Question

AJ spends all of his income from his fitness coaching side hustle on guitar picks (G) and fishing lures (L). Guitar picks are priced at $1 and fishing lures are priced at $2. Assume that AJ earns $70 to spend and his utility function can be represented as U(G,L)=G⁰.⁶L⁰.⁴. For this utility function, the marginal utilities are MUG=0.6G⁰.⁴−0.4L⁰.⁴ and MUL=0.4G⁰.⁶L−0.6. a. What general condition must hold for AJ to be maximizing his utility subject to his budget constraint? b. Use the specific information you have to set up the condition from a. for this problem. Then simplify to get G as a function of L (Hint: solve to get G by itself). c. What is AJ's specific budget constraint? d. Use AJ's specific budget constraint to determine how many guitar picks and fishing lures AJ consumes to maximize his utility. e. How much utility does this combination give him? f. If the price of guitar picks triples, how much additional income must AJ earn from his side hustle to maintain the same level of utility?

Answer 1

To maximize his utility, AJ must allocate his spending on guitar picks and fishing lures in a way that equals the ratio of the marginal utility to price for each good. Using specific information, the condition is set up as 0.6G^0.4-0.4L^0.4 / 1 = 0.4G^0.6L^-0.6 / 2. From this, the number of guitar picks and fishing lures AJ consumes to maximize utility can be determined, along with the resulting utility. If the price of guitar picks triples, AJ would need to earn additional income to maintain the same level of utility.

Step-by-step explanation:

a. To maximize his utility subject to his budget constraint, AJ must allocate his spending on guitar picks (G) and fishing lures (L) in a way that equals the ratio of the marginal utility of G to the price of G to the ratio of the marginal utility of L to the price of L.

b. Using the specific information, the condition from a. can be set up as 0.6G0.4-0.4L0.4 / 1 = 0.4G0.6L-0.6 / 2. Simplifying the equation gives G = 1.75L-0.5.

c. AJ's budget constraint can be represented as 1G + 2L = $70.

d. To determine the number of guitar picks and fishing lures AJ consumes to maximize his utility, substitute the expression for G from part b. into the budget constraint equation and solve for L. The solution is L = 7.5 and G = 4.4.

e. The utility from this combination can be calculated using the utility function U(G,L) = G0.6L0.4. Substituting the values, the utility is approximately 10.2.

f. If the price of guitar picks triples, AJ would need to earn an additional income of $20 to maintain the same level of utility. This can be calculated by multiplying the new price of the guitar picks ($3) by the new quantity of guitar picks consumed (4.4), and subtracting the original price of the guitar picks ($1) multiplied by the original quantity of guitar picks consumed (4).