired labor, it also can be calculated that the self-employed workers in agricultural production received only about 73 percent of the market wage rate for the first decade and about 87 percent for the second decade covered in this study. But since 1970, the two estimates are fairly close, which implies that the self-employed farmers receive approximately the market wage rate. The Manufacturing Sector, 1957—1974 The estimation of wage income share for the manufacturing sector is relatively less complicated than for the agricultural sector, where 57 many resources, particularly family labor and land, do not receive a market return or have a market-determined price. The wage rates used for the estimate are the market rates actually paid to the workers. The estimate for the manufacturing sector is based on the value added series and the results are presented in Table 2. Table 2 shows that the wage share for the manufacturing sector has an increasing trend for the period with a few exceptional years, particularly the three years 1962-64. The major reason for the low wage income share for the three years was the high inflation rate of the output price. The average annual inflation rate of the three years was about 25 percent, compared with about 12 percent as an average inflation rate for the whole period covered in this study. One thing to note in the above estimate is that the value added series employed for the estimate is derived as the difference between gross income and intermediate cost. But the intermediate cost did not include capital consumption and business taxes. Thus the residual shares, or the differences between total value added and the wage income share, are not net return to capital. The residual shares include the depreciation cost, business taxes, interest, rent, and business profit. Thus a decrease or increase in the wage income share in our estimate does not necessarily mean an increase or decrease in the net return to capital. From the estimate, we can observe the significantly different trends of the wage income shares between the two sectors. For the agricultural sector, it has decreased about 15 percentage points during 58 .nuooa you whom xuoa ommuo>o an hon hanuaoa owwuo>w Una wcwww>av kn pouwaaoaoo no? oumu owwa 0:8 .oounou o>ono can do wouuomou who nowgs m .msoa :.xoonuoow aooauosuwum coouox: .mmm as wouuomou nucoa you what xuoa.owwuo>o was moohoaaao mo Hogans onu aouw vauoaaofino ow gowns .hov xuoa mo use: woods Hoowmhsm onu an mandamus mo3.unmaw noan msaa :.wouox aw uaoodH Hocosuuz: .Mom aoum .oouaom oawm onu Bonn and camp ooaua may .moofluo ucmumcoo osmH c« wouamwoa mob vowed oon> N .ooa-osa .aa a oaao. oma.a o.o~o osao.a asa.msa.a os soon. oso o.oso sass.a oos.oom ms «mas. oms e.soo omoa.a oos.~os ss coca. .moo ~.sos ssmo.a oo~.moo as oooo. Hso m.~mm oooo.a Hao.ooo os Hsom. «as o.osm mess. omo.mso so smsm. son s.~om «coo. ooo.oom oo moss. mos o.som omss. mss.oom so moon. mos o.aos Hoos. oso.ao~ oo asom. mos s.oo~ ooso. osm.mas mo mama. ass o.ooa oooo. ~oo.ssa so ssam. oso o.aoa Home. oom.oo~ mo oNHm. NHH m.aoa comm. oom.~oa so «mom. son m.sma swam. oso.o~a Ho soon. so s.o~H ooss. sso.asa oo Noam. oo H.oaa Hoos. sss.~aa so oamm. so s.oa cuss. so~.moH on «own. ss N.Ho sous. ~oo.sm so osma. coo.so on omsa. ooo.os moms Leo so so 8 E Anmqm u no ssoosaoao Asoo an: .aazo s.anosoao sacs .Hazo you» , Ase Ago smo A>o ouano own: momo3 Nuance uonog Hooauo uaouao avowed oaao> asmaunmma .oouox .uouoom waauauoomnsmz may do ouonm uaoocH own: no mouuaduum .N wanna. 59 the period. But for the manufacturing sector, it has increased about 8 percentage points during the period. We also observe from short— run fluctuations of the trends in both sectors. Recent studies indicate that there have been various trends in wage income share in different sectors and in different countries or regions. Some of these studies are listed in the footnote.* Compar- ing our estimates with those of other countries cited there, the wage income share in Korea is significantly low, particularly in the manu- facturing sector, which may contribute to the attraction of a large inflow of foreign capital in the last decade. PreliminarygAnalysis of the Wage Income Shares and Their Related Variables From the previous section we have observed the general trends of the actual wage income shares in both the agricultural and manufacturing *For the U. S. national economy, the wage income share changed with a significant increasing trend from 55 percent for 1900—1909 to 67 percent for 1949-1957 (Kravis, 1959). For U. S. agriculture, the wage income share to net agricultural income increased fairly steadily from 58 percent for 1910-14 to 65 percent for 1945—46 (Johnson, 1954). But since 1946 it declined fairly steadily from 55 percent to 44 percent for 1954-57 (Rutan and Stout, 1960). For the U. S. manufacturing sector, the wage share has significantly increased at the rate of 0.4 percent per annum during the period 1948~1962 (Ferguson and Moroney, 1969). For the Canadian manufacturing sector, no significant trend with about 50 percent of wage share from 1926-58 (Goldberger, 1964). For Canadian agriculture, the share of labor in gross agricultural output decreased from about 51 percent for 1941-45 to 25 percent for 1961-65 (Lerohl and Maceachern, 1967). For the Israel agricultural and manufacturing sectors, the wage income share showed a steadily declining trend during the period 1952-69, at the rate of 1.3 percent and 0.8 percent per year, respectively (Fishelson, 1974). 60 sectors for the last two decades. We also indicated, in the previous chapter, the major economic forces affecting these trends, with an equilibrium assumption and a specified production function. There are a variety of relationships between the wage income share and its various determinants. This section will analyze the behavior of the major economic variables affecting the trends for the purpose of examining the consistency of our data and the assumptions made in the previous chapter. Real wage Rate and Average Labor Productivity From the definition of the wage income share, SL= (W/P)-(L/Y) = (W/P)/(Y/L), we can separate the change of S as the change of the real wage rate L (W/P) and the average labor productivity (Y/L). Assuming the marginal productivity theory of wages, then SL = MPL - AFL, where the dot over the veriables means the proportional rate of change, or growth rate, of the variables, as used in the previous chapter. Thus the rate of change in the wage share is simply the difference between the rates of change in marginal productivity and average productivity which are basically determined by the nature of the production function. It is clear that constancy of the wage share requires an equal growth rate of the real wage rate and average labor productivity. If the growth rate of the average labor productivity is faster than that of marginal productivity or the real wage rate, the share will decrease, and vice versa. Table 3 shows the behavior of both the variables--real wage rate and average labor productivity for the specified periods. In the table 61 the rates of change in output, labor input, output price and wage rate are set forth in terms of averages for the specified periods. Among the important changes during the periods are the following. The Agricultural Sector The real wage rate, which is the wage rate deflated by the price of agricultural output, showed a decreasing trend in the first decade, 1955-1964. But it showed a significantly increased rate during the late 1960's due to the relatively low inflation rate of the output price. The growth rate of output was about a 5.0 percent annual rate in the 1960's, which was relatively high compared to other periods. The labor input tended to decline at about a 2.7 percent annual rate since mid-1960's. The differential growth rate of output and labor input resulted in an increase in the average productivity of labor from 1960. Thus the growth rate of the average productivity of labor was higher than that of the real wage rate except in the first period, which led to the decrease in the relative wage income share for all the other periods. The Manufacturing Sector Since the mid-1960's, the real wage rate has increased very rapidly with about an 11 percent average annual rate. This resulted from a relatively lower inflation rate for manufacturing products and a faster increasing rate of money wage rate. The growth rate of average pro— ductivity was also significantly higher than that of the agricultural sector, which resulted from the larger differential growth rate of 62 Table 3. Rate of Changes in Some Variables Related to Wage Shares for Specified Periods in the Agricultural and Manufacturing Sectors, Korea. Annual Rate of Change (7.)1 Periods Output Labor Price Wage W_ X (Y) (L) (P) (W) P L Agriculture 1955-59 1.94 5.78 7.61 5.29 2.32 -3.84 60-64 5.56 3.48 22.90 19.13 3.77 2.08 65—69 4.46 -2.65 11.85 18.49 6.64 7.11 70-74 1.71 -2.76 20.81 19.95 .86 4.47 55-64 3.95 4.50 16.10 12.98 3.12 - .55 65-74 3.08 -2.70 15.83 19.22 3.39 5.78 55-74 3.50 .71 15.96 16.26 .30 2.79 Manufacturing 57-64 9.51 12.02 15.58 12.30 - 3.28 -2.51 65-69 21.65 13.69 8.60 20.66 12.06 7.96 70-74 20.04 10.77 12.32 22.29 9.97 9.27 65-74 20.85 12.23 10.46 21.48 11.02 8.62 57-74 16.18 12.14 12.57 17.70 5.13 4.04 1Calculated from Tables 1 and 2. Yt-1)/Yt-1' The rates of change of the variables are calculated as, for example, the rate of change of out- put Y = (Yt - 63 output and labor input. From the mid-1960's, the growth rate of the real wage rate was higher than that of the average productivity of labor, or opposite to the trend in the agricultural sector. As a result, the wage income share has been increased in the manufacturing sector for the period. During the whole period the real wage rate grew at an average of .3 percent annually for the agricultural sector and 5.1 percent for the manufacturing sector. The average productivity of labor grew at 2.8 percent in the agricultural sector and 4.0 percent in the manufactur— ing sector. As a result the growth rate of average labor productivity for the agricultural sector was greater than that of the real wage rate, but for the manufacturing sector the growth rate of average labor productivity was less than that of the real wage rate. Thus the wage income share should show a decreasing trend for the agricultural sector and an increasing trend for the manufacturing sector during the period. Relative Quantities and Prices of Factors It is also possible to examine the behavior of the wage income share, SL, by the relationships between the ratio of wage and property shares and its determinants. The relative share of labor to capital can be regarded as the product of the capital-labor quantity and price ratios, and changes in the division of income resulting from changes in these ratios.* That is, SL/SK = (w/r).