Mr. A derives utility from martinis \((M)\) in proportion to the number he drinks: $$\boldsymbol{U}(\boldsymbol{M})=\boldsymbol{M}$$ Mr. A is very 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 \(P_{c}\) and \(P_{v}\) Hint: Because this problem involves a fixed proportions utility function you cannot solve for utility- maximizing decisions by using calculus.

Step by step solution

01

a. Graphing the indifference curve and Mr. A's mixing preference

To graph the indifference curve for various levels of utility, we should start by setting \(U(G, V) = k\), where \(k\) represents the level of utility. In this case, the utility function is given by \(U(G, V) = \min(\frac{G}{2}, V)\). For each level of utility, we will have the equation: $$\min\left(\frac{G}{2}, V\right) = k$$ Whenever \(\frac{G}{2} = V\), we get the equation \(G = 2V\). This will be the indifference curve. Since the ratio of G to V is 2 to 1 regardless of the level of utility, plotting the curve on a graph will show that Mr. A will never alter the way he mixes martinis regardless of the prices of the two ingredients.

02

b. Calculating the demand functions for G and V

Since Mr. A has a fixed preference ratio for gin and vermouth, the demand functions for G and V can be determined directly from the utility function. We know that \(U(G, V) = \min(\frac{G}{2}, V)\). Therefore, the demand function for gin (G) can be expressed as: $$G = 2 V$$ Similarly, the demand function for vermouth (V) is: $$V = \frac{G}{2}$$

03

c. Determining the indirect utility function

Using the demand functions derived in part (b), we can now determine Mr. A's indirect utility function. We can substitute the demand functions back into the utility function: $$U(G, V) = \min\left(\frac{G}{2}, V\right) = \min\left(\frac{2V}{2}, V\right) = \min(V, V)$$ Therefore, Mr. A's indirect utility function is given by: $$U(V) = V$$

04

d. Calculating Mr. A's expenditure function

To find Mr. A's expenditure function, we need to calculate his total spending on gin and vermouth given their prices and the level of utility. Let \(P_G\) and \(P_V\) denote the prices of gin and vermouth, respectively. Mr. A's total spending is given by: $$E = P_G \cdot G + P_V \cdot V$$ Using the demand functions from part (b), we can substitute the expressions for G and V: $$E = P_G \cdot (2V) + P_V \cdot V$$ Next, recall that Mr. A's indirect utility function is \(U(V) = V\). Thus, we can substitute the level of utility, k, for V: $$E(k, P_G, P_V) = P_G \cdot (2k) + P_V \cdot k$$ This is Mr. A's expenditure function for each level of utility, given the prices of gin and vermouth.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!