4 Consumer Choice

The Policy Question
Hybrid Car Purchase Tax Credit—Is It the Best Choice to Reduce Fuel Consumption and Carbon Emissions?

The US government offered a tax credit toward the purchase of hybrid cars with the goal of reducing the output of carbon emissions produced by cars annually. We are using the tools of microeconomic consumer theory to study this policy and assess its effectiveness in reducing emissions.

We are now very close to being able to predict how consumers will change their driving and gasoline purchases in response to a government tax credit on hybrid cars. As we will see, this is simply a specific example of the general question we first raised in chapter 1 of how to predict consumer behavior when prices or incomes change. For our policy example and in general, we address this question by combining the budget constraint with the concept of preferences and utility maximization. Consumer theory in economics is based on the premise that each person will try to do their best given the money they have and the prices of the goods and services they like. This is what we mean by utility maximization—choosing the affordable bundle of goods and services that returns the highest utility.

Think about a consumer who goes grocery shopping. There are many ways to fill a shopping basket—ending up with many different possible bundles of goods. This chapter is concerned with how each consumer picks the best affordable good. As we will see shortly, consumers think about the income they have and the relative prices of all the possible goods they could buy and then choose among all the possible bundle combinations that their budget can support.

Exploring the Policy Question

1. What is your prediction about how consumers’ driving and gasoline-purchasing behavior will change when their income increases or decreases? When the price of gasoline increases or decreases? What implications will these behavior changes have for the hybrid car tax credit policy?

4.1 The Consumer Choice Problem: Maximizing Utility

Learning Objective 4.1: Define the consumer choice problem.

What is the consumer’s optimal choice among competing bundles? This question summarizes the consumer choice problem. To resolve this problem, we can combine our understanding of the budget constraint and preferences as represented by utility functions. The budget constraint describes all the bundles the consumer could possibly choose. The utility function describes the consumer’s preferences and relative level of satisfaction from the consumption of bundles. We put these two pieces together by answering the question, “Among all the bundles the consumer could possibly choose, which one returns the highest level of utility?”

Conceptually, we are overlaying the indifference curve map on the budget constraint and looking for the point or points of intersection, as in figure 4.1.1. Recall that there can be more than one indifference curve for bundles of goods [latex]A[/latex] and [latex]B[/latex]. The goal of solving the consumer choice problem is to get on the highest indifference curve—the curve that is the farthest to the upper right—while also satisfying the budget constraint. The highest indifference curve—the one that represents the highest level of utility or satisfaction—is the one that just touches the budget line at a single point. It is not possible to get on a higher indifference curve given the budget constraint, and though it is possible to get on a lower one, doing so necessarily means a lower level of utility or satisfaction.

Figure 4.1 summarizes the solution to the consumer choice problem: The consumer should pick the one bundle that returns the highest level of utility while also satisfying the budget constraint. This graph also shows us the two fundamental conditions that represent the solution to the consumer choice problem:

1. The consumer’s optimal choice is on the budget line itself, not inside the budget constraint. This is why we can focus on the line rather than the whole set of affordable bundles.
2. At the optimal choice, the indifference curve just touches the budget line, and so at this one point, they have exactly the same slope.

This second condition has a significant intuitive interpretation. Recall that the slope of the indifference curve is the marginal rate of substitution ([latex]MRS[/latex])—the rate at which the consumer is willing to trade off one good for the other and remain just as well off. Recall also that the slope of the budget line is the economic rate of substitution ([latex]ERS[/latex])—the rate at which the consumer is able to trade one good for the other at current prices. What figure 4.1 shows is that at the optimal choice, these two things must be equal. But why?

Consider a point, like point 1 in figure 4.2, where the indifference curve intersects the budget line. Here, the [latex]MRS[/latex] is greater than the [latex]ERS[/latex] absolute value. The [latex]MRS[/latex] tells us the amount of good [latex]B[/latex] the consumer is willing to give up to get one more unit of good [latex]A[/latex] and remain just as satisfied. In this case, the [latex]ERS[/latex] tells us how much of [latex]B[/latex] a consumer must give up in exchange for one unit of [latex]A[/latex] in the marketplace. Here, the consumer is willing to give up a lot of [latex]B[/latex] to get a little more [latex]A[/latex] and remain just as satisfied, but the [latex]ERS[/latex] indicates that the consumer does not have to give up that much of [latex]B[/latex] to get the same amount of [latex]A[/latex]. So it must be the case that if the consumer trades some of [latex]B[/latex] for [latex]A[/latex] in the market, they will be better off. In other words, a consumer who moves down along the budget line will be on a higher indifference curve.

