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Chapter 14 / Partial Equilibrium

# 14.3 Market Demand

In Part II, we analyzed the problem of a utility-maximizing consumer who could buy goods at constant, given prices. Specifically, we assumed that if they had devoted a budget of $m$ dollars to buying two goods (goods 1 and 2), and faced prices $p_1$ and $p_2$ for those goods, that their utility-maximizing behavior could be described by the demand functions $x_1^\star(p_1,p_2,m)$ and $x_2^\star(p_1,p_2,m)$. This told us their optimal bundle as a function of all prices and incomes.

Let’s zero in on the demand for a particular good; we might write this individual demand as $d(p)$, where $p$ is the market price of the good, and $d$ is the quantity demanded by the individual. To analyzing market demand for this good, we need to aggregate the quantity demanded by all individuals, to determine the market demand curve $D(p)$. To do this, we’re going to sum the demand of different individuals. For example, if we have $N_C$ consumers, we can number them consumer 1, 2, 3, etc., up to $N_C$. Then, if $d^1(p)$ is (Note: Notation alert: we’re using the superscripts here to denote which consumer we’re talking about; these aren’t exponents.) the demand function for individual 1, and $d^2(p)$ is the demand for individual 2, and so forth, we can write the overall market demand $D(p)$ as $$D(p) = d^1(p) + d^2(p) + d^3(p) + \cdots + d^{N_C}(p)$$ or, more succinctly, $$D(p) = \sum_{i=1}^{N_C}d^i(p)$$ which we can read as “the total quantity of a good demanded at price $p$ is the sum, for each $i$ from 1 to $N_C$, of the quantity demanded by each individual $i$ at that price.”

Now, as the above setup implies, economists understand that people are different. However, modeling or predicting the preferences of individuals can be hard or even impossible, let alone getting data on billions of individuals’ preferences. For that reason (among others), economists sometimes model market demand “as if” there are a lot of identical individuals, each of whom has the same “average-ish” preferences leading to the individual demand function $d(p)$. In this case the total market demand is just the number of consumers times the amount demanded (on average) by each consumer: $$D(p) = \sum_{i = 1}^{N_C}d^i(p) = N_Cd(p)$$ For example, let’s suppose that in some market, there are $N_C$ consumers who each have preferences that may be represented by the Cobb-Douglas utility function $$u(x_1,x_2) = \alpha \ln x_1 + (1 - \alpha)\ln x_2$$ We showed in Chapter 8 that this would imply that each individual would spend fraction $\alpha$ of their income on good 1, leading to the individual demand function $$d(p) = {\alpha m \over p}$$ The following diagrams show this situation. The diagram on the left shows the demand curve for each individual; the diagram on the right shows the market demand.

Try changing the preferences parameter $\alpha$, the income $m$, and the number of consumers to see how the two graphs are affected. Which factors affect both the amount demanded by an individual, and the market? Which only affect market demand, and not the individual demand?

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