(L/K) = (w/r)/(K/L). *Once SL/SK is known, the wage income share SL can be calculated since the sum of the wage and property shares must equal 1, SL/SK = SL/(l-SL). Thus SL can be found by substituting the numerical value of SL/SK and solving for SL. 64 Thus the rate of change in the share ratio is the sum of the rates of change in w/r and L/K, or the difference of the rates of changes in w/r and K/L. Since the ratio of the rate of change in the relative quantity (K/L) to the rate of change in the relative price (w/r) is equal to the elasticity of substitution, we are dealing with the proposition that changes in relative shares depend upon the elasticity of sub- stitution. Assuming that the relative price (w/r) and quantity (K/L) would move in same direction, the possibility for factor substitution would serve as a stabilizing force for changes in relative shares. When the rates of change in relative quantity (K/L) and price (w/r) are equal--in other words, the elasticity of substitution is unity-relative shares will of course remain unchanged. The behavior of both variables is represented in Table 4 for both the sectors for specified periods. For both sectors, our data showed the same direction of movement for the relative quantity (K/L) and price (w/r), which are consistent with our assumption. For the agricultural sector, the relative wage rate (w/r) has moved much more slowly than the increase in the relative quantity (K/L). But for the manufacturing sector, it has moved faster than the relative quantities. The negative differential growth rate of the relative price and quantity lead to a decreasing trend of the wage share in the agricultural sector, while the positive differential growth rate leads to an increasing trend of the wage share in the manufacturing sector. 65 Table 4. Rate of Changes in Relative Quantities and Prices of Factors for Specified Periods in the Agricultural and Manufacturing Sectors, Korea Annual Rate of Change (7.)1 12 {a Periods Relativquuantity Relative price ( L ) / (.r ) K , w_ ( L ) \ r ) Agriculture 1955-59 - 2.053 - 1.573 1.305 60-64 2.238 1.730 1.294 65-69 12.762 5.362 2.380 70-74 6.698 3.484 1.923 55-64 .331 .262 1.263 65-74 9.730 4.423 2.200 55-74 5.278 2.452 2.153 Manufacturing 1957—64 - 2.246 - 2.093 1.073 65-69 4.280 5.838 .733 70-74 3.540 5.312 .666 65-74 3.910 5.575 .701 57-74 1.374 2.418 .568 1See the footnote to Table 3. 66 As explained above, the ratio of the growth rate of the relative quantity and price is interpreted as the elasticity of substitution between labor and capital. The ratios are shown for the specified periods in Table 4. From the results, we can see the ratios are greater than one in the agricultural sector and less than one in the manufacturing sector, which is consistent with the movements of the wage share in the two sectors. For the explanation of the behavior of the wage share, we need to explain why the relative quantity and price have moved as we observed in the above. Under the equilibrium assumption, the reasons must be found from the nature of supply and demand conditions of the factors. A variety of assumptions can be made to answer the question. Let us assume that the relative marginal productivities were unchanged for any given ratio of capital to labor, a situation which describes the nature of the demand condition or production relation- ships. If both factors were perfectly elastic in supply, there would be proportionate increases in both, and no change in relative prices or wage shares would occur. If one were more elastic in supply than the other, the relative quantity of the more elastic factor would increase, and the relative price would change depending on the degree of substitutability between the factors. We can also consider the case in which the marginal productivity of one factor has improved relative to that of the other at any given factor ratio. In other words, the technical progress is biased. The changes in the relative quantity and price are also related to the supply conditions. If technical progress is biased to the inelastic 67 factor, then even the smaller increase in the demand for it would result in large price increases. As explained above, the relative quantity and price are determined by the difference in the supply elasticities and the marginal rate of substitution between the factors. If the marginal rate of substitution were constant at an equilibrium position over time, there could be no change in relative prices, and only relative quantities would change. But as we observed, the relative price also changed significantly during the periods, thus it indicates that the changes in the marginal rate of substitution have been an important factor influencing the behavior of the wage share. With no reliable information on the supply side, this study has put more emphasis on explanation of the demand side. If we could assume that the supply curve for labor had remained fairly stable during the period, the observed behavior of prices and quantities reflect the movement of the demand curve for labor. As we explained in the previous chapter, there are two sources causing the changes in the relative marginal productivity. One is the relative quantity of the factors, and the other is the technical progress. The responsiveness of the relative marginal productivity to the changes in the relative quantity and technical change is the main parameter to be estimated in the next chapter. Growth Rate of Output, Input, and Aggregate Productivity As seen in Table 3, the average productivity of labor has sig- nificantly changed during the period. It grew at about a 2.8 percent 68 annual rate for the agricultural sector and 4.0 percent for the manu- facturing sector. We also observed that the average productivity of labor has been an important factor causing the changes in the wage share for both sectors. By definition, the rate of change in the average productivity is simply the difference between the rate of changes in output and labor input. The sources of the changes in output are changes in inputs and technical change. Assuming that factors are paid their marginal products and linear homogeneity of production function, the rate of change in output can be segregated as the effect of input change and technical change.* (Solow, 1957). Under the assumptions, the rate of change in output is the sum of the share weighted input growth rate and the rate of technical change. In other words, the rate of technical change is just the difference between the growth rate of output and share weighted inputs. Tables 5 and 6 show the growth rate of output and inputs for both SECtOI‘S o *Assume the production function as Y = A(t)f(K,L), where A(t) is a shift factor which reflects the pull of all the forces of technical change. Differentiating the production function with respect to time and dividing by Y, we obtain - . BY K - BY L o Y-A+-§EYK+3L YL. If factors are paid their marginal products, then Y= A + Sk K + SL L. And assuming constant returns to scale, Y - L= A.+ Sk(K~L) or (L) = A + skci), where A is the rate of productivity growth. 69 Table 5. The Growth Rate of Output, Inputs, and Aggregate Productivity in the Agricultural Sector, Korea, 1955-1974 Annual Growth rate (A) Productivitys Year Output1 Capital1 Labor1 Aggregate2 growzhtindex ( Y ) ( K ) < L ) productivity (A) 1955 1.0000 56 - 7.12 - 2.40 5.54 -2.615 .9745 57 8.85 9.97 6.49 1.870 .9931 58 6.38 3.84 5.32 1.215 1.0053 59 - .36 3.49 5.76 5.939 1.0687 60 - .84 3.23 9.57 -9.995 .9716 61 11.78 3.70 5.10 6.847 1.0430 62 - 5.81 4.66 -7.21 - .645 1.0363 63 7.01 4.41 6.65 .765 1.0443 64 15.66 12.57 3.27 10.104 1.1617 65 1.54 21.85 -2.69 - .663 1.1540 66 10.17 8.15 .20 8.318 1.2587 67 - 4.62 5.38 - .56 -5.285 1.1955 68 2.32 4.34 -6.84 6.985 1.2853 69 12.87 10.90 -3.34 12.735 1.4729 70 - .87 4.02 -4.08 1.270 1.4918 71 2.20 2.77 1.01 .754 1.5031 72 - .44 2.58 -4.75 2.263 1.5379 73 4.00 6.67 - .94 2.711 1.5808 74 3.64 3.68 -5.03 5.743 1.6771 1 The growth rates of output and inputs are calculated from Table 1 and the growth rate was calculated as Y = (Yt- Yt-l) / Yt-l‘ 2The growth rates of aggregate productivity A are calculated by the difference between the growth rate of output and share (SL1) weighted inputs growth rate. 3A(t) is calculated from.A by At = At-l (1 + A). 70 Table 6. The Growth Rate of Output, Inputs, and Aggregate Productiv- ity in the Manufacturing Sector, Korea, 1957—1974 Growth rate (Z) 3 1 1 1 Productivity Year Output Capital Labor Aggregate growth index productivity ( A(t) ) (V) (K) (L) (A) 57 1.000 58 9.09 7.80 8.23 - 1.15 1.015 59 9.22 10.31 16.70 - 3.21 .982 60 8.18 7.47 9.16 .127 .983 61 3.10 2.19 4.50 .116 .984 62 13.16 9.04 7.62 4.564 1.029 63 17.29 11.20 28.45 .610 1.036 64 6.53 13.04 9.51 - 5.368 .990 65 19.95 29.39 21.06 - 6.584 .925 66 17.12 14.13 8.82 4.920 .970 67 22.77 16.45 17.42 5.952 1.028 68 27.02 19.98 14.68 9.021 1.121 69 21.39 9.90 6.47 12.818 1.264 70 18.39 13.38 4.63 8.515 1.372 71 17.71 14.80 3.64 7.441 1.474 72 15.71 14.44 14.77 1.137 1.491 73 30.93 16.88 23.32 11.484 1.662 74 17.46 13.67 9.11 5.668 1.756 1Calculated from Table 2 by the same procedure as in Table 5. 2See footnote 2 of Table 5.4 3See footnote 3 of Table 5. 71 Using the estimates of the wage income share from Tables 1 and 2, the residuals, or the rates of aggregate productivity change, are cal- culated and presented in Tables 5 and 6. There may be some possible errors in the specifications of technical change and the underlying assumptions. But the estimate can give us at least some information about the importance of technical change for the output growth and average productivity increase. One observation from the estimates is that the rate of aggregate productivity growth fluctuated greatly year to year for both sectors. It fluctuated relatively more in the agricultural sector, which may result from the fact that agricultural production has more random elements, such as weather conditions. There may be various sources of errors for the estimates, but from the fluctuations one may question particularly the measurement of capital input. Note that we measured capital input in stock concept for the manufacturing sector. Thus the fluctuations of investment affect largely the measures of capital input, which also affect the estimates of At as can be seen in Table 6. The results show that the average rate of aggregate productivity change during the whole period was about 2.5 percent per year for the agricultural sector and 3.5 percent for the manufacturing sector. There is some evidence that the productivity growth rate may have accelerated from the mid—1960's. And it is also calculated that about 73 percent of the agricultural output growth and 22 percent of the manufacturing output growth* for the period are explained by the *Solow estimated the contribution of technical change to the growth rate of average labor productivity in the U. S. nonfarm private sector for the period 1909-1949 as 87.5 percent of the total growth rate (Solow, 1957). 72 productivity growth. Thus it is clear that productivity growth should be included as an important factor in the explanation of production relationships. CHAPTER IV ESTIMATION OF FACTOR SUBSTITUTABILITY AND THE BIAS OF FACTOR EFFICIENCY GROWTH In estimating the parameters of a specified production function, it is possible to fit either the function directly or the marginal productivity conditions.* This chapter will attempt to estimate the parameters by using the marginal productivity relations.