Question

Consider a representative consumer who has preferences over an aggregate consumption good c and leisure l. Her preferences are described by the utility function: U(c,l) = ln(c) + ln(l) The consumer has a time endowment of h hours which can be used to work at the market or enjoyed as leisure. The real wage rate is w per hour. The worker pays a proportional wage tax of rate t, so the worker’s after-tax wage is (1−t)w. The consumer also has dividend income of π. (Note that there is no lump-sum tax here.) (a) Set up the consumer’s utility maximization problem. (b) Analyze what would happen to consumption and leisure if the tax rate t increased: (a) show the effect of an increase in t on a graph. (b) Explain the economic intuitions for your results

Answer 1

Solution:

1.a):

Utility function: U=lnC+lnL

Budget constraint: C+L=(1-t)w+Pi

The consumer’s utility maximization problem will look like as follows:

Maximize Utility lnC+lnL

Such that C+L=(1-t)w+Pi

The lagrange will be: L’=lnC+lnL+D[(1-t)w+Pi]

Where, D is lamda

b)

Calculate the first order conditions and put them equal to 0 to get the optimal values of C and L

An increase in the tax rate will affect both consumption and leisure through income and substitution effect.

Economic intuition: Income effect will make the income less for the consumer because of increased tax, thereby encouraging consumer to work more.

Substitution effect will make consumer substitute work for leisure, as the opportunity cost of work will increase.