So the graph of the budget line and indifference curves illustrates the two conditions that define the consumer’s optimal choice:

[latex]P_{A}A+P_{B}B=I[/latex]

and

[latex]MRS=ERS:\frac{MU_{A}}{MU_{B}}=\frac{P_{A}}{P_{B}}[/latex]

Note that the negative signs cancel.

With these two conditions, we can solve the consumer choice problem mathematically.

Note that the [latex]MRS=ERS[/latex] condition can be rearranged to [latex]\frac{MU_{A}}{P_{A}}[/latex].

Economists sometimes call this the “equal bang for the buck” condition because the equation indicates that at the optimal choice, the consumer must be getting exactly the same marginal utility per dollar from the consumption of goods A and B. Why? Think of a case where these two goods are not equal. If you’re at a pizza parlor eating slices of pizza and bowls of salad, suppose that

[latex]\frac{MU_{Pizza}}{P_{Pizza}}< \frac{MU_{Salad}}{P_{Salad}}[/latex]

This equation indicates that the marginal utility per dollar of pizza is less than the marginal utility per dollar of salad at your current level of consumption. You cannot optimize your utility. Why not?

For simplicity, let’s assume the price of both a slice of pizza and a small salad is exactly $1, so

[latex]MU_{Pizza}< MU_{Salad}[/latex]

Now let’s take $1 away from pizza and spend it instead on salad. You lose the marginal utility of that last slice of pizza, but you gain the marginal utility of one more serving of salad. From the condition above, we know that total utility must have increased, since

[latex]MU_{Pizza}< MU_{Salad}[/latex]

We also know that the marginal utility for a normal preference diminishes when you consume more of a good and therefore increases when you consume less. Therefore, the marginal utility of pizza will increase, and the marginal utility of salad will decrease, getting you closer to equality. As long as this condition is not met (as long as there is an inequality), a consumer can always make these types of trades to become better off.

4.2 Solving the Consumer Choice Problem

Learning Objective 4.2: Solve a consumer choice problem with the typical utility function.

Formally, the consumer’s optimal choice problem looks like this:

[latex]\begin{matrix} max\,U(A,B) \\(A,B) \\ \text{subject to }P_AA+P_BB \leq I \end{matrix}[/latex](4.1)

As we discussed in section 4.1, the second line of this problem (the “subject to” part) is the budget constraint. Since we have the more-is-better assumption, the consumer will always spend all of their budget on goods [latex]A[/latex] and [latex]B[/latex]. So we can rewrite the consumer choice problem as

[latex]\begin{matrix} max\,U(A,B) \\(A,B) \\ \text{subject to }P_AA+P_BB=I \end{matrix}[/latex](4.2)

Let’s use the Cobb-Douglas utility function and solve this problem analytically. Our problem is now

[latex]\begin{matrix} max\,A^\alpha B^\beta\\(A,B) \\ \text{subject to }P_AA+P_BB=I \end{matrix}[/latex](4.3)

To solve this problem, we will apply what we know from section 4.1: at the optimal solution, we have two conditions, [latex]MRS=ERS[/latex], and we are on the budget line.

[latex]MRS=ERS[/latex] in this case requires that we find the marginal utility of bundles [latex]A[/latex] and [latex]B[/latex]. This requires determining the partial derivative of the utility function for good [latex]A[/latex] and then for good [latex]B[/latex].