‘ In the next chapter more attention will be given to direct estimation procedures for the specified production function. The indirect estimation procedure, which is based on marginal productivity relations, is commonly used in many empirical researches because of the difficulty of direct estimation when the production function is nonlinear with respect to its parameters. However, the validity of the indirect procedure will depend on how close the pro- duction behaviors of the economy are to the assumption. Basic Estimating Equations with Marginal Productivity Relations For the purpose of the study we are mainly interested in estimating the substitution parameter p and factor augmenting parameters azuxiB or *However, if the error terms of the production function and marginal productivity relations are jointly distributed, the two sets of relations are interdependent, and thus the estimation, based on a separated relation, will suffer from simultaneous equations bias. But it was proved that under certain assumptions, the single estimation pro- cedures are consistent and they are also unbiased if the error terms of the production function and marginal productivity relations are independent (Hobges, 1969; Zellner, et a1., 1966). 73 74 their difference. For this reason the production function (2-11), specified in Chapter II, can be rewritten as 1 v = [cl(c°‘1<)"p + c2(tBL)‘p]"E' (4—1) where C1 and C2 are constants. From the above production function, the marginal product of each factor can be derived as 02 t'pB (V/L)p+l cl 5"“ (V/K)p+l (4—3) MPL (4-2) MPK Under the assumption of perfect competition and profit maximiza- tion, the marginal product of each factor should be set equal to the price of the factor. Thus rearranging the marginal productivity relationships and taking them in logarithms we get 1n (V/L) = 0 In W + (1-o)8 ln t —t/(K/L)t—1 - [(K/L)t/(K/L)t_l (4—11) whereA.is an adjustment coefficient and 0 < A :_1. Under the above lagged adjustment assumption, the capital-labor ratio of Equation (4-8) can be interpreted as the desired capital-labor ratio. Thus by combining the Equations (4-8) and (4-11), we obtain the equation as 1n (K/L)t = 10 ln(w/r)t + A(B-a)(l-o) 1n t + (1—1) 1n (K/L)t-l + 10 1n (Cl/CZ) (4-12) Using the same assumption, the equation system (4—10) also can introduce the lagged adjustment process. Thus it can be modified as 78 _ .4 b1 b2 1n (V/L) 1n w ln t 0 ln(V/L) _ 1 0 b t = t t 1 3 (4_13) 1n (V/K)t 1n rt 0 1n t ln(V/K)t_l 0 1 b4 b5 Lbs- where b1 = 10 b2 = l(l-o)8 b3 = A(l-o)a b4 = (1-1) b5 = 10 1n C2 b6 = 10 1n Cl In the above, we derived various estimating equations which are commonly used in empirical work, except that we have introduced a somewhat different assumption of technical progress. However, the validity of the equations depends on the marginal productivity assump- tion. And as seen, most of the equations used the factor price or the ratio of the prices as an exogenous variable. This choice of exogenous variable may be reasonable when the regression is performed on the individual firm data. But when the regression is applied at an aggregate national level, the choice of exogenous variable is not clear because of simultaneous determination of factor price and employment. We will discuss this problem further in the next section. 79 Initial Estimation with a Static Equilibrium Assumption As an initial estimation for the parameters, three basic estimating equations have been fitted to the data for both the agricultural and manufacturing sectors. They are Equations (4-4), (4-5), and (4—8), which were derived in the preceding section. Equations (4—4) and (4-5) are the equilibrium condition of labor and capital markets, respectively, and Equation (4-8) is the production expansion path equation which is the ratio of the other two equations. Application of the three different equations is basically aimed at a test of the validity of the assumptions underlying the estimation equations--homogeneity of degree one and perfect competition. Thus we hope to find the general directions of our efforts for a better estima— tion. If the assumptions were valid, then the estimates of parameters should not be significantly different across the different estimation equations. The sources and descriptions of the data employed for the estimation were explained in more detail in the first section of the previous chapter. The definitions of the data used for measurement are not identical between the sectors mainly because of the lack of availability. For the agricultural sector, the measure of labor input is an adult-man- day equivalent unit, which was corrected by age and sex. Accordingly, the wage rate was also measured per adult-man-day equivalent unit. But for the manufacturing sector, the measure of labor input was not corrected, partially due to the lack of data and partially due to 80 the conceptual difficulty.‘ Age and sex may not be important factors for the ability of labor for the manufacturing production. Rather certain specialized skills or experiences may be more important ability correction factors, but no such relevant data for this correction was available. The wage rate is the average rate per uncorrected physical day unit. Capital inputs were measured in 1970 constant prices. For agri— culture, the measure of capital input was the flow service concept, which included the depreciation cost on durable capital goods such as machinery and farm equipment, farming building, perennial trees and livestock, irrigation charges, and all variable costs of intermediate inputs. The price of capital was calculated by dividing the total current expenditures on nonlabor inputs by the measure of capital input. Total expenditures consisted of the current value of all service charges and interest on the stock value of capital goods. But for the manufacturing sector, the capital input was measured by the stock value of all tangible fixed capital goods. The price of capital was derived by dividing total returns to capital by the measure of capital input. Total returns to capital were calculated by the residual concept, that is, V - wL. Thus the deriving procedure of capital prices is different between sectors, but if the equilibrium assumption is satisfied, the two procedures should give similar results. The measure of output is the value added concept for the manufactur— ing sector but the gross output concept for the agricultural sector. The different specification of the output variables is not based on data availability, but it is rather based on a theoretical argument. 81 The value added data are more commonly used as a measure of output in most empirical studies. These data are based on the assumption that there is no substitution between primary inputs and intermediate inputs. Thus it is convenient to subtract the intermediate input from both sides of the production function. But it is often argued that some intermediate inputs, particularly for agricultural production, are substitutes for primary inputs rather than being in a fixed proportion relation (Griliches, 1964). In Korean agriculture, it seems true that some intermediate inputs such as fertilizers, weed killers and other agri- cultural chemicals can be substituted for labor input more smoothly than machinery. It should be also pointed out that the process of technical change must be viewed in the context not only of capital and labor inputs, but also of intermediate inputs as a whole. Thus for agriculture, the output variable was measured by gross output, and the intermediate input was included in the measure of capital input. Before fitting the data prepared with the above concepts to the estimating equation, we need to discuss some of the characteristics of the estimating equations. Equations (4-4) and (4-5) assumed profit maximization and constant returns to scale. But Equation (4-8) assumed only cost minimization for a given rate of output and does not require the assumption of constant returns to scale. However, if one is not interested in estimating the constant term, or the logarithm value of the ratio of distribution parameter, the above equations are also valid for the estimation even under the weaker assumption that each factor price is just proportional to its marginal product. 82 Another point we need to discuss is the choice of exogenous variables. It may be argued that both variables--employment level or factor input ratio and the prices of factors or the ratio of the prices-may be endogenous or determined simultaneously in some larger system. And it was also pointed out that, based on empirical study, the estimates of the parameters of the production function are very sensitive to the classification of the variables into exogeneous and endogenous groups (Nerlove, 1967). The argument is basically saying that both variables are subject to random error, or possibly the variable we choose as exogenous is correlated with the disturbance term in the regression equations. If this is the case, the simple ordinary least squares(OLS) estimation procedure will not provide even the desirable large sample properties. As can be seen in the equations, we choose the factor prices as exogenous variables. Since the regression equations pertain to sectors, it may be plausible to assume that the factor prices, or their ratios, are exogenous variables, which may be determined at the more aggregate level. Thus the factor employment level is the decision variable that is changed by the production decision units of each sector in response to changes in factor prices. But this may not be quite true if we consider that both sectors are relatively large compared to the total economy. Thus any change in the factor demand in one of the sectors would likely affect the factor price. There are also some other reasons for our choice of exogenous variables. Considering the fact that the price of capital has been 83 largely controlled by the government development policy for the periods observed, it is desirable to consider the factor price variable as exogenous. Another justification for the choice of capital-labor ratio or capital input as a dependent variable is that considering the nature of the data, the capital input data are more likely to contain substantial errors of measurement. This being the case, econometric considerations indicate the desirability of having the errors of measurement occur in the dependent variable and not among the explanatory variables of the regression, which will lead to inconsistent estimates of our parameters. With the above specifications, the three basic marginal productivity relations are fitted to the data for both sectors. Assuming that the error term, u, is normally and independently distributed with zero mean and constant variance and is independent of all the explanatory variables, the ordinary least squares estimation method will give at least consistent estimates. But the first results of the OLS application to the regressions showed positive autocorrelation in most of the equations, which is one of the general problems of time series data. Thus the regression equations, except Equations (4-5) and (4-8) for the agri- cultural sector in which the D—W statistics are in the inconclusive region, were recomputed by the iterative two-stage procedure. The first stage computes the OLS residuals ignoring all complica- tions of the covariance matrix. Next, compute the ratio of the mean product of the successive residuals to the OLS variance estimator which is to be regarded as an estimator of autocorrelation coefficient, 0. 84 The second stage uses the estimated value of p, constructs new variables (yt - 6 yt_1) and (Xt - 6 Xt-l)’ and applies OLS to the new variables. Again compute the second round residuals and estimate the second round 5. These procedures are to be iterated until the values of the estimators converge. The procedure is convergent and the final round estimates coincide with the maximum likelihood estimates (Kmenta, 1971, p. 288).