Divergences and Social Factor Prices

Under competitive market conditions, the prices of fully employed factors will reflect marginal value products, unless divergences in the factor market are present. If D represents the price equivalent value of factor market divergences, observed factor prices can be expressed as

where w = factor price, P = output price, and MPP = marginal physical product (OQ / AL). A superscript P is used to denote that the variable is observed under private market conditions.

The observed values of PP and MPPP in equation 1 might also be distorted from their social values. If t is used to represent the divergence in output prices and OMPP to represent the divergence in marginal physical products, equation 1 can be rewritten as

where a superscript w is used to indicate the value of a variable under world prices for outputs, represents the effect of commodity market distortions on factor productivity, and represents the effect of factor market distortions on factor productivity. Equation 2 can be rearranged to yield

The social factor price, w_S, is the value of the marginal product measured at world prices; this is the first term on the right-hand side of equation 3, (P_w*MPP_w).

If equation 3 is rearranged and the interaction terms t x OMPP are ignored, the following equation results:

Equation 4 shows that the difference between private (observed) factor prices and their social values can be accounted for by two categories of effects. The term represents the impact of commodity market divergences on factor prices. The term describes the effects of factor market divergences; these divergences influence factor prices directly and perhaps indirectly as well, through their effects on input productivity. Fixing high factor prices by legislative decree, for example, encourages producers to reduce use of the input and results in artificially high marginal productivities.

The disaggregated view of factor price divergence presented in equation 4 provides an organizational framework for the evaluation of shadow prices. One begins with observable private market wages, rates of return to capital, and land rents, then identifies various divergences, and finally makes judgments on their quantitative significance. Ideal conditions for the calculation of social factor prices arise when empirical estimates of factor demand and supply curves are available for each industry in the economy. Private market prices and quantities can be combined with information about divergences to determine the shifts or movements along the demand and supply curves in each factor market. When such information is not available or when estimates are considered unreliable, less precise estimates of social values must be formulated.

The following sections consider the two categories of divergences in equation 4: factor market divergences and commodity market divergences. The discussion focuses on computational issues. Subsequent sections consider two groups of indirect influence on the factor prices. Macroeconomic distortions affect factor prices through exchange-rate induced influences on commodity prices or through direct impacts on the price of domestic capital resources. Input substitution effects represent responses to changes in relative factor prices and account for part of the observed changes in marginal physical products described in equation 4.

Factor Market Divergences

Adjustments for the impact of factor market distortions are easiest for proportional taxes or subsidies. In that case, the analyst need only decide whether the taxes or subsidies have been passed on to the factor. For example, social security taxes are often levied on employers in an attempt to increase remuneration to labor. If laborers compete for employment, however, money wages would fall by the full amount of the tax. Assuming that workers eventually receive the value of the social security tax, total compensation remains constant. Only the temporal pattern of wage receipts has changed; labor forgoes some current income in favor of increased payments during retirement or illness-induced absence from work.

When taxes have been applied to all sectors of an economy, decisions on the treatment of employer taxes are based on the consideration of employment levels in the presence of the factor tax. Full employment implies that the tax reduces money wages. When the tax is applied only to certain sectors of the economy (industry but not agriculture, for example), full employment is not sufficient evidence to disregard the tax as a distortion; policy may have caused excess labor to migrate from the taxed sector to the untaxed sector, resulting in full employment. Such movements should cause differences between sectoral factor prices. Comparisons of tax-inclusive wage rates with wage rates in sectors of the economy that do not pay factor taxes indicate whether the taxes should be treated as a divergence (if the two prices are unequal) or ignored (if the two prices are equal).

Adjustments for regulations that fix the absolute level of prices in the factor markets are more difficult. Private market values are sometimes available from parallel markets, and wage rates and land rental rates can often be compared to official prices in order to determine whether the regulations are enforced. But unless parallel markets are large, their prices will not be close to social (non-regulated) values. For example, interest rates in the parallel market can be difficult to relate to the rate of return on investment, particularly if capital markets are subject to fragmentation or other divergences besides rationing.

Market failures are the final category of factor market divergences. These failures are often identified by regional comparisons of prices for the factors. If factors are mobile between regions, integration of the factor market is possible and factor prices in one area may be linked to factor prices in another area. But in general, more direct confirmation of market failures is needed, because competitive factor market circumstances may also explain factor price differentials.

If the costs of migration-transportation and moving costs as well as psychic costs-from one area to another are positive, even perfectly competitive factor prices can differ. Consequently, social prices for the factor need not be equal in all regions. Figure 7.2 illustrates this point. The labor market in region 1 is compared to that in region 2; the wage rate is represented by w*. The costs of migrating from region 1 to region 2 are w* - w_L; the costs of migrating to region 1 are w_U - w*. Initially, local demand and supply in region 1 are assumed to be in equilibrium at the wage rate w*, which is equal to the wage rate in region 2. But this equality need not be maintained. Following an increase in local demand in region 1, caused by higher prices of region 1 outputs, demand expands to D' and the wage rate rises to w₁. This rise has no effect on the wage rate in region 2, because the costs of migrating to region 1 are larger than the regional wage difference. Migration will begin only if demand causes the wage rate to rise above w_U. At that point, regional wages will begin to move together. Analogous results follow for the case when demand shifts backward and the local wage rate declines. Regional wage rates become linked only when the wage rate falls below w_L, the wage rate that motivates out-migration from region 1.

