Question

2) Chimichanga Fest Your utility function is given by U-X,X, where xi s your consumption of Chimichangas and x, is your consumption of all the other goods in the economy. Yes, you spend 60% of your budget on Chimichangas, which is totally reasonable after the Dumpling House tragedy. a) Solve the utility maximization problem, finding the uncompensated demand for x, & x, and the indirect utility function in terms of p,, p, and Y. b) Solve the expenditure minimization problem, findig the compensated demand for x, & x, and the expenditure function in terms of p, p, and c) Find the consumption bundle that maximizes utility when pi = 1.5, p2-1 and Y = 10 (we normalized the price of all other goods to 1). Calculate your utility level then replace it in the expenditure function to make sure you obtain the initial budget (Y) d) Suppose the government decides to impose a S 0.50 unit tax on Chimichangas and p now goes from S1.5 to $2 he values of p, and Y don't change i. Calculate the Cost of Living Adjustment (COLA) that allows you to buy the same consumption bundle as before. What utility level you achieve after COLA? Is it higher or lower than before? ii. Calculate the equivalent variation (EV) which is how much income you lose with this tax (or how much you would pay to avoid it). Start by using the indirect utility function to calculate the new utility level after the price change, then use the expenditure function to calculate the income needed to buy this new utility level iii. Calculate the Compensating Variation (CV), whiclh is how much income you would need to go back to your previous utility level (before the price change) Start by using the indirect utility function to calculate the utility level before the price change, then use the expenditure function to calculate the income needed to buy this utility level after the price change e) How much money did the government collect with this tax? i. Multiply the consumption of Chimichangas by the unit tax to calculate the amount of tax collected. ii. Compare the tax collected to EV and respond: Is it better to tax Chimichangas or income?

Answer 1

U = x₁³x₂²

p₁x₁+p₂x₂ = Y

U = (Y-p₂x₂ / p₁)³x₂²

for utility maximization, dU/dx₂=0

3(Y-p₂x₂ / p₁)²x₂²(-p₂/p₁) + 2(Y-p₂x₂ / p₁)³x₂

₌ (Y-p₂x₂ / p₁)²x₂ ( 2Y/p₁ -5p₂x₂/p₁)

=> x₂ = p₁Y/p₂ , 0, 2Y/5p₂

maxima will occur for

x₂ = 2Y/5p₂

=> x₁ = 3Y/5p₁

U = (3Y/5p₁)^3 * (2Y/5p₂)^2

U = (108/3125)*Y^5 / p₁³p₂²