* But actually, the iterative procedure was reduced by stopping after obtaining the second round estimates. The results estimated by the procedure are presented in Table 7. Various observations can be made from the results of our initial estimates. For agriculture, the estimates are largely inconsistent across different estimation equations, but Equation (4-8) is relatively better than the other two equations in a statistical sense. There may be various sources which lead the estimates to be biased, such as errors of observation in the explanatory variables, the simultaneity problem in the more complete system, some factors causing market imperfections, and so on. From the results shown in Table 7, however, an important question can be raised as to the assumption of profit maximization and/or constant returns to scale for agricultural production, which was made for the derivation of Equations (4-4) and (4-5). Considering the small scale production, with a low marketing rate for their products, the self— food supply objective may be more important for Korean agriculture *It was pointed out that there is some possibility that the likeli- hood function may have multiple local maxima. Hence the iterative pro- cedure is subject to the risk that the local maximum obtained may not be a general maximum. But the empirical examples showed that it is very rare case (Hildreth and Lu, 1960). 85 during the period, perhaps implying that the cost minimization assump- tion is better than profit maximization. The assumption of constant returns to scale is also suSpect when we consider the fact that land, the most important factor for agriculture, is a very limiting input, and that it was fairly constant for the time periods. Thus decreasing returns to scale may be more likely than constant returns to scale. If this is the case, the estimate of the elasticity of substitution from Equations (4-4) and (4-5) will be biased. The directions of the bias will be discussed in a later part of this section. From the above discussion, for the estimation of the parameters for the agricultural production it seems relatively reasonable to rely more on the estimates of Equation (4-8) than on Equations (4-4) and (4-5). If this is acceptable, it is possible to argue that the estimates of the elasticity of substitution parameter provided by Equations (4-4) and (4-5) are significantly low. There may be various possibilities by which the estimates from both equations are biased, particularly negative bias, for the explanation of low value of the estimate. One possibility comes from the lagged behavior of factor price to product price change, or the degree of market imperfection. For example, from Equation (4—4), assuming that the money wage rates are lagged to price change, which means that the price increase is negatively correlated with the real wage rates, then the estimate of the substitution parameter_will be downward biased if the price increase is positively related with the output (V), which agrees with our data. 86 Table 7. Initial Estimates of Elasticities of Substitution and Factor Efficiency Growth Parameters, 1955-1974 Parameters Estimation —2 Equation 0 B 8-0 0 R ** Agriculture (4-4) -.1516 1.2706 (.1621)* (.5204) .844 (4-5) .0406 .6592 (.1052) (.4986) .786 (4-8) —.1129 1.7738 (.1818) (.4718) .914 Manufacturing (4-4) .0153 .6719 (.1310) (.1362) .9387 (4—5) .4537 .7024 (.3142) (.3842) .8224 (4—8) -.5243 .7872 (.1092) (.1562) .9012 *The numbers in the parentheses are the standard errors of the regression coefficients. The standard errors of the parameters, a, B and 8-0, which are nonlinear functions of random variables are cal— culated by a Taylor's approximation procedure. Let y = y(x) be a differentiable function of x where x is a vector of random variables. Then the approximated variance of y can be expressed as Var (y) = j'Zj where j = 3y/3x and 2 is the covariance matrix of x (Goldberger, 1964, p, 125). Thus for example, since C 1 2 . C1 2 A 83“ ='——*—' in Equ. (4‘82) V(8-a) = C————) V(C ) + [‘-_:‘—‘] V(C ) 1-0 1—6 1 (1-0 )2 2 2 2 2 61 . . +-2 ———:-—§- cov 0. (For the details of the proof, see the original paper.) 89 argument may not be satisfied in our data, or there may be some other reasons for which we have no insight. FromTable 7, it may be observed that the estimate of the elasticity of substitution is greater than one for the agricultural pro- duction and less than one for the manufacturing production, and the factor efficiency growth is biased to capital augmenting for both sectors. Thus in Hicksian terminology the nature of the technical progress is capital- using for agriculture and labor—using for manufacturing during the periods covered in this study. However, the results are very tentative considering the various possible sources of the bias which are discussed above. Thus more detailed discussions about the magnitudes of the estimated values of the parameters are reserved until we get more reliable evidence. With the above very tentative observations from the initial estimation results, more efforts will be directed in the remaining part of this chapter to the improvement of the estimates by introducing variuos assumptions and estimation methods. In the following section we will concentrate our efforts on estimating the parameters of agri— cultural production; a partial adjustment assumption in the expansion path equation will be introduced. In the final section the efforts will be toward the improvement of the estimates for the manufacturing pro- duction. In this discussion we will maintain the profit maximization assumption and will apply different estimation methods with different Specifications of the error term in the hope of increasing the efficiency of the estimates. Application of the direct estimation method will be 9O discussed in the next chapter. This method requires only the acceptance of a specified production function. Particularly for agriculture, we may suspect the equilibrium assumption underlying the indirect estima— tion procedure as pointed out above. Estimation from the Expansion Path Equation for the Agricultural Sector The main effort of this section is to improve the estimates of the parameters for agricultural production. Based on the results drawn from the simple basic regressions, attention will be given to the production expansion path equation. On the same ground, the equilibrium relations of each marginal productivity equation will not be given any further attention for agricultural production. Equation (4-8) related the actual level of capital intensity to the relative factor price ratio. But it may be interpreted as indicat- ing the relationship between the desired or optimal capital intensity and relative factor prices. There may be various reasons causing the lagged behavior of the actual input adjustment to optimum combinations. They may include technical constraints, institutional rigidities, the fixity of some factors, persistence of habit, and so on (Griliches, 1967). Thus it should not be assumed that such adjustments are completed instantly so that the desired capital intensity is always identical to the actual capital intensity of production in each period of time. In order to take account of the lagged behavior of the input adjustment or the short-run inflexibility of factor proportions, Equation (4-8) was reformulated by replacing the optimal input 91 combination, (K/L)t*’ for the actual inputs ratio, (K/L)t° The modified estimation equation was derived in Equation (4-12) by specifying the adjustment process of (K/L)t as an exponentially distributed lag of a Nerlove type. But assuming that the optimum combination of factors is not known in advance, the specification of the adjustment process described in Equation (4-11) may not be appropriate. Thus the adjustment process may be revised by assuming that the current input ratio (K/L)t is adjusted by comparing (K/L)t-l with (K/L):_1 rather than with (K/L):. Thus, the adjustment process can be expressed as (K/L)t/(K/L)t-l = [(K/L):_l/ (v(62) + v one .HH pow o u mud pom H Mom mHH. n mus omega .H uAmIdv .m 64669 acne q .H magma scum .AmmmH pom Hue nuH3v vaHHwQ m>Huooamou one mo Some mo ucfiompwa mau mo mm :oxmo Aumnavu an amumasoamu b HID 0 .HH sou ama.a u 6 cam H Lou amam.a u 6 606:3 .w. 4-6 IIIAAmIHVI za wouMHDUHmU m N H 121 menm. wac.o «OH.mu nm¢.u dom.Hu HHH.1 nm¢.u an.Hu anuoan mama. NoN.NH owH.~u «qw.u mmN.Nu HNN.- qu.u qHO.Nu moumomH men. me.~ NNo.mu umH.1 ode. 1 mmm.n de.u own. a «ouoomH Homm. mmo.mu mN¢.N qHH. mad. I oen.u qHH. Hum. mmummmH mmoa. omn.m qu.wn mwo.u omw.Ha HoN.- mwo.1 mmo.Hu «unmomH mHHm. Hmm. mmN.Hu mHo.- own. . oq¢.1 mHo.u one. 1 umnmmmH mwmq. wHN.m ohm.Ha mmm.- qmo.Hu mNN.a nmm.1 mom. 1 unummmH HH H HH H HH H ANV uwomzo N + H HmoHnnoou AHV wchoaoow Am w. MH65uo¢ «commwn hp HHmanmo hp leaoouomv AHWM.oumnm owmz aH mwamso mo comm Hmsod< Hmaowuuomoum aema - mmma .uouuom Hausuasoauwa osu aH meowuom voHHHoomw Mom moumsm oEoonH owwz wouownoum new Hmsuo< :H owdmso mo moumm .w oHamH 122 result of the estimation Equation (4-15) which was based on the pro- duction expansion path (estimate I), and the other was the result of Equation (5—2) which was based on Kmenta's approximation procedure (estimate II). The theoretical relation predicts the wage income share to be decreased at an average annual rate of 1.03 percent from estimate I and .34 percent from estimate II, which are compared with the 1.97 percent of the actual rate of change during the whole period. Based on estimate 1, the rate of the change in wage income share is separated into two sources as explained above. The results show that, of the 1.03 percent of the annual rate of the change, the capital deepening or the term A contributes .80 percent and the biased technical change or the term B contributes .23 percent. Thus_the capital deepen- ing factor appears to be the major source of the change with an elastic substitutability between capital and labor for the period. For other specified five- or ten-year periods, the theoretical rates of the change were also calculated. The results show that the actual rates of the changes were consistently faster than the predicted rates except for the period 1965-1969. And the predicted rates of change which were based on estimate II were slower than those based on estimate 1. As can be seen from Table 8, the actual and predicted rates of change are at least in the same direction, but there are large differences in actual magnitudes of the rates. In the average term for the whole period, it is calculated that our theoretical relation can explain only about 52 percent of the decreasing trend of the 123 actual share based on estimate I and only 17 percent based on estimate II.