17.3 Step 1: Firm Optimization

As discussed in Chapter 19, profit-maximizing firms will choose the point along the PPF that sets $MRT$ equal to the price ratio.

Let’s model this economy “as if” it were comprised of two price-taking firms, one of which produces good 1, and the other of which produces good 2, each with the production function \(Y = f(L) = 10\sqrt{L}\) Furthermore, assume that there are $\overline L = 100$ units of labor available in the economy. The equation of the PPF is therefore \({Y_1^2 \over 100} + {Y_2^2 \over 100} = 100\) or \(Y_1^2 + Y_2^2 = 100^2\) The MRT of this PPF is \(MRT(Y_1,Y_2) = {Y_1 \over Y_2}\) Setting the MRT equal to the price ratio $p_1/p_2$ gives us the tangency condition \({Y_1 \over Y_2} = {p_1 \over p_2}\) or \(Y_2 = {p_2 \over p_1}Y_1\) Substituting this into the PPF constraint yields \(\begin{aligned} Y_1^2 + \left[{p_2 \over p_1}Y_1\right]^2 &= 100^2\\ (p_1^2 + p_2^2)Y_1^2 &= p_1^2 \times 100^2\\ Y_1^\star(p_1,p_2) &= {100p_1 \over \sqrt{p_1^2 + p_2^2}} \end{aligned}\) and therefore, \(Y_2^\star(p_1,p_2) = {p_2 \over p_1}Y_1^\star(p_1,p_2) = {100p_2 \over \sqrt{p_1^2 + p_2^2}}\) The resulting GDP, $M$, from this optimal bundle is \(\begin{aligned} M(p_1,p_2) &= p_1Y_1^\star(p_1,p_2) + p_2p_1Y_1^\star(p_1,p_2)\\ &= {100p_1^2 \over \sqrt{p_1^2 + p_2^2}} + {100p_2^2 \over \sqrt{p_1^2 + p_2^2}}\\ &= 100\sqrt{p_1^2 + p_2^2} \end{aligned}\)

Notice, by the way, that if you double both prices (or cut both in half), the optimal bundle and “GDP budget line” remain unchanged. In other words, the outcome of this model depends only on the price ratio, not on individual prices. Indeed, we can rewrite the expressions for $Y_1$ and $Y_2$ as \(\begin{aligned} Y_1^\star(p_1,p_2) &= {100 \over \sqrt{1 + \left({p_2 \over p_1}\right)^2}}\\ Y_2^\star(p_1,p_2) &= {100 \over \sqrt{1 + \left({p_1 \over p_2}\right)^2}} \end{aligned}\)