17.2 Conditions for General Equilibrium

In Unit I, Chuck’s optimal bundle was the point along the PPF where his MRS was equal to his MRT:

In Chapter 16, we solved for the equilibrium economy-wide quantities as functions of the prices of all goods: that is, $Y_1^\star(p_1,p_2)$ and $Y_2^\star(p_1,p_2)$. We found that economy would choose the point along the PPF where the MRS was equal to the price ratio, $p_1/p_2$, as shown in the left-hand graph below. In this chapter, we’ll give the resulting income to consumers and find their optimal quantity of goods to consume, $X_1^\star(p_1,p_2)$ and $X_2^\star(p_1,p_2)$, as shown in the right-hand graph:

We will then find the prices which result in all markets clearing: that is, where $X_1^\star(p_1,p_2) = Y_1^\star(p_1,p_2)$ and $X_2^\star(p_1,p_2) = Y_2^\star(p_1,p_2)$.

We will proceed in steps:

1. Firm profit maximization. Find output as a function of prices: \(Y_1^\star(p_1,p_2)\) \(Y_2^\star(p_1,p_2)\)From that output, determine GDP: \(M(p_1,p_2) = p_1Y_1^\star(p_1,p_2) + p_2Y_2^\star(p_1,p_2)\)
2. Consumer utility maximization. Find the consumer’s optimal quantities of goods 1 and 2, facing prices $p_1$ and $p_2$ and having $M(p_1,p_2)$ from step 1 as their income: \(X_1^\star(p_1,p_2) = x_1^\star(p_1,p_2,M(p_1,p_2))\) \(X_2^\star(p_1,p_2) = x_2^\star(p_1,p_2,M(p_1,p_2))\)
3. Market clearing. Set demand equal to supply: \(X_1^\star(p_1,p_2) = Y_1^\star(p_1,p_2)\) \(X_2^\star(p_1,p_2) = Y_2^\star(p_1,p_2)\) and solve for the equilibrium prices of goods 1 and 2.

What we’ll find is that when the system is in equilibrium, we will have found the point along the PPF where $MRS = MRT$: that is, the exact same point Chuck would have chosen if he were just maximizing utility subject to his PPF.