Social Prices in a General Equilibrium Model

The two-good, two-factor model of international trade provides the simplest framework in which to establish a basis for social price determination. Figure 6.1 illustrates an economy capable of producing two goods, Q₁ and Q₂, with fixed supplies of two domestic-factor inputs, labor and capital. Both inputs are necessary in the production of either good, and both may be shifted from one industry to the other. The production of both goods takes place with constant returns to scale; for example, doubling the amount of each input in an industry results in doubling the amount of output. These two assumptions ensure that the production possibilities curve is at least as far from the origin as the straight line EZF. Putting all labor and capital in industry Q₁ results in an output of E, whereas placing all inputs in industry Q₂ results in an output of F. Using half of each

input in each industry results in the output combination at point Z, which represents half the maximum outputs of Q₁ and Q₂

World Prices and Maximum Consumption

The economy is likely to be capable of better performance than that indicated by line EZF. If inputs have different productivities in the two industries, total output can be increased by allowing different input allocations between the commodities. When inputs are reallocated between the two industries, the output of Q₁ can be maintained and the output of Q₂ can be increased, resulting in an output combination represented by Y. The assumption of diminishing marginal returns to each input means that this set of maximum production points will have a shape that is concave with respect to the origin. When the economy is completely specialized in production of the first commodity (point E), labor and capital resources can be withdrawn with relatively little impact on the output of good 1 and a relatively large impact on the output of good 2. But with diminishing marginal productivities, the incremental tradeoffs become progressively less attractive. Successively larger amounts of good 1 must be sacrificed to attain a one-unit increase in the output of good 2. The maximum production possibilities frontier is represented by the curve EBYF.

Movements along the production possibilities curve express the opportunity cost of one good's production in terms of the other good. The curve can be interpreted also as the consumption possibilities for an economy that is entirely self-sufficient. But the introduction of international trading opportunities expands the consumption possibilities set beyond EBYF. If the country is too small to influence commodity prices, the trading opportunities of the economy can be represented by straight lines that intersect the relevant production point. One such line is ABGC. All points along this line represent combinations of goods 1 and 2 that have an equal value in international markets. Therefore, ( Q₁ x P₁) = - ( Q₂ x P₂). Rearranging the terms gives ( Q₁/Q₂This result shows that the slope of the trading opportunities line_, Q₁/Q₂can be expressed also as the negative ratio of world prices, -P₂/P₁. The choice of which good to import and which to export then depends on domestic consumer preferences. Consumption at point G implies exports of Q₁equal to HI and imports of Q₂ equal to JK. The choice of a point along segment AB would imply imports of Q₁and exports of Q₂.

The line ABGC is the maximum consumption possibilities frontier for the economy. No other trade opportunity line (of slope -P₂/P₁) would include a point on the production possibilities curve and still allow such large amounts of the two commodities to be consumed. Because production income equals expenditure on consumption, this maximum can be measured by evaluation of either consumption choices at world prices or production choices at world prices. The evaluation of production involves a unique point (B). But the economy can choose to consume at any point on ABGC because these points are all of equal value.

Factor Prices

World prices are the social prices for tradable commodities because their use allows the economy to reach the maximum consumption possibilities frontier. The remaining social prices needed for the simple model are the rental rate (interest plus depreciation) for capital, r, and the wage rate for labor, w. Under the assumption that factor supplies are fixed, these input prices are determined by the prices for outputs and production technology. The assumption of constant returns to scale means that so long as both outputs are produced, knowledge of the particular amounts of Q1 and Q₂ is not necessary for the estimation of social input prices. Only world prices and technology matter.

If competition for the services of domestic factors eliminates excess prcfits, total costs and total revenues can be expressed as an equality, as in equation 1:

Equation 1

where _L and _K represent quantities of labor and capital. Division by Q, and _Q2 yields equation 2:

Equation 2

The ratios _L/Q and _K/Q represent the use of the inputs per unit of output. Equation 3 reformulates equation 2, using l; and k_i as input-output coefficients:

Equation 3

Equation 3 is termed the zero-profit condition. It is meant to represent a state that the economy would occupy if prices and technologies did not change over time. In reality, industries rarely demonstrate zero profits. Instead, output prices and technologies change frequently, with the result that total costs are usually greater than or less than total revenues. The use of the zero-profit condition as a basis for social price determination thus measures incentives under long-run equilibrium, assuming a continuation of the conditions prevailing during a particular time period.

Rearrangement of the zero-profit condition (equation 3) shows how the prices of inputs are dependent on the output prices and the input-output coefficients:

Equation 4

These factor costs represent the social opportunity costs of factors used by a new commodity system, Q₃. If production of Q₃ cannot profitably compensate domestic factors at prices w and r, national income cannot be increased from the introduction of Q₃. The economy would be better off, in the sense of maximum consumption possibilities, by remaining with the production of goods Q, and Q₂. The calculation of PAM usually treats the system under study as a new commodity system, so that social opportunity costs of factors are determined by the other commodities in the economy. Social profit then represents the net contribution of the commodity system to national income.

Generalization of the Simple Model

The input-output equations of the simple model can be expanded to provide a model of any desired degree of detail. Equation 5 illustrates this general model of the economy:

Equation 5

where

_{wi =} Price _of input i
_{pi =} Price _of output _j
Q_j = Quantity _of output _j
Zij = Quantityof input i used in production of output j

Dividing through the ith equation by Qj yields the analog of equation 3

Equation 6

Some of the inputs used in the production of goods will also be tradable goods; fertilizers, seeds, and agricultural chemicals are examples of commodities available on world markets. If Z₁ through Z₄ are assumed to be domestic factor inputs and Z₅ through Z_m are assumed to be tradable inputs, the equations can be rewritten so that domestic factor inputs are segregated from tradable outputs and inputs, as shown in equation 7:

Equation 7

where w₅, w_{6, . . .,} w_m are replaced by world prices P₅, P₆, . . ., Pm because these inputs are also tradable outputs.

The right-hand side of the equation now represents value added rather than output price. In matrix form, equation 7 can be written as follows:

Equation 8

Equation 8 is another form of the zero-profit condition. The prices of domestic factors, times the relevant input-output coefficients, exactly equals each VA_j, the value-added in production. Given world prices and input-output coefficients, the domestic factor prices can be determined in a manner analogous to that used in the simple model. But in order for the system of equations to generate a solution, the matrix of input-output coefficients must be square. The number of commodities must equal the number of factors. This requirement might appear to compromise the generality of the approach, as most economies are unlikely to attain such an equality. In equation 8, for example, only four commodities are needed to determine the factor prices.

Counting numbers of inputs and outputs is a difficult and ultimately arbitrary exercise. The number of domestic factor inputs can be made almost infinitely large if one recognizes different types of labor, capital, and land. The number of commodities can also be expanded if sector output is divided into commodity groups, specific commodities, or brand names and qualities of a particular commodity. But social factor prices can still be determined, as long as the number of goods produced in the economy exceeds the number of domestic factors engaged in their production. The presence of more goods than factors thus implies redundant information for the determination of social factor prices. Calculation of the social factor prices involves a search for the core set of production activities that will result in the maximum income in the economy. Any activity that pays domestic factors less than their values attained under the maximum income for the economy will be eliminated by competition for the use of factor inputs. The circumstance of more commodities than factors appears reasonable for the majority of empirical circumstances.