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Chapter 4: Problem 5

Mr. A derives utility from martinis \((m)\) in proportion to the number he drinks: \\[U(m)=m.\\] \(\mathrm{Mr}, \mathrm{A}\) 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).\\] a. Graph Mr. A's indifference curve in terms of \(g\) and \(v\) for various levels of utility. Show that, regardless of the prices of the 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? d. Calculate Mr. A's expenditure function; for each level of utility, show spending as a function of \(p_{8}\) and \(p_{v}\). Hint: Because this problem involves a fixed-proportions utility function, you cannot solve for utility-maximizing decisions by using calculus.

Short Answer

Expert verified
Mr. A's indirect utility function is: \(U(M, p_g, p_v) = \frac{M}{p_g + 2p_v}\).

Step by step solution

01

Identify the Utility Function

Mr. A's utility function for martinis is: \(U(g, v) = \min\left(\frac{g}{2}, v\right)\).
02

Find the Equation for the Indifference Curve

To graph the indifference curves, we need to find the equation that represents them. Since Mr. A prefers a fixed proportion of gin to vermouth, the indifference curves will be linear and have the form \(v = \frac{g}{2}\).
03

Graph the Indifference Curves

Plot the equation \(v = \frac{g}{2}\) for various levels of utility. The graph will show that indifference curves have a constant slope of 2, which means that Mr. A always uses the same proportion of gin to vermouth.
04

Prove Mr. A Will Never Alter the Mix

Since the indifference curves have a constant slope and Mr. A always uses the same proportion of gin to vermouth, it means that he will never change the mix regardless of the prices of the two ingredients. #b. Calculating Demand Functions for Gin and Vermouth#
05

Use the Budget Constraint

Mr. A's budget constraint is given by the equation: \(p_gg + p_vv = M\), where \(p_g\) and \(p_v\) are the prices of gin and vermouth, respectively, and \(M\) is Mr. A's income.
06

Substitute the Equation for the Indifference Curve

Since from step 2 of part a, \(v = \frac{g}{2}\), use this to rewrite the budget constraint as: \(p_gg + p_v \frac{g}{2} = M\).
07

Solve for the Demand Functions

To find the demand functions for gin and vermouth, solve for \(g\) and \(v\) from the budget constraint: \(g = \frac{2M}{p_g + 2p_v}\) \(v = \frac{M}{p_g + 2p_v}\) #c. Finding Mr. A's Indirect Utility Function#
08

Substitute the Demand Functions into the Utility Function

Substitute the demand functions for gin and vermouth found in part (b) into the utility function: \(U(g,v) = \min\left(\frac{g}{2}, v\right)\) \(U\left(\frac{2M}{p_g+2p_v}, \frac{M}{p_g+2p_v}\right) = \min \left(\frac{2M}{2(p_g + 2p_v)}, \frac{M}{p_g + 2p_v}\right)\).
09

Simplify the Expression

Simplify the expression for the indirect utility function: \(U\left(M, p_g, p_v\right) = \frac{M}{p_g + 2p_v}\) #d. Calculating Mr. A's Expenditure Function#
10

Solve for M using the Indirect Utility Function

From the indirect utility function, we can solve for the income \(M\) as a function of the prices of gin and vermouth and the utility level: \(M = U(p_g, p_v) (p_g + 2p_v)\).
11

Find the Expenditure Function

Replace \(M\) in the budget constraint with the expression found in step 1: \(E(U, p_g, p_v) = U(p_g, p_v) (p_g + 2p_v)\). This is Mr. A's expenditure function. It shows his spending on gin and vermouth as a function of the prices of gin and vermouth and his utility level.

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Most popular questions from this chapter

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?

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.

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 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.)

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.
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