Question

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

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

Step-by-step solution

Identify the Utility Function

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

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 \(v = \frac{g}{2}\).

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.

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#

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.

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: \(p_gg + p_v \frac{g}{2} = M\).

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#

Substitute the Demand Functions into the Utility Function

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

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#

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: \(M = U(p_g, p_v) (p_g + 2p_v)\).

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.