# 16.6 Profit Maximization and the PPF

Now that we’ve derived an equation for the PPF, we need to figure out which point along it we want to produce.

In the next chapter we’re going to think about preferences and demand, but for now let’s consider a simpler question: what’s the most valuable combination of outputs? That is, given output prices $p_1$ and $p_2$, how can we maximize the *total* value of goods produced, or GDP:
\(GDP(Y_1,Y_2) = p_1Y_1 + p_2Y_2\)
If maximizing GDP is our objective, our constraint is that we’re along the PPF: that is we could write the problem as
\(\begin{aligned}
\max_{Y_1,Y_2}\ & p_1Y_1 + p_2Y_2\\
\text{s.t. }& c_1(Y_1) + c_2(Y_2) = \overline C
\end{aligned}\)
The Lagrangian for this maximization problem is
\(\mathcal{L}(Y_1,Y_2,\lambda) = p_1Y_1 + p_2Y_2 + \lambda(\overline C - c_1(Y_1) - c_2(Y_2))\)
which gives us our first-order conditions
\(p_1 - \lambda MC_1 = 0\)
\(p_2 - \lambda MC_2 = 0\)
\(c_1(Y_1) + c_2(Y_2) = \overline C\)
Setting the $\lambda$’s equal to one another gives us the tangency condition
\(\frac{p_1}{p_2} = \frac{MC_1}{MC_2}\)
or, since we know that $MRT = MC_1/MC_2$,
\(MRT = \frac{p_1}{p_2}\)
In other words, society maximizes its GDP when it chooses the point along the PPF where the MRT is equal to the price ratio.

Visually, we can see the “objective function” of the social planner just like a “perfect substitutes” utility function with the “utility weights” being the prices. We might call the level sets of this objective function “iso-GDP lines” and plot them in a diagram with the PPF; in this case, the GDP-maximizing combination of outputs is the highest achievable iso-GDP line subject to the PPF:

So that’s what the “social planner” would do. What happens if we have individual firms making their own decisions? That is, what if goods 1 and 2 are produced by different competitive firms? Well, from Chapter 15, we know that each firm maximizes its own profit by producing up until the point where its marginal costs equals the market price of its good: so we have \(\left. \begin{matrix}\text{Firms in market 1 set } & p_1 = MC_1\\ \text{Firms in market 2 set } & p_2 = MC_2\end{matrix} \right\} \Rightarrow \frac{p_1}{p_2} = \frac{MC_1}{MC_2}\)

So, firms maximizing *their own profits* will result in a point at which the MRT of the *economy as a whole* equals the price ratio, thereby maximizing the economy’s GDP.

But how will they “see” the PPF? This comes from the fact that the labor market has to clear. In particular, if firms in market 1 have a labor demand given by $LD_1(p_1,w)$ and firms in market 2 have a labor demand given by $LD_2(p_2,w)$, then the market wage will adjust until \(LD_1(p_1,w) + LD_2(p_2,w) = \overline L\) This is the resource constraint we used to derive our PPF! And it’s the condition in the guns and butter example which causes an increase in labor demand from one industry to result in higher wages, and therefore lower employment in equilibrium in the other industry.

Intuitively, the factor that limits how much the economy can produce is how much labor it has available to it. The wage rate “signals” to firms how scarce labor is, and firms competing for workers will bid up wages until the total amount of labor firms want to hire is equal to the total amount of labor available in the economy.

The homework problem goes through a concrete example of how this process works.