Another influence on competitive factor price differentials is the duration of employment alternatives. Labor demands and wages may vary seasonally, and employment alternatives may comprise long periods of low-wage work in one region versus shorter periods at higher wages in another region. Factor prices then differ across regions, but market failures need not be present. Total compensation over the course of a production cycle is not different enough to induce migration.

Identification of capital market failures requires a different definition of migration costs and market segmentation. The physical costs of transferring financial capital among regions are extremely small and would not generate any substantial difference in regional rates of return. A single social rate of return, therefore, would be representative for all regions. But social (and private) rates of return can differ more substantially if investment risks vary among potential borrowers. Then the relevant market boundaries are not geographical regions but types of borrowers and commodities. If small borrowers or particular commodities have relatively high probabilities of financial default, rates of return in those sectors must be higher to account for the increased costs of lending. Similarly, some portion of the transaction costs of lending and borrowing are independent of the amount of loan; in percentage terms, small investments must earn higher rates of return than large investments. These differential risks and transaction costs remain even

if capital markets are integrated. Social rates of return need not be equal for all commodity systems.

Commodity Market Divergences

Figure 7.3 is used to illustrate the impact of commodity divergences on factor prices. Figures 7.3a and 7.3b present input-output productivity curves for two industries that use the same input, unskilled labor. These industries are assumed to be the only employers of the factor. In industry 1, the marginal productivities of input use diminish sharply as increasing amounts of labor are applied in the production process. In industry 2, the marginal productivities are almost constant, giving the input-output productivity curve a nearly linear shape. In both cases, the firm-level productivity curves are drawn under the assumption that all other input levels are held fixed. Variations in the amounts of nonlabor

inputs would generate productivity curves for labor that differ from those illustrated in the figure.

The demand of the firm for unskilled labor is determined by the profitability of input use. If labor exhibits diminishing marginal returns (as is assumed in the figure), the most profitable use of labor will result when the marginal value of production just equals the incremental cost of labor input: (delta Q₁)(P₁) = (delta Q_L)(w). Rearranging these terms yields a relationship between the input-output productivity ratio and the input-output price ratio: delta Q₁ / delta Q_L = w / P₁. The slope of the input-output productivity curve (delta Q₁ / delta Q_L) equals the input-output price ratio at the maximum profit point. The assumption of diminishing marginal productivities implies that delta Q₁ / delta Q_L exceeds w / P₁, whenever input use is less than this profit-maximizing level.

The same assumption also implies that larger quantities of labor will be used by the firm as wages decline, yielding a firm (and industry) labor demand curve that is downward sloping, as shown in Figure 7.3c. The industry 2 demand curve (Figure 7.3d) is more elastic than the industry 1 curve because the marginal productivity of labor is assumed to change more slowly as increased amounts of labor are used. Aggregation of the industry demand curves D₁ and D₂ yields the market demand for labor, illustrated in Figure 7.3e. The equilibrium wage rate, w*, results from the intersection of the market demand curve with the available supply of labor, Q*. The allocation of labor between the two industries is indicated in the industry demand diagrams as Q*_L1 and Q*_L2.

If the interactions between labor inputs and other inputs (whose use levels are also affected by changes in the output price) are ignored, the same marginal productivity curve can be used to determine input demand under varying output prices. If the output price increases from P’₁ to P’₁’; the firm-level labor demand increases from q₁ to q₂, and the marginal productivity of labor in industry 1 declines. Changes in the number of firms in each industry will also affect the factor demand curve. As P₁ increases, industry 1 expands and industry 2 contracts.

The net shift in aggregate labor market demand could thus be positive or negative, depending on the relative intensity of factor use in the two industries. In the example, industry 1 is more labor-intensive than industry 2, and total labor demand shifts outward, from D* to D**. The equilibrium wage rises from w* to w**. Relative to the initial position, labor use has increased in industry 1 (Q*_Ll to Q**_L1) and declined in industry 2 (Q*_L2 to Q**_L2). Marginal physical products in industry 2 have had to increase to justify the higher wage. If industry 2 were the labor-intensive industry, aggregate demand would shift backward, offsetting the wage-increasing impact of industry 1 expansion. Wages and marginal physical products in both industries would decline.

As the number of commodities is increased, the relationships between factor prices and commodity price divergences become more obscure. Divergences that increase commodity prices increase factor demands by firms; divergences that decrease commodity prices shift firm factor demands backward. Changes in the numbers of firms in each industry cause further back-and-forth movements in aggregate factor demand. If protection is not biased toward any factor (if, for example, capital intensive industries do not receive more protection than labor-intensive industries), the output price divergences will not have a substantial impact on factor prices. The factor price effect of a divergence in one commodity market would be offset by the factor price effect of a divergence in another output of opposite factor intensity. But when output prices are biased by factor intensity, factor price effects can arise.

