25.3 How Firms Collectively Maximize GDP

In the last section, we showed that the amount of two goods produced by firms in general equilibrium depends on three factors: the amount of labor, which determines the location of the PPF; and the prices of the two goods. However, is that point along the PPF the “best” point in any meaningful sense of the word? In this section we’ll show that, in fact, firms choose the point along the PPF which maximizes the economy’s GDP: that is, the market value of all final goods and services. In other words, even though they are not coordinating (in fact, they’re competing!), and even though no firm sees the PPF, the result of the competitive forces is to collectively maximize GDP subject to the PPF.

The Social Planner’s Solution

Let’s solve that problem as if we were a social planner. That is, given output prices $p_1$ and $p_2$, how can we maximize the total value of goods produced, or GDP, given the resources available to us?

Let’s make the simplifying assumption that there is just one competitive firm producing good 1, and one competitive firm producing good 2. In fact, the arguments go through with lots and lots of firms in each industry, the math is just easier this way. To help us keep track of the fact that we’re talking about what an industry produces, though, let’s introduce a bit of new notation: \(\begin{aligned} Y_1 &= \text{total amount of good 1 produced by all firms in industry 1}\\ Y_2 &= \text{total amount of good 2 produced by all firms in industry 2} \end{aligned}\)

In this case, the market value of producing some combination of goods $(Y_1,Y_2)$ is the value of those goods: \(GDP(x_1,x_2) = p_1Y_1 + p_2Y_2\) Note that this is just like the expression we used for the cost of a bundle $(x_1,x_2)$ when analyzing a budget line; only in this case it’s the value of the bundle, since we’re thinking of it from the firms’ perspective.

As usual, the equation of the PPF is derived from the fact that the total amount of labor used to produce good 1 by firms in industry 1, plus the total amount of labor used to produce good 2 by firms in industry 2, has to add up to the total amount of labor available in the economy: \(L_1(Y_1) + L_2(Y_2) = \overline L\) The social planner’s problem, therefore, is \(\begin{aligned} \max_{Y_1,Y_2}\ & p_1Y_1 + p_2Y_2\\ \text{s.t. }\ & L_1(Y_1) + L_2(Y_2) = \overline L \end{aligned}\) The Lagrangian for this maximization problem is \(\mathcal{L}(Y_1,Y_2,\lambda) = p_1Y_1 + p_2Y_2 + \lambda(\overline L - L_1(Y_1) - L_2(Y_2))\) which gives us our first-order conditions with respect to $Y_1$ and $Y_2$ as \(\begin{aligned} {\partial \mathcal{L} \over \partial Y_1} = p_1 - \lambda {dL_1 \over dY_1} = 0 \Rightarrow \lambda = p_1 \times {dY_1 \over dL_1} = p_1 \times MP_{L1}\\ {\partial \mathcal{L} \over \partial Y_2} = p_2 - \lambda {dL_2 \over dY_2} = 0 \Rightarrow \lambda = p_2 \times {dY_2 \over dL_2} = p_2 \times MP_{L2} \end{aligned}\) Equating these values of $\lambda$ we get \(p_1 \times MP_{L1} = p_2 \times MP_{L2}\) Cross-multiplying gives us the tangency condition \({MP_{L2} \over MP_{L1}} = {p_1 \over p_2}\) Notice that the left-hand side of this is just our expression for the MRT, or the slope of the PPF! And, since the third first-order condition just gives us back the equation of the PPF, this means that the social planner would choose 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:

[ See interactive graph online at https://www.econgraphs.org/graphs/scarcity/general_equilibrium/gdp_max ]

The social planner gets to the highest possible iso-GDP line along the PPF by finding the point along the PPF where the PPF is just tangent the iso-GDP line: in other words, where \(MRT = {p_1 \over p_2}\) So how, do competitive firms do this?

Competitive Equilibrium

Remember from Lecture 21 that if we write each firm’s profit maximization problem as \(\pi(L) = pf(L) - (wL + rK)\) when we take the derivative with respect to $L$ and set it equal to zero, we get \(p \times MP_L = w\) In other words, the wage rate must be equal to the marginal revenue product of labor for every firm. But this is just the same as the FOC above, just with $w$ instead of $\lambda$; so if all firms are setting $p \times MP_L$ equal to the same wage, then for any two firms 1 and 2, it must be the case that \(p_1 \times MP_{L1} = p_2 \times MP_{L2}\) In other words, the wage rate in a competitive market is the same as the Lagrange multiplier in the social planner’s problem. (And, in fact, it’s the dollar value of an additional unit of labor, which makes sense!)

Another way to think about this is to remember that we can write marginal cost (when labor is the only input) is \(MC = w \times {1 \over MP_L}\) Therefore the ratio of marginal costs in two industries will be \({MC_1 \over MC_2} = \frac{w \times {1 \over MP_{L1}}}{w \times {1 \over MP_{L2}}} = {MP_{L2} \over MP_{L1}}=MRT\) In other words, another way of writing MRT is as the ratio of marginal costs, $MC_1/MC_2$! Since each firm will set its marginal cost equal to the price in its industry, we have \(\left. \begin{matrix}\text{Firms in market 1 set } & p_1 = MC_1\\ \text{Firms in market 2 set } & p_2 = MC_2\end{matrix} \right\} \Rightarrow \frac{p_1}{p_2} = \frac{MC_1}{MC_2} = MRT\) 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 do firms “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 $LD_1(w\ |\ p_1)$ and firms in market 2 have a labor demand given by $LD_2(w\ |\ p_2)$, then the market wage will adjust until \(LD_1(w\ |\ p_1) + LD_2(w\ |\ p_2) = \overline L\) This is the resource constraint we used to derive our PPF; so when the labor market is in equilibrium, all the labor is being used and in its most productive capacity, so we must be operating along the PPF.

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.