[latex]MU_A=\frac{\partial (A^\alpha B^\beta )}{\partial A}=\partial A^{\left (a-1 \right )}B^\beta[/latex]

[latex]MU_B=\frac{\partial(A^\alpha B^\beta)}{\partial B}=\beta A^\alpha B^{\left (\beta -1 \right)}[/latex]

The [latex]MRS[/latex] is the ratio of the two marginal utilities:

[latex]MRS=-\frac{MU_A}{MU_B}=-\frac{\frac{\partial \left ( A^\alpha B \right )}{\partial A}}{\frac{\partial (A^\alpha B^\beta )}{\partial B}}=\frac{\partial A^{(a-1)}B^{\beta }}{\beta A^{a}B^{(\beta -1)}}=-\frac{\alpha B}{\beta A}[/latex](4.4)

The [latex]ERS[/latex] is the ratio of the prices:

[latex]MRT=-\frac{P_A}{P_B}[/latex](4.5)

Putting these two equations together gives us

[latex]MRS=MRT\Rightarrow -\frac{\alpha B}{\beta A}=-\frac{P_A}{P_B}\Rightarrow \frac{\alpha }{\beta }\frac{B}{A}=\frac{P_A}{P_B}[/latex](4.6)

The second part of the consumer choice problem, the budget constraint, as we are on the budget line or the “subject to” part, is straightforward:

[latex]P_AA+P_BB=I[/latex](4.7)

At this point, solving the problem is a matter of simple algebra. We have two equations with two unknowns, good [latex]A[/latex] and good [latex]B[/latex]. We can solve these equations by repeated substitution: solve equation 4.4 for good [latex]A[/latex] or [latex]B[/latex], and substitute the result into Equation 4.5. If we solve equation 4.4 for bundle [latex]B[/latex], we get

[latex]B=\frac{\beta }{\alpha }\frac{P_A}{P_B}A[/latex]

Substituting into equation 4.5 gives us

[latex]P_AA+P_B\left ( \frac{\beta }{\alpha } \frac{P_A}{P_B}A\right )=I[/latex](4.8)

Simplifying this equation gives us

[latex]\frac{\left ( \alpha +\beta \right )}{\alpha}P_AA=I[/latex]

Solving for [latex]A[/latex], we get

[latex]A=\frac{I}{P_A}\frac{\alpha }{\left ( \alpha +\beta \right )}[/latex](4.9)

Plugging this equation into equation 4.6 gives us

[latex]A=\frac{I}{P_B}\frac{\beta }{\left ( \alpha +\beta \right )}[/latex]

Simplifying this equation gives us the solution for good [latex]B[/latex]:

[latex]B=\frac{I}{P_B}\frac{\beta}{(\alpha+\beta)}[/latex](4.10)

Remember that the prices, the income, and the parameter values for [latex]\alpha[/latex] and [latex]\beta[/latex] are all just numbers that are given. Therefore, these solutions for goods [latex]A[/latex] and [latex]B[/latex] are simply a specific amount of both—one specific bundle of [latex]A[/latex] and [latex]B[/latex] consumption that is the very best choice a consumer has among all the possible choices.

Equations 4.9 and 4.10) are the demand functions for goods [latex]A[/latex] and [latex]B[/latex], respectively. Demand functions are mathematical functions that describe the relationship between quantity demanded and prices, income, and other things that affect purchase decisions. We can use these demand functions to predict what will happen to the consumption of both goods when prices and incomes change. From the demand functions, it is easy to predict that increases in the price of the good will lead to lower consumption and increases in income will lead to greater consumption. We will return to the examination of these demand functions in the next chapter.

4.3 Corner Solutions and Kinked Indifference Curves

Learning Objective 4.3: Solve a consumer choice problem with utility function for perfect complements and perfect substitutes.

So far, we have considered the optimal consumption bundle for a consumer who has “well-behaved” preferences, meaning that they have indifference curves that are smooth, curved in, and not touching the vertical or horizontal axes. The optimal choice for these well-behaved preferences is characterized by the [latex]MRS=ERS[/latex]. The solution to the consumer choice problem with these preferences is always an interior solution: a utility maximizing bundle that has a positive amount of both goods.

But we know that there are other relatively common preference types—such as perfect complements and perfect substitutes—that have indifference curves that are shaped differently. The solution to the consumer choice problem for these preference types cannot be characterized by the [latex]MRS=ERS[/latex] condition.

