Chapter 4, Utility Maximization and Choice Video Solutions, Microeconomic Theory: Basic Principles and Extensions

Problem 1

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)?

Nick Johnson

Numerade Educator

05:32

Problem 2

Bordeaux $\left(w_{F}\right)$ at $\$ 40$ per bottle and a less expensive 2005 California varietal wine $\left(w_{\mathrm{C}}\right)$ priced at $\$ 8 .$ If her utility is $$U\left(w_{F}, w_{C}\right)=w_{F}^{2 / 3} w_{C}^{1 / 3}$$ then how much of each wine should she purchase?
b. When she arrived at the wine store, our young oenologist discovered that the price of the French Bordeaux had fallen to $\$ 20$ a bottle because of a decrease in the value of the euro. If the price of the California wine remains stable at $\$ 8$ per bottle, how much of each wine should our friend purchase to maximize utility under these altered conditions?
c. Explain why this wine fancier is better off in part (b) than in part (a). How would you put a monetary value on this utility increase?

Angela Guo

Numerade Educator

02:13

Problem 3

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?

Andrew Davis

Numerade Educator

02:09

Problem 4

a. $\mathrm{Mr}$. Odde Ball enjoys commodities $x$ and $y$ according to the utility function $$U(x, y)=\sqrt{x^{2}+y^{2}}$$ Maximize Mr. Ball's utility if $p_{x}=\$ 3, p_{y}=\$ 4,$ and he has $\$ 50$ to spend. Hint: It may be easier here to maximize $U^{2}$ rather than $U$. Why will this not alter your results?
b. Graph Mr. Ball's indifference curve and its point of tangency with his budget constraint. What does the graph say about Mr. Ball's behavior? Have you found a true maximum?

Nick Johnson

Numerade Educator

10:21

Problem 5

Mr. A derives utility from martinis $(m)$ in proportion to the number he drinks: $$U(m)=m$$ Mr. A is particular about his martinis, however: He only enjoys them made in the exact proportion of two parts gin ( $g$ ) to one part vermouth ( $v$ ). Hence we can rewrite Mr. A's utility function as $$U(m)=U(g, v)=\min \left(\frac{g}{2}, v\right)$$
two ingredients, Mr. A will never alter the way he mixes martinis.
b. Calculate the demand functions for $g$ and $v$
c. Using the results from part (b), what is Mr. A's indirect utility function?
problem involves a fixed-proportions utility function, you cannot solve for utility-maximizing decisions by using calculus.

Oluwadamilola Ameobi

Numerade Educator

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Problem 6

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?

Oluwadamilola Ameobi

Numerade Educator

02:49

Problem 7

The lump sum principle illustrated in Figure 4.5 applies to transfer policy and taxation. This problem examines this application of the principle.
a. Use a graph similar to Figure 4.5 to show that an income grant to a person provides more utility than does a subsidy on good $x$ that costs the same amount to the government.
b. Use the Cobb-Douglas expenditure function presented in Equation 4.52 to calculate the extra purchasing power needed to increase this person's utility from $U=2$ to $U=3$
c. Use Equation 4.52 again to estimate the degree to which good $x$ must be subsidized to increase this person's utility from $U=2$ to $U=3 .$ How much would this subsidy cost the government? How would this cost compare with the cost calculated in part (b)?
d. Problem 4.10 asks you to compute an expenditure function for a more general Cobb-Douglas utility function than the one used in Example $4.4 .$ Use that expenditure function to re-solve parts (b) and (c) here for the case $\alpha=0.3,$ a figure close to the fraction of income that low-income people spend on food.
e. How would your calculations in this problem have changed if we had used the expenditure function for the fixedproportions case (Equation 4.54 ) instead?

Akash M

Numerade Educator

00:01

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?

Oluwadamilola Ameobi

Numerade Educator

03:03

Problem 9

expenditure function for this utility function. Hint: The expenditure function will have kinks at various price ratios.

Breanna Ollech

Numerade Educator

03:03

Problem 10

In Example 4.1 we looked at the Cobb-Douglas utility function $U(x, y)=x^{\alpha} y^{1-\alpha},$ where $0 \leq \alpha \leq 1 .$ This problem illustrates a few more attributes of that function.
a. Calculate the indirect utility function for this Cobb-Douglas case.
b. Calculate the expenditure function for this case.
c. Show explicitly how the compensation required to offset the effect of an increase in the price of $x$ is related to the size of the exponent $\alpha$

Breanna Ollech

Numerade Educator

03:28

Problem 11

The CES utility function we have used in this chapter is given by $$U(x, y)=\frac{x^{0}}{\delta}+\frac{y^{0}}{\delta}$$
a. Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion $$\frac{x}{y}=\left(\frac{p_{x}}{p_{y}}\right)^{1 /(\delta-1)}$$ b. Show that the result in part (a) implies that individuals will allocate their funds equally between $x$ and $y$ for the CobbDouglas case $(\delta=0),$ as we have shown before in several problems.
c. How does the ratio $p_{x} x / p_{y} y$ depend on the value of $\delta$ ? Explain your results intuitively. (For further details on this function, see Extension E4.3.)
d. Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated.

Karol Hajduk

Numerade Educator

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Problem 12

Suppose individuals require a certain level of food $(x)$ to remain alive. Let this amount be given by $x_{0}$. Once $x_{0}$ is purchased, individuals obtain utility from food and other goods $(y)$ of the form $$U(x, y)=\left(x-x_{0}\right)^{\alpha} y^{\beta}$$
where $\alpha+\beta=1$
a. Show that if $I>p_{x} x_{0}$ then the individual will maximize utility by spending $\alpha\left(I-p_{x} x_{0}\right)+p_{x} x_{0}$ on good $x$ and $\beta\left(I-p_{x} x_{0}\right)$ on good $y$. Interpret this result.
b. How do the ratios $p_{x} x / I$ and $p_{y} y / I$ change as income increases in this problem? (See also Extension E4.2 for more on this utility function.)

Oluwadamilola Ameobi

Numerade Educator

01:49

Problem 13

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.

R M

Numerade Educator

01:33

Problem 14

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 $U_{2}\left(c_{2}\right)=c_{2}$

Jennifer Stoner

Numerade Educator