Cyprian is a Grenadian. His preferences are described by the utility function u(x, y) = ln(k)

Cyprian is a Grenadian. His preferences are described by the utility function

u(x, y) = α ln(k) + β ln(y),                                         (1)

where k is the number of Kit Kats and y is spending on everything else. He’s a loyal citizen and pays his taxes. Let’s consider that since y represents all other spending, the price of y is normalised to 1. Cyprian’s income is fixed at m.

(a) Write Cyprian’s budget constraint.

(b) Solve for Cyprian’s optimal choice of k and y.

(c) The Grenadian government imposes a tax on the consumption of Kit Kats. For every Kit Kat he consumes, he must pay tk dollars in tax in addition to the price pk.

i. Write Cyprian’s new budget constraint.

ii. Solve for Cyprian’s new optimal choices of k and y given the new budget constraint.

iii.  What is the difference between Cyprian’s optimal choices with and without the tax?

(d) In addition to being taxed, Cyprian now gets a lump-sum rebate of T from the government.

i. Write his new budget constraint.

ii. What are his new optimal choices?

iii. What is the difference between Cyprian’s optimal choices with and without the rebate?

iv. In words, describe what the government has done graphically to Cyprian’s budget line by imposing the tax and giving him a rebate.

v. Suppose the rebate Cyprian gets is exactly the same amount of money he paid in taxes. Would he choose the optimal bundle that he chose before either taxes or the rebate were imposed? Why?

(e) Assume α = 0.5 and β = 0.5. Choose random values for pk, tk, and m. Ignore the rebate. Using Cyprian’s demand functions:

i. Calculate the amount of money you’d need to compensate Cyprian to hold his purchasing power constant when the tax is imposed.

ii. Calculate the substitution effect of the tax on the demand for Kit Kats.

iii. Calculate the income effect of the tax on the demand for Kit Kats