In this question we will study how workers change their labor supply. Imagine there is a
consumer/worker with preference over consumption C and leisure & given by the equation below:
U(C,) = log(C) + log(l)
1. Assume the consumer faces a wage rate of w and consumption price P. She
also has one unit of available time to spend working or resting. Solve for the consumer's
optimal choice of consumption and hours worked.
2. Now assume a pandemic hits the economy and the consumer receives more
utility from leisure. We model this by changing the preferences to:
U(C,) log(C) + 0 log(l)
0
with > 1. Solve for the hours worked and consumption under these new preferences.
Compute the elasticity of labor supply and consumption with respect to real wages. Compare
your answer with the previous part (when 01). Does the consumer wants to work more or
less?
3. The government wants to increase hours worked, and so it enacts a law
that includes a subsidy on wages. This subsidy is proportional to labor income. Explain why
the new budget constraint can be expressed as: PC = (1+7)w(10), where is the subsidy
(with 7 > 0). Solve for the optimal hours worked and consumption and compare your answer
with that of the previous question. What is the effect of this subsidy on hours worked?
4. Now the government wants to try a different fiscal policy. Instead, it creates
a new lumpsum tax that the consumer has to pay regardless of income or consumption.
Explain why the new budget constraint can be written as: PC w(1)T, where T
is the new lumpsum tax. Assume 0
consumption. Solve for hours worked and consumption. Compare with part 2.

5. Imagine the government wants to choose the tax amount T such that the
worker supplies exactly the same number of hours worked as in part 1. Find this tax amount.
Assume < 1+0. Is the consumer better off? Relate your answer to the First Welfare
Theorem, assuming the given prices clear markets in a competitive equilibrium.