* Using the estimated rates of the change in five-year average term, the wage income share index is derived for both the actual and predicted shares. The base of the index is 1974 as 1.0 for actual share and .91 for predicted shares, which is the average ratio of the estimated marginal product to the actual wage rate for the period 1970-1974. (See Table 12.) The derived indexes are presented in Table 9. The difference between the actual and predicted indexes will show the degree of deviation from the marginal productivity theory, assuming no logical errors in our analysis, no specification errors in our models, and no sampling errors in our estimates. From Table 9 we can observe that the actual wage income share index is considerably above that of the predicted share for the earlierl period. But the difference between the two indexes has been significantly decreased over time as can be seen in the last two columns of Table 9. Before 1960, the predicted share was about 75 percent of the actual share from estimate I and only 65 percent from estimate II. But after 1970, both of the predicted indexes are about 88 percent of the actual *In our specification, the bias of technical change B is a function of t. In Tables 8 and 10, we calculated B as an average concept for the specified periods by taking t in each case at the middle point of the respective period. For comparison, we also calculated the values of B for the specified periods by calculating B for each year using the value of t for that year, and then averaging these annual B values for the respective periods. The results were not substantially different from the values calculated by the procedure used in Tables 8 and 10. 124 Table 9. Actual and Predicted Wage Income Share Indexes in the Agricultural Sector, 1955 - 1974. Year .Actuall Predicted2 I/ II/ (A) I II A A 1955 1.422 1.132 .964 .796 .678 56 1.457 1.126 .966 .773 .663 57 1.494 1.120 .967 .750 .647 58 1.531 1.115 .968 .728 .632 59 1.569 1.109 .969 .707 .618 1960 1.608 1.104 .970 .687 .603 61 1.531 1.097 .969 .717 .633 62 1.458 1.090 .968 .748 .664 63 1.388 1.083 .967 .780 .697 64 1.322 1.076 .965 .814 .730 1965 1.259 1.069 .964 .849 .766 66 1.232 1.045 .956 .848 .776 67 1.205 1.023 .948 .849 .787 68 1.180 1.000 .940 .847 .797 69 1.155 .978 .932 .847 .807 1970 1.130 .957 .924 .847 .818 71 1.096 .944 .920 .861 .839 72 1.063 .931 .915 .876 .861 73 1.031 .918 .911 .890 .884 74 1.000 .910 .910 .910 .910 1The base of the index is 1974 as 1.0. The indexes are derived by using the 5 year average rates of the change which are presented in column 7 of Table 8. The calculation procedure is It-l = (1+SL)"1 It’ where It is the share index at time t. 2Derived with the same procedure as above. The base of the index is 1974 as .91 which is the average ratio of the marginal product and actual wage rate for 1970-1974. (from Table 12). 125 index. This trend in the difference between the actual and predicted indexes may imply that there were some other factors playing an important role in the change of the actual wage share for the period. One explanation for the trend may be that, particularly in the earlier period, there was a relatively large amount of unemployed or underemployed labor in Korean agriculture, which made possible the overemployment in the agricultural sector by self-employed workers. Thus the marginal product of self—employed workers may be fairly low compared to the actual wage rate. But as the economy expanded, or the amount of unemployed labor decreased, the agricultural labor market moved toward equilibrium. However, this explanation may be risky for various reasons which we will discuss later in this chapter. The Increasing Trend of the Wage Income Share in the Manufacturing Sector The manufacturing sector showed an increasing trend in its actual wage income as a share of the total value added for the sample period, 1957-1974, which is quite an Opposite trend to that for the agricultural sector. The share has increased from 33 percent for 1957-1959 to 41 percent for 1972-1974, some 8 percentage points, or about one- fourth. Following the same procedure used in the previous section, this section will analyze the sources of the increasing trend of the actual wage share in the manufacturing sector. It has increased at an average annual rate of about 1.5 percent during the whole sample period. 126 The trend was accelerated over the period 1965-69 with an annual growth of 3.1 percent. For the other specified five-or-ten year periods, the average rates of change were also calculated and are presented in column 7 of Table 10. As seen in Table 4 of Chapter III, the capital-labor ratio has also significantly changed with an average annual rate of increase of 1.4 percent during the whole sample period. But this rate of growth of the capitalélabor ratio was much slower than that in the agricultural sector. For the same period, the capital-labor ratio in the agricultural sector grew about 5.3 percent annually. For the manufacturing sector, we estimated the elasticity of substitution 0 as .68 from the estimation Equation (4-16), which is based on the marginal productivity relation (estimate I) and .83 from the Equation (5-6), which is based on the nonlinear estimation directly from the specified production function (estimate II). The estimate of the difference in the factor augmenting parameters between capital and labor, a-B, was about .53 from Equation (4-16) and .30 from Equation (5-6). With the above estimation results, the bias of technical change, B, was calculated by the Equation (6-3) used in the previous section. The value of B (at the sample period midpoint, i.e., t=10) turned out to be -2.49 percent based on estimate I and -l.09 percent based on estimate II. Thus the bias of technical progress has been labor-using and it appears to be more significantly biased compared with the agri- cultural sector. Thus, at a given capital-labor ratio, the technical change increases the marginal product of labor much faster than that of 127 capital, and the difference in the annual growth rate of marginal pro— ductivity between labor and capital is about 1.1-2.5 percent. As seen above, for both the sectors, technical change was characterized by a capital augmenting bias, which means that the capital augmenting rate exceeds that of labor. But technical change is capital— using in the agricultural sector and labor-using in the manufacturing sector. In other words, at a given capital-labor ratio, technical change increased marginal productivity of capital relative to that of labor in the agricultural sector, thus leading to a decrease in wage if: income share in the sector. But for the manufacturing sector, technical change increased the marginal productivity of labor relative to that of capital, thus leading to increase in the relative wage income share for the manufacturing sector. The underlying relations of the above explanation can be seen more clearly as follows. The capital augmenting biased technical change increased the quantity of efficiency units of capital relative to those of labor, thus increasing the capital-labor ratio in efficiency unit terms. But the inelastic nature of factor substitutability resulted more in increase in relative marginal productivity of labor than in the decrease in the relative quantity of labor. In other words, the increase in the relative quantity of capital in efficiency units is not enough to offset the decrease in the relative marginal productivity or the relative price of capital. For this reason, the technical change reduced the relative marginal productivity of capital in physical unit terms at a given capital-labor ratio, and thus decreased the relative 128 share of capital and correspondingly increased the relative share of labor. The underlying economic relationships may be interpreted as follows. In the labor market, the labor-using technical change will shift the demand for labor to the right at a given volume of output and factor price ratio. If there were no change in the supply of labor, the result would be an increase in the wage rate and employment, and accord- ingly an increase in the wage share. But as we observed in Chapter III, the total output continually increased during the whole sample period, and the supply curve of labor may also shift to the right with a significant movement of labor from the agricultural sector to the manufacturing sector. The result would be a decrease in wage rates and an increase in employment. In fact, actual data show a rapid increase in employment with an annual growth of about 12 percent. And the real wage rate also significantly increased, with an annual rate of about 5 percent. Using the relation expressed in Equation (6-1), the theoretical rate of change in the wage income share was predicted in average terms for the specified periods. The rate of the change was also separated into two sources--capital deepening and biased technical change. The results are presented in Table 10. The results show that the predicted growth rate of the wage income share based on the marginal productivity theory is about 1.7 percent annually from estimate I, which is fairly close to the 1.5 percent actual rate of change for the whole sample period. But estimate II 129 .q magma sous c .N mHan Eoumm .HH now com. u m - a sum H “on Hmmq. u m I a mums; .H u Amuav Hmm.AQm-Hvu an umumesoamo I I N .HH new Hmm. n 0 saw H you ammo. u 0 means .m..wmm Asmuavu so magmasoamoa ooo¢. oqm.m qmu.H hum. mmn.H oom. me. h~¢. ohm. «snome mmom. om~.¢ wmo.m qu. mmd.~ emu. ¢o~.H m¢m. mmN.H acumemH mmmm. o¢~.Nu moo. nmq. omm.H mow. mho.u oom.n mao.u qounmoH nnwm. OHm.m HhH.N mHn. Noo.N mmm. 0mm. owe. NHH.H ohumoaH wHom. onm.H qdm.H mom. cNN.H mum. ¢Hm.H wnH. now. ¢nnnmmH H A HH H HH H HH H m m M. mHmauo< ANV mwam:o a N + H Hmoaafiumu AHV wawcoamov wmwmwn kn Houwmmo %n N H mpOHuom Auaouumav AHmv madam mwmz :H owamso mo mumm Hmsaa< chowuuomoum qeaa - smma .uouoom wawuauommsamz wsu GH monHmm cmeHommm you moum5m maooaH mwmz vmuowcmum can Hanuo< cw mowamso mo moumm .oH mHan 130 predicted the rate of change as only .50 percent annually for the period, which was much lower than the rate of actual change. For other specified periods, the rates of the change were also calculated, and they are compared with the actual rates of the change. In most of the periods, the rates of changes predicted by estimate I are fairly consistent with the actual rate of the change. But the results predicted based on estimate II are significantly lower than the actual rates, even though the direction of the changes is consistent for all the Specified periods. As a whole one may conclude that the marginal productivity theory can explain the behavior of the wage income share in the manufacturing sector relatively better than in the agricultural sector. The predicted rate of the change is separated by the two sources-- capital deepening and biased technical progress. With estimate I, it is calculated that the capital deepening factor has contributed to the growth of the wage share at an annual rate of .41 percent, while the labor-using biased technical progress has increased the share at an annual rate of 1.32 percent. Using estimate II, about .18 percent of the annual rate was contributed by capital deepening and .35 percent of the annual rate by biased technical progress. Thus for the manu- facturing sector, the biased technical change has been more important in changing the wage income share than was the capital deepening. Following the same procedure used in the previous section, the wage income share indexes were derived for both the actual and pre- dicted shares with the 5 year average rates in the changes. The base of 131 the index is 1.0 for the actual share in 1974. The base index for the predicted share is set as 1.537 in 1974, which is the ratio of measured marginal product and the actual wage rate for the period 1970-74. The derived indexes