16.4 Supply and the Resource Allocation Problem
In the last section we saw that an increase in demand for one good would result in an increase in the quantity of that good, and a decrease in the quantity of the other, as resource prices were bid up and labor was reallocated toward the good with higher demand.
Let’s think now about what this means for the PPF that we introduced in Chapter 3 to ask a broader question: how do the relative prices in an economy result in the overall allocation of resources across goods?
To answer this question, we’ll introduce some new notation: let $Y_1$ and $Y_2$ be the total amount of goods 1 and 2 produced by all firms in the economy. Then, for a given level of productive resources (e.g., a total amount of labor in the economy), let’s define two new functions: \(Y_1^\star(p_1,p_2)\) \(Y_2^\star(p_1,p_2)\) as the equilibrium quantities produced by all firms producing goods 1 and 2, as a function of the prices of all goods. Note that this is distinct from the individual quantity of one good produced by one firm as a function of its own price.
Essentially, we’re studying a similar resource allocation problem to Chuck’s problem of how to allocate labor across goods 1 and 2; but instead of satisfying his utility, the economy is going to be trying to maximize the monetary value of goods we produce. Chuck’s solution was characterized by two equations:
- Tangency condition: $MRS = MRT$
- Constraint condition: $L_1 + L_2 = \overline L$
For the rest of this chapter, we’ll show that equilibrium across goods and labor markets will be characterized by two conditions:
- Profit maximization in market 1: $p_1 = MC_1$
- Profit maximization in market 2: $p_2 = MC_2$
- Labor market clearing: $LD_1(p_1,p_2) + LD_2(p_1,p_2) = \overline L$
We’ll show that $MC_1/MC_2$ is, in fact, another expression for the MRT, so that the first two conditions here actually imply the tangency condition $p_1/p_2 = MRT$; and that the labor market clearing condition shows how wages communicate the “constraint” of the PPF to millions of firms.
To start out, let’s show why firm profit maximization implies $p_1/p_2 = MRT$.