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Lecture 10: Shifts in Demand; Indirect Utility

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As we saw on Wednesday, demand functions describe a consumer’s optimal bundle as a function of prices and income. For example, we can write the demand for good 1 as \(x_1^*(p_1,p_2,m)\) These demand functions represent three relationships:

When we investigate any of these relationships, the natural thing to do is to hold the other variables constant. Therefore, when we plot the demand curve for good 1, we plot $x_1$ as a function of $p_1$, holding $p_2$ and $m$ constant (the ceteris paribus assumption); a change in $p_1$ is therefore reflected in the diagram as a movement along the demand curve for good 1. If $p_2$ or $m$ affect the relationship between $x_1$ and $p_1$ – that is, if the prices of other goods or a consumer’s income affect how that consumer responds to the price of good 1 – then changes in $p_2$ or $m$ will cause a shift of the demand curve for good 1.

The key economic concepts we’ll be analyzing will have to do with the direction of the responses to changes, and should be familiar from Econ 1. Specifically:

In the first part of this lecture, we’ll examine how different utility functions may be used to model these kinds of behaviors. In the second part of this lecture, we’ll be talking about how to measure the degree to which demand changes, using the concept of elasticity.

Complements, Substitutes, and Independent Goods

The way the demand curve shifts in response to the price of another good depends on the relationship between those two goods:

Let’s look at each of these in turn, by examining the effect of a price increase in one flavor of jelly on the quantity demanded of peanut butter, strawberry jam, and t-shirts.

Complements

If two goods are complements, an increase in the price of either good will result in a decrease in the quantity bought of both goods. For example, if you enjoy sandwiches made with peanut butter (good 1) and grape jelly (good 2), an increase in the price of jelly increases the price of making a PB&J sandwich; so you might have fewer such sandwiches and do something else for lunch.

Substitutes

If two goods are substitutes, an increase in the price of one good will result in a decrease in the quantity bought of that good, and an increase in the quantity of the other. For example, if you view strawberry jam (good 1) and grape jelly (good 2) as substitutes, then an increase in the price of grape jelly will cause you to use more of the relatively cheaper strawberry jam in recipes which could use either.

Independent Goods

Finally, let’s think about goods like t-shirts (good 1) and grape jelly (good 2), which have no obvious connection. For such goods, we would not expect a change in grape jelly to affect the quantity of t-shirts bought at all:

One utility function we’ve seen that exhibits this behavior is Cobb-Douglas, in which the consumer will spend a given fraction of their income on each good. For example, with the Cobb-Douglas utility function $u(x_1,x_2) = x_1x_2$, the consumer’s demand for good 1 is $x_1^\star(p_1,p_2,m) = m/2p_1$, which doesn’t depend at all on $p_2$.

The CES Function

One useful family of utility functions for analyzing complements and substitutes is a CES utility function, of the form \(u(x_1,x_2) = (ax_1^r + bx_2^r)^{1 \over r}\) In this case the $r$ parameter measures how complementary or substitutable goods are.

Negative values of $r$ correspond to complements. As $r$ approaches $-\infty$, the preferences approach the “perfect complements” or Leontief functional form we’ve become familiar with. However, for any negative value of $r$, the goods are complementary; we sometimes call these “weak complements” to distinguish them from “perfect complements,” but just “complements” works too.

Likewise, positive values of $r$ correspond to substitutes. When $r = 1$, the function is just linear; so the goods are “perfect substitutes.” However, a value of $r$ between 0 and 1 represents goods which are substitutes but not perfect substitutes — again, we sometimes call these “weak substitutes” or just “substitutes.”

Finally, when $r = 0$, the goods are neither substitutes nor complements, but independent goods.

The graph below allows you to play with changes in $p_1$ and $p_2$ for different values of $r$, for the simple CES utility function \(u(x_1,x_2) = (x_1^r + x_2^r)^{1 \over r}\) The initial value of $p_2$ is 4; try changing that to a smaller or larger value to see how the demand curve shifts, and the optimal bundle shifts:

See interactive graph online here.

Normal and Inferior Goods

If an increase in income causes a consumer to buy more of a good, we say that good is a “normal good.” Conversely, if an increase in income causes a consumer to buy less of a good, we say that good is an “inferior good.”

