Question

Consider the consumer’s optimization problem in the one-period model. Utility is:

u ( c , l ) = ln ⁡ ( c ) + γ ln ⁡ ( l )

where c is consumption of goods, l is leisure, and γ is a constant. The consumer’s nominal budget constraint is:

P c = W ( h − l )

where P is the price of goods; W is the nominal wage; and h is the total hours in the period. Note that the consumer does not receive any nominal profits and, for now, she does not pay any taxes.

a). Write the Lagrangian and take the first order conditions. Derive the optimality condition that relates the marginal rate of substitution to the relative price of consumption and leisure.

b). Consider the case in which the government decides to impose a lump-sum tax (T) on the consumer. Her nominal budget constraint becomes:

P c = W ( h − l ) − T

Use a diagram to show how this change affects the consumer’s optimal choice of consumption and leisure. Your diagram should have leisure on the x-axis and consumption on the y-axis. Be sure to label the old and new optimal points clearly. What explains the change in consumer behavior?

c). Now instead of a lump-sum tax, the government imposes a consumption tax. The consumer’s nominal budget constraint becomes:

P ( 1 + τ c ) c = W ( h − l )

where τ c > 0 is the consumption tax rate so that if the consumer wishes to consume c, she must pay P ( 1 + τ c ) c . What is the slope of the new budget line? Use a diagram to show how the consumer’s optimal choice of consumption and leisure changes relative to the no-tax case. What explains the change in consumer behavior?

Answer 1