As we saw in chapter 1, perfect complements have indifference curves that are kinked at ninety-degree angles, and perfect substitutes have indifference curves that are straight lines that begin and end on the axes. For both perfect complements and perfect substitutes, the solution to the consumer choice problem is the one consumption bundle that puts the consumer on the highest indifference curve possible. But in both cases, this does not have the tangency condition of the MRS equaling the ERS.

Since perfect complements have indifference curves that are kinked, they have abrupt changes in slope at a single point. At this kink, the [latex]MRS[/latex] is not defined because there is no slope. The solution to the consumer choice problem with perfect complement preferences is interior—you definitely need some of both goods to get any utility—but at the kink, there is no [latex]MRS[/latex] to equate to [latex]ERS[/latex].

The solutions to consumer choice problems with perfect complement preferences are usually corner solutions: a utility maximizing bundle that consists of only one of the two goods—in other words, a consumption bundle that is located at one corner of the budget constraint. Corner solutions are typical when preferences are perfect substitutes but can occur for many other preference types that we will not study. Let’s see how to find the utility maximizing bundle for these two preference types, starting with perfect complements.

Perfect Complements

In the case of perfect complements with strictly positive prices, the optimal bundle is always the one at the ninety-degree kink in the indifference curve, as shown in figure 4.3. We can use this observation similarly to how we used the fact that [latex]MRS=ERS[/latex] in the previous section. If the optimal bundle is always at the kink and is always on the budget line, we can once again find two equations with two unknowns to solve.

Recall that the utility function for perfect complements looks like this:

[latex]U(A,B)=min[\alpha A, \beta B][/latex]

A parable is a fixed value given outside the model, one that never changes. A variable is a value that can change, such as prices, income, and so on.

Consuming at the kink implies that [latex]\alpha A = \beta B[/latex]. This condition makes intuitive sense: since the utility function takes on only the value of the smaller of the two choices, spending money on some of good [latex]A[/latex] without the corresponding increase in good [latex]B[/latex] will not raise utility at all but will cost the consumer money. So this choice cannot possibly be optimal. The only optimal decision is to spend money on goods [latex]A[/latex] and [latex]B[/latex] in the precise proportions that they are consumed. With the condition [latex]\alpha A = \beta B[/latex] and the budget constraint [latex]P_AA+P_BB=I[/latex], we can easily solve through repeated substitution. Note that [latex]A=\frac{\beta}{\alpha}B[/latex] substitutes into the budget constraint as follows:

[latex]P_A\frac{\beta}{\alpha}B+P_BB=I[/latex]

Solving for [latex]B[/latex] gives us

[latex]B=\frac{I}{P_A\frac{\beta}{\alpha}+P_B}[/latex], or [latex]B=\frac{\alpha I}{\beta P_A+\alpha P_B}[/latex]

Solving for [latex]A[/latex] using [latex]A=\frac{\beta}{\alpha}B[/latex] gives us

[latex]A=\frac{\beta}{\alpha}[\frac{\alpha I}{\beta P_A+\alpha P_B}]=\frac{\beta I}{\beta P_A+\alpha P_B}[/latex]

Again, we see that the quantity demanded of each item decreases with an increase in its price and increases with increases in income. But now we also have the interesting result that the quantity demanded of one good decreases as the price of the other good increases. This makes intuitive sense because perfect complements are goods that are only consumed together, such as hot dogs and hot dog buns.

Perfect Substitutes

With perfect substitutes, the optimal bundle is generally at either one corner of the budget constraint or the other. If you stop and think about it for a moment, the intuition behind this observation becomes clear. If you like Coke and Pepsi equally well and think of them as perfect substitutes for one another, you will logically buy only the one that has the lower price. If Coke costs $1 and Pepsi $1.50, and you like them equally well, why would you ever buy any Pepsi at all? There is one exception to this all-of-one-or-the-other rule. When Coke and Pepsi have the same prices, you can get all of one, all of the other, or any combination of both. In this case, there is not just one optimal bundle. In this chapter, we will concentrate on the case where there is one optimal bundle, and so you, the consumer, have to purchase all of one or the other.

