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Chapter 4: Problem 8

Two of the simplest utility functions are: 1\. Fixed proportions: \(U(x, y)=\min [x, y]\) 2\. Perfect substitutes: \(U(x, y)=x+y\) a. For each of these utility functions, compute the following: \(\bullet\) Demand functions for \(x\) and \(y\) \(\bullet\) Indirect utility function \(\bullet\) Expenditure function b. Discuss the particular forms of these functions you calculated-why do they take the specific forms they do?

Short Answer

Expert verified
Short answer: For a fixed proportions utility function, the demand functions for goods x and y are equal, indicating a constant proportion between the two goods: \(x = \frac{I}{p_x + p_y}\) and \(y = \frac{I}{p_x + p_y}\). The indirect utility function is \(V(p_x, p_y, I) = \frac{I}{p_x + p_y}\), and the expenditure function is \(E(p_x, p_y, u) = u(p_x + p_y)\). For a perfect substitutes utility function, the consumer only consumes the good with a lower price per unit, resulting in the following demand functions: \(x = \frac{I}{p_x}\) if \(p_x \leq p_y\) and \(0\) otherwise, and \(y = \frac{I}{p_y}\) if \(p_y \leq p_x\) and \(0\) otherwise. The indirect utility function for this case is \(V(p_x, p_y, I) = \max\left(\frac{I}{p_x}, \frac{I}{p_y}\right)\), and the expenditure function is \(E(p_x, p_y, u) = u \min(p_x, p_y)\). The specific forms of these functions result from the particular properties of their respective utility functions and the optimization of utility under budget constraints.

Step by step solution

01

1. Fixed proportions utility function calculations:

Let's start with the fixed proportions utility function (\(U(x, y) = \min[x, y]\)). To find the demand functions for goods \(x\) and \(y\), we need to consider the budget constraint, which is given by \(p_x x + p_y y = I\), where \(p_x\) and \(p_y\) are the prices of goods \(x\) and \(y\) respectively, and \(I\) is the consumer's income. The following steps will help us compute the demand functions for \(x\) and \(y\):
02

1.1 Optimal choice of goods x and y:

To achieve the highest utility, the consumer will expend all their income on both goods, such that \(x = y\). Thus, \(p_x x + p_y x = I \Rightarrow x (p_x + p_y) = I\) Now, we can find the demand function for good \(x\) as: \(x = \frac{I}{p_x + p_y}\). Similarly, the demand function for good \(y\) is given by: \(y = \frac{I}{p_x + p_y}\).
03

1.2 Indirect utility function:

Using the demand functions, we can compute the indirect utility function by replacing \(x\) and \(y\) in the utility function with their demand functions: \(V(p_x, p_y, I) = \min \left[\frac{I}{p_x + p_y}, \frac{I}{p_x + p_y}\right] = \frac{I}{p_x + p_y}\).
04

1.3 Expenditure function:

The expenditure function \(E(p_x, p_y, u)\) is the minimal cost needed to achieve a specific utility level \(u\). We can use the demand function to find this by solving for \(I\) using the indirect utility function: \(I = u(p_x + p_y)\) Substitute this back into the demand functions for goods \(x\) and \(y\), and we get: \(E(p_x, p_y, u) = up_x\frac{u}{p_x + p_y} + up_y\frac{u}{p_x + p_y} = u(p_x + p_y)\).
05

2. Perfect substitutes utility function calculations:

Next, let's move on to the perfect substitutes utility function (\(U(x, y) = x + y\)). We'll follow the same steps as earlier:
06

2.1 Optimal choice of goods x and y:

For this utility function, we observe that increasing the quantity of either \(x\) or \(y\) while maintaining the budget constraint will increase utility. Hence, the consumer will only consume the good with the lower price per unit. In case of equal prices, they can consumer any combination of the two goods. Therefore, the demand functions for goods \(x\) and \(y\) are: \(x = \frac{I}{p_x}\) if \(p_x \leq p_y\) and \(0\) otherwise \(y = \frac{I}{p_y}\) if \(p_y \leq p_x\) and \(0\) otherwise
07

2.2 Indirect utility function:

By plugging the demand functions back into the utility function, we can compute the indirect utility function: \(V(p_x, p_y, I) = \max\left(\frac{I}{p_x}, \frac{I}{p_y}\right)\).
08

2.3 Expenditure function:

Finally, we can find the expenditure function \(E(p_x, p_y, u)\) by solving for \(I\) in the indirect utility function: \(I = u \min(p_x, p_y)\) Hence, the expenditure function is given by: \(E(p_x, p_y, u) = u \min(p_x, p_y)\).
09

Discussion:

In both cases, the specific forms of the demand, indirect utility, and expenditure functions are derived from the properties of the utility functions. For fixed proportions, since the utility is based on the minimum of \(x\) and \(y\), the demand functions are equal, indicating a constant proportion between the two goods. In contrast, for perfect substitutes, the consumer is only concerned with the total number of goods, so they will only consume the good with a lower price per unit. This is reflected in the demand functions for perfect substitutes utility function. Overall, the forms of these functions are a direct consequence of the properties of each utility function and their subsequent optimization and budget constraints.