are presented in Table 11. The results show that both of the predicted indexes are far above the actual share indexes as can be seen from Table 11. The indexes predicted by estimate I are about 48 percent higher than the actual indexes for the period 1957-1969 and about 52 percent higher than the actual indexes for 1970-1974. Thus the difference between the two indexes has slightly increased over time. And the indexes predicted by estimate II are about 79 percent higher than the actual indexes for 1957-1959 and 56 percent higher than the actual share indexes for 1970-1974. The difference between the two indexes has a significantly decreasing trend based on estimate II. The general results of the above two sections show that the pre- dicted and actual shares have moved in the same directions but that there are large differences in the actual magnitudes. But the explanation for the differences in the actual magnitudes is not clear concerning whether our estimates are biased in certain directions or whether some other vairables, which were ignored in the marginal productivity theory of distribution, have played an important role in the determina- tion of wage income shares for the period. One more complication is that one set of our estimates of parameters used for the analysis is based on the marginal productivity relationship. Assume for a moment that our estimate of parameters reflects the true flH .,132 Table 11. Actual and Predicted Wage Income Share Indexes in the Manufacturing Sector, 1957-1974 1 1 Year Actual Predicted I / II / ( A ) I II A A 1957 .772 1.134 1.383 1.469 1.791 58 .777 1.149 1.389 1.479 1.788 59 .782 1.165 1.395 1.490 1.784 60 .787 1.181 1.402 1.501 1.781 61 .792 1.198 1.408 1.513 1.778 62 .797 1.214 1.415 1.523 1.775 63 .802 1.231 1.421 1.535 1.772 64 .808 1.248 1.427 1.545 1.766 65 .813 1.269 1.434 1.561 1.764 66 .839 1.290 1.446 1.538 1.723 67 .865 1.322 1.458 1.528 1.686 68 .893 1.355 1.470 1.517 1.646 69 .921 1.388 1.483 1.507 1.610 70 .950 1.422 1.495 1.497 1.574 71 .962 1.457 1.508 1.515 1.568 72 .974 1.483 1.518 1.523 1.559 73 .987 1.510 1.527 1.530 1.547 74 1.000 1.537 1.537 1.537 1.537 1 Used the same procedure as in Table 9 (see footnote to Table 9). 133 value of the production relationship. Then the results of the analyses imply that we could not rely on the marginal productivity theory to explain the behavior of the wage income share in the Korean economy. Then the validity of the estimation procedure which is based on the marginal productivity relationship is also questionable. Thus estimate I, which is based on the relationship has a problem in its reliability. In another way, if we assume that the marginal productivity theory has been truely valid for the period, then both of our estimates are biased by some other sources of errors. Thus in any ways, the reliability of our estimate I is questionable. However, remember that, as pointed out in Chapter IV, we could still obtain unbiased parameter estimates with the marginal productivity relationships if the disequilibrium factors are not correlated with the regression variables under the given assumptions. But as seen, the gap between the actual and predicted shares derived from both of the estimates, or the degree of disequilibrium, has been changed significantly over time. This fact makes a more serious problem for the reliability of estimate 1. For this reason, one may give more credit to estimate II, which is not based on the marginal productivity relations. Accordingly one comes to the conclusion that for this period the disequilibrium factors played an important role in the Korean economy with a different direction between sectors, but with the degree of disequilibrium being significantly reduced over time for both the sectors. 134 Marginal Products and Actual Returns to Labor In the previous two sections, we analyzed the behaviors of the relative wage income shares with the estimation results of production parameters for both the agricultural and manufacturing sectors. The analyses showed that the general directions of the predicted rates of change in the wage shares, based on the marginal productivity theory, are consistent with the actual rates of the changes for both the sectors, but there were large differences in actual magnitudes. Thus the results may indicate that the theory we used could not be a complete explanation of the changes in the wage income shares for the period under study in the Korean economy. With this tentative observation, this section will attempt to estimate the marginal product of labor by using the estimates of pro- duction parameters and to examine the extent of variations between the estimated marginal product and the actual wage rate over time and between sectors. The marginal product of labor is found simply by differentiating the specified production function with respect to labor. The variables that determine the marginal product are the same as those in the pro- duction function. If the economy is in equilibrium and there are no economies or diseconomies of scale, the value of the marginal product and actual returns to labor should be equal.* *The statement that, in equilibrium, value of marginal product is equal to the wage rate is, strictly speaking, applicable to the individual firm in purely competitive markets. The discussibn which follows assumes that taking the partial derivative of our aggregate production function gives a marginal product which is equal to the individual firms' marginal products. 135 From the specified production relation, we can derive the marginal product of labor as p 0 - —+ _ MPL = y Vu-a) c 8" vt'(°"3)p<§)'p]‘l <6-5) X. L where all notations are the same as previously. From Equation (6-5), we can see more clearly how the production parameters and variables affect the marginal product of labor. At a given capital-labor ratio, a larger value of v and smaller value of 6 will give the higher marginal product of labor. And the larger value of 9 will yield a larger marginal product of labor. When the value of v *From the production function v = [6(taK)-p + (1-6)(tBL)_p]-?\5)' (6-6) we can derive the expression as B. p - - - - vV/L" = Wt <1—6)c Bp+6t ”6%) D1 1 (6-7) and the Equation (6-4) can be written as -0 _ o MPL = y Vu-am: 39(vV/L°)% (6—8) Substituting (6-7) into (6-8), we obtain —1.Y. MPL = v[l+(igg)t-(a-B)p(%?-p] L Multiplying L-in both sides of above equation we also obtain the expression as MPL-L _ _ __Qfi -(a-B)p EL-o -1 V - SL - v[l+(l_6)t (L) ] .136 is unity and the value of p is zero, Equation (6-5) reduces to the expression'MPL = (1—6)-% which is a Cobb-Douglas case. With Equation (6-4), empirical estimates of the marginal productivity of labor are calculated by using the estimates of production parameters and data of the production function. As explained in the section on data description, the output was measured in units of million won at 1970 constant price, and labor input was measured in units of million days of work. Thus the unit of calculated marginal product of labor is won per day of work (or 8 hours of work). The empirically estimated marginal product of labor for both the sectors is presented in Tables 12 and 13. For the agricultural sector, the results show that the actual wage rate was far above the estimated marginal product of labor particularly until the early 1960's. However, the difference between the marginal product and actual wage rate decreased over time. The actual wage rate was about 50 percent higher than the estimated value of the marginal product of labor for the first five-year period 1955-59, but it was fairly close to the actual wage rate for the last five-year period, 1970-74. In other words, it is calculated that the values of the marginal product were only about 67 percent and 93 percent of the actual wage rate for the respective periods. Thus the returns to the self-employed labor in agricultural production appear to have been far below the actual wage rate, but have largely improved over time. The marginal product of labor for 8 hours of work in the agricultural sector rose from 316 won in the period 1955-56 to 558 won in the period 137 Table 12. 'Marginal Products and Actual Returns to Labor for the Agricultural Sector, 1955-74 Year MPL1 VMPL2 W3 W/VMPL (Won) (Won) 1955 316 51 76 1.481 56 315 59 85 1.438 57 332 65 92 - 1.420 58 336 61 92 1.518 59 319 59 93 1.574 60 329 60 96 1.589 61 310 73 106 1.452 62 324 88 _ 115 1.308 63 336 113 143 1.263 64 349 178 199 1.121 65 363 176 221 1.254 66 394 206 256 1.245 67 381 247 307 1.242 68 413 296 381 1.288 69 474 396 463 1.169 70 483 483 579 1.169 71 494 597 695 1.164 72 514 746 803 1.070 73 536 850 886 1.042 74 580 1,232 1,141 .926 1The unit is 1970-constant won. 2VMPL = P * MPL. 3From Table 1. 138 Table 13. Marginal Products and Actual Returns to Labor for the Manufacturing Sector, 1957-74 Year MPL VMPL w W/VMPL (Won) (WOn) 57 560 138 77 .558 58 561 148 84 .568 59 566 150 86 , .573 60 558 154 92 .597 61 550 174 104 .598 62 582 207 112 .541 63 520 228 128 .561 64 532 304 179 .589 65 549 333 205 .616 66 538 411 265 .645 67 566 441 295 .669 68 636 534 347 .650 69 742 676 444 .657 70 858 858 571 .666 71 837 870 683 .785 72 1,003 1,187 798 .672 73 1,096 1,394 878 .630 74 1,167 1,881 1,194 .635 1The unit is 1970-constant won. 2VMI’L = P * MPL. 3From Table 2. .139 1973-74 in 1970 constant prices. Thus in the last two decades, the marginal product of labor has increased about 77 percent. However the growth of marginal productivity of labor was not significant until the early 1960's, but it has accelerated since the mid-1960's as can be seen in Table 12. Quite unlike the agricultural sector, the data for the manufactur- ing sector shows that the estimated marginal product of labor was far above the actual wage rate for the whole sample period. The actual wage rate was only about 56 percent of the value of the marginal product in 1957-58 and 63 percent in 1973-74. But recall that the measure of the actual wage rate did not include any side benefits as noted in the data explanation. Even considering this fact, it seems clear that there was also a large gap between the actual wage rate and the marginal product. However the wage rate is approaching the value of the marginal product over time. The marginal product of manufacturing labor rose from 611 won in 1957—1958 to 1,132 won in 1973-1974 in 1970 constant prices. Thus in the last two decades the marginal product of labor has about doubled compared with the 77 percent increase for the agricultural sector. As in the agricultural sector, the marginal product of manufacturing labor also remained fairly stable until the mid-1960's, but it has rapidly grown in the ten-year period, 1965—1974. As seen in the above, the data indicate that there were significant differences between the actual returns and marginal products of labor in both the sectors. But the directions of the differences or disequilibriums 140 were not the same between the sectors. For the manufacturing setor, the value of marginal product of labor was far above the actual wage rate, thus the wage earners were paid too little compared to their actual contributions to the production. But the value of the marginal product of agricultural labor was lower than the wage rate. Thus the self-employed agricultural workers were left with low returns to their labor. However, the disequilibrium or the gap between the value of the marginal product and the actual wage rate has been decreased significantly over time for both sectors. The above results may partially explain the fact that the growth rate of the capital-labor ratio was significantly lower in the manufactur- ing sector than in the agricultural sector for the whole sample period. As seen in Table 4 of Chapter III, the annual growth rate of the capital- labor ratio was about 1.4 percent in the manufacturing sector and 5.3 percent in the agricultural sector. A different rate of substitution between factors is basically determined by the nature of production. But with the wage rate far below the marginal product, the optimal amount of capital in the manufacturing sector is less than it would be if labor were paid its marginal product. Thus the pressure to substitute capital for labor is relatively reduced. With a reverse situation, the agricultural sector has had more incentive to substitute capital for labor than it would be in an equilibrium case. It is also interesting to compare the estimated results between the sectors. In Table 14, we have calculated the ratios of the estimated marginal products, the values of the marginal product, and the actual wage rates between the sectors for specified periods. 