Consider figure 4.4, which illustrates a generic budget line (in black) for goods [latex]A[/latex] and [latex]B[/latex] and a series of indifference curves (in blue) representing preferences for the perfect substitutes of goods [latex]A[/latex] and [latex]B[/latex]. The slope of the indifference curves (the [latex]MRS[/latex]) is always greater than the slope of the budget line (the [latex]ERS[/latex]). As you can see from figure 4.5, this means that consumers always do better when they consume more [latex]A[/latex] and less [latex]B[/latex]—consumers move to a higher indifference curve.

Specifically, consider bundle 1 in the graph. This bundle contains both [latex]A[/latex] and [latex]B[/latex]. If we compare bundle 1 to the bundle in the corner of the budget constraint labeled “optimal bundle,” you can see that the optimal bundle is on a higher indifference curve, which means that this bundle is better. In fact, this is true of any other bundle in the budget constraint—all other bundles place consumers on a lower indifference curve.

Finding this optimal bundle is relatively easy mathematically. Since we know it has to be at a corner, we just need to check the two corners of the budget constraint and see which one yields the higher utility. In general, utility functions that represent perfect complements look like this:

[latex]U(A,B)=\alpha A + \beta B[/latex]

Note that the additive form of the utility function is the key—you can always get to the same utility by taking away some of one good and adding some of the other (even if there is none of the other good in the bundle). Note also that the [latex]MRS[/latex] is

[latex]MRS=\frac{\alpha}{\beta}[/latex]

Since [latex]\alpha[/latex] and [latex]\beta[/latex] are parameters, this means the slope of the indifference curve is a constant, and so the indifference curves for perfect substitutes are straight lines that intersect the axes.

Solving for the optimal consumption bundle for perfect substitutes starts with checking the corners, which means we ask what utility the consumer gets from spending all of their income on just one good. So if

[latex]P_AA+P_BB=I[/latex]

and the consumer decides to consume only [latex]A[/latex], then the total amount consumed of [latex]A[/latex] is

[latex]\frac{I}{P_A}[/latex]

Similarly, the total amount consumed of [latex]B[/latex] if all of the income is spent on [latex]B[/latex] is

[latex]\frac{I}{P_B}[/latex]

So all that is left to do is to check if

[latex]U(A,0)=\alpha\frac{I}{P_A} > \beta\frac{I}{P_B}=U(0,B)[/latex]

if the opposite is true, or if they are equal.

If the above is true, then we know immediately that consuming only [latex]A[/latex] is the optimal choice. Similarly, if the opposite is true, we know consuming only[latex]B[/latex] is optimal. Finally, if they happen to be equal, any combination of [latex]A[/latex] and [latex]B[/latex] consumption along the budget line is optimal. Figure 4.5 illustrates these three possibilities.

4.4 Policy Example
Hybrid Car Purchase Tax Credit—Is It the Best Choice to Reduce Fuel Consumption and Carbon Emissions?

Learning Objective 4.4: The Hybrid Car Tax Credit and Consumer Choice

Suppose you are a policymaker considering proposals to reduce the consumption of fossil fuels and carbon emissions. With the tools from chapters 1 through 4, you can now conduct an economic analysis to analyze the likely outcome of a tax credit for electric and hybrid cars. Begin by considering the consumer choice problem prior to the introduction of the tax credit.

First, combine the indifference curve mapping from chapters 1 and 2 with the budget constraint from chapter 3, as shown in figure 4.6. Remember, these are simply graphical representations of the mathematical equations 4.4 and 4.5.

We are looking for the point on the budget line that puts the consumer on the highest indifference curve possible. This is of course the tangency point between the indifference curve and the budget line. This tangency is characterized by the equality of the marginal rate of substitution between miles driven and other goods consumed and the price ratio of the same two goods, the economic rate of substitution. Figure 4.6 shows that given our assumption of convex indifference curves, there is a unique point that defines the number of miles driven and the amount of money for other goods consumed that make up the optimal bundle for this “typical” consumer.

Mathematically, we can solve for the tangent point by remembering that it is characterized by two things:

1. The equality of the marginal rate of substitution to the price ratio
2. The budget constraint holding with equality

The first equation ensures that the bundle is at a point of tangency between the indifference curve and lines with the same slope as the budget line. But remember that the price ratio describes the slope, not the specific budget line, which is why we need the second equation to make sure we are on the specific line that describes the consumer’s budget.