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Most popular questions from this chapter

Michele, who has a relatively high income \(I\), has altruistic feelings toward Sofia, who lives in such poverty that she essentially has no income. Suppose Michele's preferences are represented by the utility function $$U_{1}\left(c_{1}, c_{2}\right)=c_{1}^{1-a} c_{2}^{a}$$ where \(c_{1}\) and \(c_{2}\) are Michele and Sofia's consumption levels, appearing as goods in a standard Cobb-Douglas utility function. Assume that Michele can spend her income either on her own or Sofia's consumption (through charitable donations) and that \(\$ 1\) buys a unit of consumption for either (thus, the "prices" of consumption are \(p_{1}=p_{2}=1\) ). a. Argue that the exponent \(a\) can be taken as a measure of the degree of Michele's altruism by providing an interpretation of extremes values \(a=0\) and \(a=1 .\) What value would make her a perfect altruist (regarding others the same as oneself)? b. Solve for Michele's optimal choices and demonstrate how they change with \(a\). c. Solve for Michele's optimal choices under an income tax at rate \(t .\) How do her choices change if there is a charitable deduction (so income spent on charitable deductions is not taxed)? Does the charitable deduction have a bigger incentive effect on more or less altruistic people? d. Return to the case without taxes for simplicity. Now suppose that Michele's altruism is represented by the utility function $$U_{1}\left(c_{1}, U_{2}\right)=c_{1}^{1-a} U_{2}^{a},$$ which is similar to the representation of altruism in Extension \(\mathrm{E} 3.4\) to the previous chapter. According to this specification, Michele cares directly about Sofia's utility level and only indirectly about Sofia's consumption level. 1\. Solve for Michele's optimal choices if Sofia's utility function is symmetric to Michele's: \(U_{2}\left(c_{2}, U_{1}\right)=c_{2}^{1-a} U_{1}^{a} .\) Compare your answer with part (b). Is Michele more or less charitable under the new specification? Explain. 2\. Repeat the previous analysis assuming Sofia's utility function is \(U_{2}\left(c_{2}\right)=c_{2}\)

Suppose that a fast-food junkie derives utility from three goods-soft drinks \((x),\) hamburgers \((y),\) and ice cream sundaes \((z)-\) according to the Cobb- Douglas utility function $$U(x, y, z)=x^{0.5} y^{0.5}(1+z)^{0.5}$$ Suppose also that the prices for these goods are given by \(p_{x}=1, p_{y}=4,\) and \(p_{z}=8\) and that this consumer's income is given by \(I=8\) a. Show that, for \(z=0\), maximization of utility results in the same optimal choices as in Example \(4.1 .\) Show also that any choice that results in \(z>0\) (even for a fractional \(z\) ) reduces utility from this optimum. b. How do you explain the fact that \(z=0\) is optimal here? c. How high would this individual's income have to be for any \(z\) to be purchased?

In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by $$U(x, y)=\left(x-x_{0}\right)^{\alpha} y^{\beta, }$$[in this function the elasticity of substitution \(\sigma=1 /(1-\delta)]\) a. Show that the indirect utility function for the utility function just given is $$V=I\left(p_{x}^{r}+p_{y}^{r}\right)^{-1 / r}$$ where \(r=\delta /(\delta-1)=1-\sigma\) b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by $$E=V\left(p_{x}^{r}+p_{y}^{r}\right)^{1 / r}$$ f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.

a. On a given evening, J. P. enjoys the consumption of cigars ( \(c\) ) and brandy ( \(b\) ) according to the function $$U(c, b)=20 c-c^{2}+18 b-3 b^{2}$$ How many cigars and glasses of brandy does he consume during an evening? (cost is no object to J. P.) b. Lately, however, J. P. has been advised by his doctors that he should limit the sum of glasses of brandy and cigars consumed to \(5 .\) How many glasses of brandy and cigars will he consume under these circumstances?

Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies ( \(t\) ) and soda ( \(s\) ), and these provide him a utility of utility \(=U(t, s)=\sqrt{t s}\) a. If Twinkies cost \(\$ 0.10\) each and soda costs \(\$ 0.25\) per cup, how should Paul spend the \(\$ 1\) his mother gives him to maximize his utility? b. If the school tries to discourage Twinkie consumption by increasing the price to \(\$ 0.40,\) by how much will Paul's mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a)?
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