141 Table 14. Comparisons of Actual Wage Rates, Marginal Products, and Value of Marginal Products of Labor Between the Agricul- tural and Manufacturing Sectors for Specified Periods . l Periods MPLA VMPLA WA /MPLM /VMPLM /WM 1955-59 .583 .423 1.123 60-64 .604 .466 ' 1.064 65-69 .670 .546 1.048 70-74 .580 .628 1.002 1955—74 .597 .516 1.059 1Calculated from Tables 12 and 13. Subscripts A and M stand for the agricultural sector and the manufacturing sector respectively. Marginal products are in value units at 1970 prices. 142 The estimated marginal product of agricultural labor was only about 60 percent of that of manufacturing labor. The relative marginal product of agricultural labor has been improved from 58 percent in 1957—1959 to 67 percent in 1965-1969, but it went back down to 58 percent for 1970-1974. The relative value of the marginal product of agricultural labor to manufacturing labor was also calculated for the specified periods. It shows that the value of the marginal product of agricultural labor was about 52 percent of that of manufacturing labor as an average for the whole period, which implies that there was also a large disequilibrium between the sectors. However, the disequilibrium between the sectors has been significantly lessened over time. The relative value of the marginal product of agricultural labor rose from 42 percent in 1957-1959 to 63 percent in 1970—1974. The improvement in the relative value of the marginal product of the agricultural sector is attributed partially to the increase in the relative price of the agricultural output to manufacturing output and partially to the increase in the relative physical marginal product of agricultural labor, as seen in the above. From the above results, or the presence of a large disequilibrium between the sectors, one would expect that the large amount of migration which was actively taking place in the last decade from rural to urban areas will continue at least in the near future until the disequilibrium is adjusted. Even though there were large differences in the value of the marginal product of labor in the two sectors, the actual wage levels 143 were fairly close.* The data indicate that the actual wage rates of both the sectors fall within the range between the values of marginal products of both sectors during the whole period. Thus manufacturing workers were paid less than the values of their marginal products. And self-employed agricultural workers earned less return than they could earn in other hired work. ' As a Summary, the results of this section seem to lead to a conclusion that disequilibrium within and between the sectors has been significant in the Korean economy for the last two decades. But both of disequilibriums appear to be adjusted over time. However, the conclusion is very tentative and it is valid only when the data and estimates used for the analyses reflect true values. As explained in previous chapters, there may be various possibilities that the data and estimates are biased. The weakness in the data and possible errors in estimation could mean that the economy may not be in disequilibrium as indicated by the above results. More work on this possibility is needed. If such a disequilibrium does in fact exist, there have been various possible causes in the process of rapid growth of Korean economy in the last decade. The analysis of these causes will be another important subject for investigation. *Note that the measurement units of wage rate are not the same between sectors. As explained in the section of data description, WA was measured by adjusted man—equivalent unit and WM was measured by unadjusted physical labor unit. 144 However, our analysis could make at least one clear conclusion. If one believes the assumption that a competitive equilibrium or marginal productivity theory has truly operated in the Korean economy during the period, most of our estimates of production parameters are biased. For the agricultural sector, the directions of the bias will be down— ward in either one or both of the estimates,<5and<“—;-l->[(f<-f.> + (a-mc’ll where o is the elasticity of substitution between factors. From the expression, it is clear that the rate of change in wage income share depends on the changes in the capital-labor ratio and the differential growth rate of factor efficiency at a given technical condition of factor substitutability. Using the theoretical relationships described above, this study attempted to explain the behavior of the actual wage income share for both the sectors. Using various sources of data, the actual wage income share was estimated during the last two decades. The results showed that 147 the behavior of the share has been significantly different between the sectors. For the agricultural sector, three different estimates of the actual wage income share were obtained; two were based on a hired wage rate and the other one was based on the residual concept. All of the estimates showed a consistently decreasing trend during the period as a whole. Agricultural wage income as a share of total gross agricultural income (where the wage share calculation was based on the hired wage rate) has decreased from 51 percent for 1955-1957 to 35 percent for 1972-1974, some 16 percentage points, or about one-third. It was also calculated that the wage income share has decreased at an average annual rate of about 2.0 percent. But the rate of change in the agricultural sector fluctuated widely compared to the rate in the manufacturing sector. For the first ten-year period, 1955-1964, the annual rate of change was -1.3 percent, but it accelerated in the second ten-year period to a —2.6 percent annual rate of change. For the manufacturing sector, the estimate of the actual wage income as a share of the total value added showed an increasing trend during the period, 1957-1974, or contrary to the trend in the agricultural sector. The share has increased, from 33 percent for 1957-1959 to 41 percent for 1972-1974, some 8 percentage points, or about one-fourth. The average annual rate of change is calculated as 1.5 percent during the whole period. For the explanation of the actual behavior of the wage income share, we estimated the parameters determining the wage share. 148 One of the major difficulties for the estimation was the availability and reliability of basic data. The definitions of data used for the measurement are not identical between the sectors partially because of the lack of availability and partially because of the conceptual differences. For the agricultural sector, output was measured by the total gross output concept. The measure of the labor input was an adult-man-day equivalent unit adjusted by age and sex, and the capital input was measured by the flow service concept. For the manufacturing sector, output was measured by the total value added concept. The measure of the labor input was unadjusted physical units of labor, and the capital input was measured by the stock concept. After experimenting with various different estimation procedures, we derived two sets of estimates for each sector. For both sectors, one set of the estimates (estimate I) was based on the marginal pro- ductivity relationships and the other one (estimate II) was the result estimated directly from the specified production function by a linear approximation or nonlinear least squares procedure. For the agricultural sector, estimate I was based on the estimat- ing equation which was derived from the production expansion path with a partial adjustment assumption. Estimate II was based on a linear approximation of a specified CES function by expanding the logarithm of the function. The Hildreth-Lu estimation procedure was applied for both the estimating equations with a first order autoregressive scheme. 149 For the manufacturing sector, estimate I was based on the two— stage Zellner—Aitken's efficient estimation method, with the assumption that the disturbance terms of each marginal productivity relation were mutually correlated. Estimate II was based on the nonlinear least squares method, and Bard's version of the Gauss-Newton method was applied for the solution. The major estimation results are summarized as follows. First, the elasticity of substitution is greater than unity for the agri— cultural sector and less than unity for the manufacturing sector. For the agricultural sector, the point estimates of the elasticity of substitution were 1.392 and 1.134 in estimates I and II respectively. But for the manufacturing sector, it was .683 in estimate I and .831 in estimate II. Second, for both the sectors, the estimated capital augmenting parameter 0 turned out to be greater than that of labor 8. For the agricultural sector, the estimate of 0-8 was .175 (estimate I), and for the manufacturing sector, the estimates of a and B were .599 and .064 respectively (estimate I). Thus, the results seem to indicate that during the sample period the growth in productivity has been mainly the result of the efficiency growth of the capital input, which has been rapidly expanded in the last decade. Third, the point estimate of the scale parameter was consistently close to unity for both the sectors, which agreed with the assumption used in deriving the marginal productivity theory of distribution. For the agricultural sector, the estimate of the adjustment coefficient, 150 A = .409, was significantly below unity, which implies that the input adjustment process to the price change has been fairly slow for Korean agricultural production during the period. But for the manufacturing sector, the adjustment coefficient, A = .848, was relatively close to unity. The confidence interval estimates, or significance tests, for the estimated parameters were also made. However, the usual significance test of the ratio of regression coefficients presented a difficulty because the distribution of the ratio of two normal variables with non-zero means is unknown. For this reason, we derived a formula to 7?! find the maximum and minimum values of the ratio of two normal variables within the joint confidence region of the variables with a specified probability. Using