Specifically, if the utility function that represents the “typical” consumer’s preferences over miles driven (M) and all other consumption (C) is

[latex]U(M,C)=M^\alpha C^\beta[/latex]

then we can solve for the MRS:

[latex]MRS=\frac{MU_M}{MU_C}=\frac{\frac{\alpha U}{\alpha M}}{\frac{\alpha U}{\alpha C}}=\frac{\alpha }{\beta }\frac{C}{M}[/latex]

As noted above, this is one condition that characterized the optimal consumption bundle. The other is the budget line:

[latex]\text{Income} = (Price_{Miles Driven})(\text{Miles Driven}) + \text{Dollars for Other Consumption}[/latex]

We can solve this system of equations by the process of repeated substitution to come up with the description of the precise optimal bundle for this consumer.

Now let’s consider what happens after policymakers pass the tax credit into law. From chapter 3, we know that the effect of the tax credit on the budget constraint is to make the cost of miles consumers drive less expensive, which represents a drop in the price of miles driven. This will cause the budget line to expand along the vertical axis, but it will not change the horizontal intercept. The consumer must now choose a new optimal bundle, and we must compare the new bundle to the old one. Let’s first examine this graphically in figure 4.7.

In this graph, it is clear that the new bundle represents more of both goods. The consumer will choose to consume more of other goods with the extra buying power that the more fuel-efficient car affords as well as to consume more miles driven due to the lower cost of the activity. So the policy turns out to have an unintended consequence: in an effort to decrease fuel consumption, the policy actually lowers the marginal cost of driving, inducing consumers to drive more.

By making some fairly basic assumptions about typical consumer preferences and modeling the consumer choice problem, economic theory suggests that we should expect an increase in miles driven as a result. This application illustrates the power of models. By simplifying reality into a model framework, we can discover something about the world and human behavior that was not obvious.

But this is just a theory, so it is suggestive rather than definitive. It is important to note that the assumptions we made may not be completely accurate, and so our prediction may be inaccurate. For example, depending on the precise indifference curve mapping, we could actually see decreased miles driven after the hybrid car tax credit.

We now need to test the theory by evaluating real-world data. So what does the data suggest? Studies have shown that

1. An increase in MPG increases how much consumers drive and
2. This effect is in the range of 10 to 30 percent, meaning that a 10 percent increase in MPG would increase how many miles consumers drive by between 1 and 3 percent.

In the case of fuel consumption, economists generally prefer that policymakers increase the gas tax–the subsidy approach–and other indirect approaches, like the corporate fuel economy standard known as CAFE.

The tax on fuel raises the cost of consumption and, as we have now seen, decreases fuel consumption. A number of studies have shown that an increase in the gas tax is a more cost-effective way to decrease gasoline usage.

With our analysis of the policy based on both theory and evidence, we are now in a position to answer the original question posed at the beginning of section 1.1.

Suppose that a hybrid car tax credit was wildly successful and succeeded in doubling the average fuel economy of all cars on US roads. What do you think would happen to the fuel consumption of all US motorists? Should the government expect the fuel consumption and carbon emissions of cars driven in the United States to decrease by half in response?

The answer is clearly no. Even the most conservative estimates suggest that a 100 percent increase in average fuel efficiency will result in an increase of miles driven by 10 percent, meaning that the decrease in fuel consumption and carbon emissions will fall but by less than half.

Exploring the Policy Question

1. Suppose the hybrid car tax credit was accompanied by a tax increase on gasoline. How would the analysis of the policy change? Could you make a prediction about the change in carbon emissions now?
2. How would the analysis change if consumers’ preferences change and they begin to lower their gas consumption by carpooling, walking, and biking more instead of driving? How could you show this in figure 4.7?
3. The phenomenon of increased energy efficiency leading to increased consumption is known among economists as the rebound effect, and it is common in other practical contexts besides vehicles and driving. What other examples of rebound effect would you expect to see in the real world?

LEARN: KEY TOPICS

Terms

Consumer choice problem

What is the consumer’s optimal choice among competing bundles? Among all the bundles the consumer could possibly choose, which one returns the highest level of utility?

