Question

Suppose the government wishes to tax a utility maximizing consumer to obtain a certain amount of tax revenue. A utility maximizing consumer has utility function u(x,y)= square root(x+y). The price of x is $1, the price of y is $4 and the consumers income is $120.

a) Suppose the government imposes sales tax t=1 on good x per unit. What is the optimal consumption for good x and good y for the consumer under the sales tax? What is the utility level that the consumer achieves under the sales tax? How much revenue does the government collect by imposing the sales tax?

b) Suppose the government imposes income tax instead of sales tax to collect the same amount of revenue that could be earned from imposing sales tax in part (a). What is the optimal consumption for good x and good y for the consumer under the income tax? What is the utility level that the consumer achieves under the income tax? Compare the utility level under the in come tax with the utility level under the sales tax in part (a)

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Answer 1

Concept used: The given function is a monotonic transformation of the utlity function of substitutes. Here, the optimal amount of the two goods is found by comparing the Marginal Rate of Substitution with the price ratio. So the three cases are:

(1) MRS > Px/Py

Here, the consumer spends all of his income on good x. So, x = m/Px and y = 0.

(2) MRS = Px/Py

Here, the budget line coincides with the indifference curve. So the entire budget line shows the optimum combination of goods the consumer can consume.

(3) MRS > Px/Py

Here, the consumer spends all of his income on good y. So, y = m/Py and x = 0.