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Chapter 14 / Partial Equilibrium

# 14.6 Equilibrium with Different Consumers and Firms

In the previous example, we analyzed a situation in which there are a lot of identical consumers and firms. Such a model is called a “representative agent” model, and economists often use this model for situations in which it is reasonable to talk about “average preferences” or “average production technologies.” However, the world is a gloriously varied place, so we’d like to also be able to analyze situations in which consumers and firms aren’t all the same.

To do this, let’s work through an example in which we have two consumers with different preferences, and two firms with different levels of capital. In particular, we’ll consider a market for sandwiches in which there are two consumers (Adam and Eve) and two firms (Subway and Togo). Even though there are a small number of consumers and firms, we’re going to model this “as if” all agents are price takers.

## Demand

Our two consumers, Adam and Eve, have preferences over sandwiches (good 1) and money spent on other goods (good 2) \begin{aligned} u^A(x_1,x_2) &= 8 \ln x_1 + x_2\\ u^E(x_1,x_2) &= 4 \ln x_1 + x_2 \end{aligned} Let’s assume that sandwiches are a relatively small part of each of their budget, so we can ignore corner solutions in which Adam or Eve spends their entire budget on sandwiches. Furthermore, since good 2 “money spent on other things,” the price of good 2 is just 1; so let’s write the price of a sandwich as $p$. Finally, let’s write $A$ for the quantity of sandwiches Adam consumes, and $E$ for the quantity of sandwiches Eve consumes. Therefore, Adam’s $MRS = 8/A$ and Eve’s $MRS = 4/E$.

Following our procedure from Part II, Adam and Eve will each set their MRS equal to the price ratio (in this case, $p$): $$\text{Adam's optimization (tangency) condition: }{8 \over A} = p$$ $$\text{Eve's optimization (tangency) condition: }{4 \over E} = p$$ Solving for the quantity demanded in each case, Adam’s demand for sandwiches will be $$d^A(p) = {8 \over p}$$ and Eve’s will be $$d^E(p) = {4 \over p}$$ The overall demand in this “market” is therefore \begin{aligned} D(p) &= d^A(p) + d^E(p)\\ &= {8 \over p} + {4 \over p}\\ &= {12 \over p} \end{aligned} Visually, we can understand this as the horizontal summation of their demand curves. In the following diagram, drag the price up and down to see how many sandwiches each of them demands at that price, and therefore what the total quantity demanded in the market is:

## Supply

Suppose our two firms, Subway and Togo’s, each have the sandwich production function $$f(L,K) = \sqrt{LK}$$ Furthermore, in the short run, suppose Subway has $\overline K^S = 4$ units of capital, Togo’s has $\overline K^T = 2$ unit of capital, the price of labor is $w = 4$, and the price of capital is $r = 2$.

Suppose $S$ is the quantity of sandwiches produced by Subway, and $T$ is the quantity of sandwiches produced by Togo’s. Using our methods from Unit III, we can derive the cost functions for Subway and Togo’s as $$c^S(S) = S^2 + 8 \Rightarrow MC^S(S) = 2S$$ $$c^T(T) = 2T^2 + 4 \Rightarrow MC^T(T) = 4T$$ Since each firm will set $p = MC$, their supply functions will be $$s^S(p) = {p \over 2}$$ $$s^T(p) = {p \over 4}$$ The overall supply in this “market” is therefore \begin{aligned} S(p) &= s^S(p) + s^T(p)\\ &= {p \over 2} + {p \over 4}\\ &= {3p \over 4} \end{aligned} Again, visually this is the horizontal summation of the two supply curves:

## Equilibrium price, quantity, and allocation

Setting market demand equal to market supply gives us the market equilibrium price: \begin{aligned} S(p) &= D(p)\\ {3p \over 4} &= {12 \over p}\\ p^2 &= 16 \\ p^\star &= 4\end{aligned} At a price of $p = 4$:

• Adam demands $8/4 = 2$ sandwiches
• Eve demands $4/4 = 1$ sandwich
• Subway produces $4/2 = 2$ sandwiches
• Togo’s produces $2/2 = 1$ sandwich

Note that competitive markets have both determined the overall quantity of sandwiches (three) and also who gets sandwiches (Adam gets more than Eve) and who produces sandwiches (Subway makes more than Togo’s). This allocation comes directly from the utility and production functions: Adam likes sandwiches more than Eve does, and Subway has more capital than Togo’s (and correspondingly lower marginal costs).

An important question, though, is is this the “right” overall quantity and allocation of sandwiches? We’ll answer that in the next chapter; before we get there, though, let’s think about what happens to our equilibrium in the presences of different kinds of government policies.

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