Corner solutions

A utility maximizing bundle that consists of only one of the two goods—in other words, a consumption bundle that is located at one corner of the budget constraint. Corner solutions are typical when preferences are perfect substitutes.

Demand functions

Mathematical functions that describe the relationship between quantity demanded and prices, income, and other things that affect purchase decisions.

Interior solutions

A utility maximizing bundle that has a positive amount of both goods.

Kinked curves

Curves that have abrupt changes in slope at a single point.

Graphs

Solution to the general consumer choice problem

Solution to the consumer choice problem for perfect complements

Solution to the consumer choice problem for perfect substitutes

Equation

Demand functions

The Demand Function for Good A

[latex]A=\frac{I}{P_A}\frac{\alpha }{\left ( \alpha +\beta \right )}[/latex]

The Demand Function for Good B

[latex]B=\frac{I}{P_B}\frac{\beta}{(\alpha+\beta)}[/latex]

The Consumer’s Optimal Choice Problem

[latex]\begin{matrix} max\,U(A,B) \\(A,B) \\ \text{subject to }P_AA+P_BB \leq I \end{matrix}[/latex]

The second line of the consumer’s optimal choice problem (the “subject to” part) is the budget constraint. Using the more-is-better assumption, the equation is rewritten to indicate the consumer will always spend all of their budget on goods A and B, as below:

[latex]\begin{matrix} max\,U(A,B) \\(A,B) \\ \text{subject to }P_AA+P_BB = I \end{matrix}[/latex]

Applying the Cobb-Douglas utility function

[latex]\begin{matrix} max\,A^\alpha B^\beta\\(A,B) \\ \text{subject to }P_AA+P_BB=I \end{matrix}[/latex]

To solve this requires taking the partial derivative. This equation looks as follows:

[latex]MU_A=\frac{\partial (A^\alpha B^\beta )}{\partial A}=\partial A^{\left (a-1 \right )}B^\beta[/latex]
[latex]MU_B=\frac{\partial(A^\alpha B^\beta)}{\partial B}=\beta A^\alpha B^{\left (\beta -1 \right)}[/latex]

The MRS is the ratio of the two marginal utilities:

[latex]MRS=-\frac{MU_A}{MU_B}=-\frac{\frac{\partial \left ( A^\alpha B \right )}{\partial A}}{\frac{\partial (A^\alpha B^\beta )}{\partial B}}=\frac{\partial A^{(a-1)}B^{\beta }}{\beta A^{a}B^{(\beta -1)}}=-\frac{\alpha B}{\beta A}[/latex]

The ERS is the ratio of the prices:

[latex]MRT=-\frac{P_A}{P_B}[/latex]

When combined, they take the following form:

[latex]MRS=MRT\Rightarrow -\frac{\alpha B}{\beta A}=-\frac{P_A}{P_B}\Rightarrow \frac{\alpha }{\beta }\frac{B}{A}=\frac{P_A}{P_B}[/latex]

The second part of consumer choice problem, the budget constraint, reads like this:

[latex]P_AA+P_BB=I[/latex]

Solving the Cobb-Douglas Equation

Solving the problem involves repeated substitution. The MRS ratio initially solved for Bundle B is as follows:

[latex]B=\frac{\beta }{\alpha }\frac{P_A}{P_B}A[/latex]

The next step is substituting the above result into the ERS ratio equation, providing the formula below:

[latex]P_AA+P_B\left ( \frac{\beta }{\alpha } \frac{P_A}{P_B}A\right )=I[/latex]

Simplified:

[latex]\frac{\left ( \alpha +\beta \right )}{\alpha}P_AA=I[/latex]

Solved for [latex]A[/latex]:

[latex]A=\frac{I}{P_A}\frac{\alpha }{\left ( \alpha +\beta \right )}[/latex]

The result of the above equation substituted into the combined [latex]MRS/ERS[/latex] equation is the demand function for good [latex]A[/latex]:

[latex]A=\frac{I}{P_B}\frac{\beta }{\left ( \alpha +\beta \right )}[/latex]

Solving again for B provides the demand function for good B:

[latex]B=\frac{I}{P_B}\frac{\beta}{(\alpha+\beta)}[/latex]