How do we see this represented using budget line and indifference curves? Well, if both goods are normal, then an increase in income will cause them to consume more of both goods; so when the consumer gets more income and their budget line shifts out, their optimal bundle should move up and to the right. However, if one of the goods is inferior, then an increase in income will cause them to consume more of the normal good, but less of the inferior good; so the optimal bundle may move up and to the left, or down and to the right, after an increase in income.

Every functional form we’ll see will have both goods be normal good. However, you should be able to illustrate what would happen in the case of an inferior good with a (non-precisely mathematical) sketch. I’ll provide one of those during class; I haven’t yet figured out how to get the computer to draw one. :)

Indirect Utility Functions

Let’s wrap up our theory of demand by looking at a kind of function that’s going to be really important next week: indirect utility functions.

The indirect utility function is a function of prices and income that describes the utility from the utility-maximizing bundle given those prices and income. That is, if a consumer has ordinary (“Marshallian”) demand functions \(x_1^\star(p_1,p_2,m)\) \(x_2^\star(p_1,p_2,m)\) then the indirect utility function $V(p_1,p_2,m)$ may be found by plugging those functions back into the utility function: \(V(p_1,p_2,m) = u(x_1^\star(p_1,p_2,m),x_2^\star(p_1,p_2,m))\) In other words, this asks: if a consumer with income $m$ faces prices $p_1$ and $p_2$, what is the maximum amount of utility they could achieve?

Indirect utility functions are often used when analyzing situations in which a consumer is choosing their income in a given situation: for example, choosing how much of their current income to save, as we’ll see in the next lecture. In that case, the choice variable is the amount of money to spend in each period; so the utility they get will depend on how happy having an additional dollar to spend in each period would make them.

The Lagrange multiplier and the indirect utility function

We’ve often referred to the Lagrange multiplier in a consumer optimization problem as measuring “bang for the buck” – that is, how much additional utility you would get from the last dollar spent on any good.

Another way of thinking about it is the increase in utility you’d get if you had another dollar to spend. Well, now we have an indirect utility function which tells you exactly how much utility you have, as a function of the amount of money you have, for given prices. So indeed, as we would expect, the partial derivative of the indirect utility function with respect to $m$ gives us our $\lambda$! Let’s see how this works with a simple Cobb-Douglas function

Example: Cobb-Douglas

Suppose someone has Cobb-Douglas preferences of the form \(u(x_1,x_2) = x_1x_2^2\) Let’s suppose that this consumer faces prices $p_1 = 3$ and $p_2 = 2$. The first-order conditions for the Lagrangian of this problem will solve to \(begin{aligned} \lambda &= {MU_1 \over p_1} = {x_2^2 \over 3}\\ \lambda &= {MU_2 \over p_2} = {2x_1x_2 \over 2} = x_1x_2 \end{aligned}\) Setting these two values of $\lambda$ equal to one another, we get $x_2 = 3x_1$; plugging this into the budget constraint gives us The optimal bundle for this utility function will be (using the Cobb-Douglas trick) \(x_1^\star(p_1,p_2,m) = \frac{m}{9}\) \(x_2^\star(p_1,p_2,m) = \frac{m}{3}\) To find the indirect utility function, we plug these optimized values back into the utility function: \(\begin{aligned} V(p_1,p_2,m) &= u(x_1^\star(p_1,p_2,m),x_2^\star(p_1,p_2,m))\\ &= x_1^\star(p_1,p_2,m) \times (x_2^\star(p_1,p_2,m))^2\\ &= \frac{m}{9} \times \left(\frac{m}{3}\right)^2\\ &= {m^3 \over 81} \end{aligned}\) So, what’s the value of an additional dollar, when you’re optimizing? It’s \({\partial V(p_1,p_2,m) \over \partial m} = {m^2 \over 27}\) Now, note that if we plug the optimized values of $x_1$ and $x_2$ back into our equation for $\lambda$, we get: \(begin{aligned} \lambda &= {x_2^2 \over 3} = {(m/3)^2 \over 3} = {m^2 \over 27}\\ \lambda &= {2x_1x_2 \over 2} = x_1x_2 = {m \over 9} \times {m \over 3} = {m^2 \over 27} \end{aligned}\) So indeed, when the consumer is optimizing, $\lambda$ is the derivative of the indirect utility function with respect to $m$!

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