the above procedure, we derived the 95 percent confidence interval estimate of the elasticity of substitution as .9041 §_0 :_2.5717 for the agricultural sector and .3849 §_0 :_.9409 for the manufacturing sector. This result implies that the usual CD function will have a specification bias at least for the manufacturing sector. The significance test based on the same procedure showed that the factor augmenting bias 0-8 for agriculture and the labor augmenting parameter 8 for manufacturing are not significantly different from zero, but the capital augmenting parameter a for manufacturing was significantly positive at the 95 percent significance level. Using the estimation results, the bias of technical progress was calculated for both the sectors. As in usual terminology, the bias of 151 technical change was defined as a differential growth rate of marginal productivity between factors at a given capital-labor ratio. In our notation, the bias (B) can be calculated as l B 292—1-(6-3) t- = MPK — MPL. At the year 1965 or t=10, the calculated B turned out to be .51 percent for the agricultural sector and —2.49 percent for the manufacturing sector, which means that the technical progress was capital-using for the agricultural sector and labor-using for the manufacturing sector. Thus at a given capital-labor ratio, technical progress results in a decrease in the wage income share for the agricultural sector, but it causes an increase in the wage income share for the manufacturing sector. It also appears that the technical progress has been more significantly biased in the manufacturing sector than in the agricultural sector during the period. Using the theoretical relationship derived from the marginal pro~ ductivity theory, we attempted to explain the behavior and sources of changes in wage income shares. For this purpose, the theoretical rate of change in the wage income share was predicted and it was also separated into two sources—-capital deepening and biased technical change. For the agricultural sector, the theoretical relationship pre- dicts the wage income share to decrease at an average annual rate of 1.03 percent with estimate I and .34 percent with estimate II, which are compared with the 1.97 percent of the actual rate of the change during the whole period. 152 Based on estimate I, the rate of the change was separated into two sources. The results shows that, of the 1.03 percent of the annual rate of change, the capital deepening factor contributed .23 percent. Thus the capital deepening factor appears to be a major source of the change with an elastic substitutability between capital and labor. The actual and predicted rates of the change are in the same direction, but there are large differences in the actual magnitudes of the rates. In average terms, it is calculated that our theoretical relationship can explain only about 52 percent of the decreasing trend of the actual share with estimate I and only 17 percent with estimate II. i. Using the estimated rate of the change, the wage income share index was derived for both the actual and predicted shares. The derived index showed that the actual wage income share index was sub- stantially above that of the predicted share for the earlier period. But the differences between the two indexes have significantly decreased over time. For the manufacturing sector, the annual predicted rate of increase in the wage income share is about 1.7 percent with estimate I, which is fairly close to the 1.5 percent actual rate of the change during the whole period. But estimate II predicts the rate of change as only .50 percent, which is much lower than the rate of actual change. Based on estimate I, the marginal productivity theory seems to better explain the behavior of the wage income share in the manufacturing sector than in the agricultural sector. With estimate I, it is also calculated that the capital deepening factor has contributed to the growth of the wage share at an annual 153 rate of .41 percent, while the labor-using biased technical progress has increased the share at an annual rate of 1.32 percent. Using estimate II, about .18 percent of the annual rate was contributed by capital deepening and .35 percent of the annual rate by biased technical progress. Thus for the manufacturing sector, the biased technical progress has been more significant in changing the wage income share than has been the capital deepening. The analysis showed that the general direction of the predicted rates of change in the wage shares, based on the marginal productivity theory, were consistent with the actual rates of the changes for both the sectors, but there are large differences in actual magnitudes. Thus the results may indicate that the theory we used could not be a complete explanation of the changes in wage income shares for the sample period in the Korean economy. However, any conclusions are tentative due to the various possible sources of bias in the estimates of the parameters. With the same estimates, this study also attempts to estimate the marginal product of labor, and examines the extent of variations between the estimated value of the marginal product and the actual wage rate over time and between sectors. For the agricultural sector, the estimated value of the marginal product of labor is far below the actual wage rate particularly until the early 1960's. However, the difference between the value of the marginal product and the actual wage rate has been considerably reduced over time. It is calculated that the value of the marginal product was r 154 only about 67 percent of the actual wage rate in 1955-1959 and 93 percent in 1970-1974. The marginal product of labor for 8 hours of work in the agricultural sector rose from 316 won in 1955-1956 to 558 won in 1973-1974 (in 1970 constant prices). Thus during the last two decades, the marginal product of labor has increased about 77 percent. Quite unlike the agricultural sector, the data for the manufactur- ing sector show that the estimated value of the marginal product of labor is far above the actual wage rate for the whole period. The actual wage rate was only 56 percent of the value of marginal product in 1957-1958 and 63 percent in 1973-1974. The marginal product of manufacturing labor rose from 611 won in 1957-1958 to 1132 won in 1973—1974 (also at 1970 constant prices). Thus during the period, the marginal product of manufacturing labor has about doubled, which is compared with the 77 percent increase in the agri- cultural sector. The data indicate that there were significant differences between the actual returns and values of marginal products of labor in both the sectors. But the directions of the differences were not the same between the sectors. The value of marginal product of manufacturing labor was far above the actual wage rate, implying that wage earners have been paid too little compared to their actual contributions to the production. But the value of marginal product of agricultural labor was lower than the wage rate, implying that the self-employed agricultural workers received low returns for their labor. However, the actual wage 155 rate has tended to approach the value of the marginal product over time for both the sectors. The data also show that the value of the marginal product of agri- cultural 1abor is only about 52 percent of that of manufacturing labor on the average for the whole period, which implies that there was also a large disequilibrium between the sectors. However, the disequilibrium between the sectors has been significantly reduced over time. The relative value of the marginal product of agricultural labor to the manufactur— ing labor rose from 42 percent in 1957-1959 to 63 percent in 1970-1974. While it is hard to judge the validity of our analysis, one may draw a conclusion that the marginal productivity theory seems to be consistent with the actual movements of wage income shares in their general directions, but there are considerable differences in actual magnitudes. This may indicate that there are some other factors, or disequilibrium factors, paying an important role in the Korean economy during the period. However, the importance of the disequilibrium factors within and between the sectors has apparently decreased over time. Thus the Korean economy may have been in the process of adjusting to an equilibrium during the period. But the conclusion is very tentative, and it is only valid when the data and the estimates of parameters used for the analysis reflect true values. There may be various possibilities the data and estimates were biased. The weakness in the data and possible errors in estimation could mean that the economy may not have been in the disequilibrium 156 indicated by the results. 'MOre work on this possibility is needed. If such a disequilibrium does in fact exist, there are various possible disequilibrium factors due to the process of rapid growth of the Korean economy in the last decade. The analysis of various possible factors or the causes of disequilibrium will be another important subject to be investigated. There may be basically two sources of differences in factor sub- stitutability between different sectors. One is the difference in the basic nature of the production technology, and the other one will Fifi be the change in the commodity structure of each sector. However, the sources which have caused the differences in the degree of substituta- bility between the sectors are not clear in our aggregated sector study. Thus more disaggregated study is needed to provide more evidences for the estimate when the data become available. APPENDIX APPENDIX In order to find the maximum and minimum values of the ratio of two normally distributed random variables b1 and b2 within the joint con- fidence region of the two variables, we need to find the slope of T1 and T2, or the coordinates of two points, 1 and 3, in Figure 2. To do this we formulate the standard maximization and minimization problem subject to a constraint which is the ellipse equation of the confidence region: b Max. and Min. -—l b2 2 2 _ S.T. Clbl + C2b2 + C3b1b2 + 0461 + C5b2 + c6 - o, where C1 = 811’ C2 = a22, C3 = 2 812’ C4 = 2(a 31181 + a12 82), _ 2 2 Cs ‘ ”2(32282 + 3128 )’ C6 ' a1181 + 82282 + 23128182- 6' The values of the aij are explained on page 98, B1 and 82 are the expected values of b1 and b2. Construct the auxiliary function by the method of Lagrange multipliers as b 1 b2 2 L- b2 + A(Clb 1 + C2b2 + C3blb2 + C4bl + C5b2 + C6). To maximize and minimize the ratio, find the values of b1 and b2 for which the partial derivatives of L are all zero. The partial derivatives are _ 1 Lbl b2 + A(ZClb1 + C3b2 + 04) (1) b1 Lb2 = —-;5 + A(2C2b2 + C3b1 + C5) (2) 2 157 158 2 2 LA — Clbl + C2b2 + C3blb2 + C4b1 + C5b2 + C6 (3) Taking the ratio of Equations (1) and (2) and rearranging the relation we get the expression as 2 2 _ 2Clb1 + 202 b2 + 2036le + c461 + C5b2 — 0 (4) Multiplying Equation (3) by 2 and subtracting it from the above Equation (4), we get a simple straight line equation as C4b1 + C5b2 + 2C6 = 0 (5) Equation (5) is the straight line equation connecting the two points which give the maximum and minimum values of the ratio. Equation (5) is M in Figure 2. Then the two equations--the straight line Equation (5) and the ellipse Equation (3)--are solves simultaneously. 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