Indirect Effects: Macroeconomic Distortion

The effects on factor prices of macroeconomic distortions will be transmitted indirectly, through commodity market prices, or directly, through a change in the cost of capital resources. The distorting effects of macroeconomic policy can be represented by the government budget deficit identity

(G-T) = (S-I) + (F_in - F_out),

where G - T is the deficit, S - I is the difference between domestic savings and investment, and F_in - F_out is net foreign-exchange inflows. If a deficit is financed through borrowing from the domestic capital market, domestic interest rates become unduly high; producers in all industries reduce capital use to increase the marginal physical product of capital and justify higher capital costs. If a deficit is financed through foreign borrowing, an overvalued exchange rate results. In the long run, the misvaluation is transmitted uniformly to the prices of all factors. In terms of equation 4, the exchange-rate effect is equivalent to a uniform (and in this case, positive) tariff on all commodities. Marginal physical products remain unchanged, because all tradable industries increase demand simultaneously.

Indirect Effects: Input Substitution

Input substitution incentives arise if the elimination of divergences causes relative factor prices to change. The producer then has incentives to try to lower costs (and alter marginal physical products) by altering the combinations of inputs used in production. Figure 7.4 illustrates the range of producer responses to changes in factor prices. In Figure 7.4a, input choices are fixed. Only one combination of inputs can be used in production, L units of labor and K units of capital. Prices of labor and capital are initially w and r, respectively, yielding isocost line AB. If the price of capital increases, the new isocost line will be A'B'. But the tangency to the production iso-quant will remain unchanged. The input combination that minimizes production costs is still L units of labor and K units of capital.

In Figure 7.4b, the options for input choice are increased. An infinite number of combinations can be used to produce one unit of output. Initial choices are represented in the diagram by L units of labor and K units of capital. A change in the factor price then results in a different least-cost input combination. By increasing the use of labor and reducing the use of capital, the producer is able to reduce the impact of factor price changes on production costs. Total production cost in Figure 7.4b is less than that in Figure 7.4a, because the increase in labor costs, (L' - L)w, is more than offset by the reduction in capital cost, (K - K')r'. As shown in Figure 7.4b, iso-cost line A"B" lies inside isocost line A'B'.

Analysis of input substitution becomes more complicated when more than two inputs are present; changes in the use of these other inputs will cause shifts in the capital-labor isoquant. Figure 7.5 illustrates this result. The input-output productivity curve for labor is initially represented by OA. This curve is drawn under conditions of fixed levels of all other inputs. If alteration of the other input prices leads to changes in their use, the productivity curve for capital and labor can be affected. In the figure, the productivity curve for labor shifts upward, from OA to OB. Such a shift would occur, for example, in the event of a reduction in the price of fertilizer. The amount of labor required to produce one unit of output is initially L_A, but this magnitude declines to L_B as fertilizer use increases. If fertilizer use exerts a similar effect on capital productivity, the unit production isoquant for capital and labor will shift inward, toward the origin. Given constant values for w and r, the new combinations of labor and capital become L’_A and K’_A.

A different complication arises if the productivity curve for capital shifts downward in response to increased fertilizer use, reflecting some strong complementarity between fertilizer and capital usage, so that the same quantity of output requires more, rather than less, capital. In this event,

the new unit isoquant may be _QB' instead of _QB. Input substitution responses to a change in fertilizer use increase capital requirements and decrease labor requirements relative to the initial values of K_A and L_A. Although such complementarity effects exist, they do not appear so widespread as to dominate an economy's response to changing input prices. In most economies, input substitution relations are expected to reflect positive cross-productivity effects.

Input substitution creates changes in marginal physical productivities; when aggregated across all industries, the factor demand curve will shift, causing effects on factor prices. Without information about input substitution possibilities, the effects on factor prices will usually have to be ignored in empirical work. But encouragement for empiricists comes from the envelope theorem, which shows that the first-order changes in production costs are accounted for by changes in input prices. A change in cost can be represented as

At the cost-minimizing level of output, the producer has chosen input combinations so that w(Al) + r(Ok) = 0. The effect on costs of an increase in labor use must be equal to the reduction in costs associated with a simultaneous reduction in capital use. If conditions were otherwise, producers could lower total costs by increasing labor use and decreasing capital use (or vice versa). At the margin, therefore, producers respond completely to factor price changes with "perfect" input substitution. The relationship between factor price changes and cost changes is the same, whether input quantities are fixed or variable. The intuitive appeal of the result is increased in the many-input case, because this case allows more ways for producers to substitute inputs and offset the cost effects of factor price increases.

In other circumstances, input substitution effects should be ignored, regardless of their magnitude. The methodology just described allows a complete assessment of the incentive effects of policy; private prices are compared with estimates of social prices that would exist if divergences were eliminated. Systems are evaluated in relation to potential (maximum) national income. But in some situations, such as foreign-exchange contributions, evaluations will be concerned with the actual contribution of the commodity system to national income. In the distorted economy, the opportunity costs of inputs to the system are determined by their social values in existing (distorted) production technologies. Thus second-best social factor prices would be calculated from world prices for commodities and the existing marginal physical productivities, allowing no role for input substitution effects.