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the problem set is mainly about labor’s utility question

problem set is attached below

ECON-UA 351 Ã¢â‚¬â€ Labor Economics
Professor: KatarÃ„Â±ÃŒÂna BorovicÃŒÅ’kovaÃŒÂ
Spring 2021
Problem Set 1
Due on February 15, 4pm ET (via NYU Classes)
to the problem set via the Assignments feature on NYU Classes by the deadline
above.
1
True or false?
For each of the following statements, decide whether it is true or false, and explain your
answer in at most two sentences using the material you learned in the course.
No points will be given without an appropriate explanation!
Question 1.1 Under usual assumption on the utility function, the marginal rate of substitution diminishes as the quantity of leisure increases, holding the utility level constant.
Question 1.2 The marginal utility of leisure is the change in leisure as hours worked
increased, holding consumption fixed.
Question 1.3 If the wage increases from below LarryÃ¢â‚¬â„¢s reservation wage to above LarryÃ¢â‚¬â„¢s
reservation wage, Larry will leave the labor force.
Question 1.4 If workerÃ¢â‚¬â„¢s non-labor income increases, his reservation wage increases as
well.
Question 1.5 Utility-maximizing workers allocate their time so that the last dollar spent
on leisure activities yields the same utility as the last dollar spent on goods.
2
Labor-leisure decision
ShellyÃ¢â‚¬â„¢s preference for consumption and leisure are given by
U (C, L) = (C Ã¢Ë†â€™ 100)(L Ã¢Ë†â€™ 40).
1
There are 110 hours in a week available to split between labor and leisure. Shelly earns \$10
per hour. She also received \$320 worth of assistance benefits each week regardless of how
much she works.
Question 2.1 Graph ShellyÃ¢â‚¬â„¢s budget constraint.
Question 2.2 What is ShellyÃ¢â‚¬â„¢s marginal rate of substitution when L = 100 and she is on
her own budget line?
Question 2.3 What is ShellyÃ¢â‚¬â„¢s reservation wage?
Question 2.4 Find ShellyÃ¢â‚¬â„¢s optimal amount of consumption and leisure.
Question 2.5 The government now imposes tax on labor earnings Ãâ€ž = 0.2. How many
hours will Shelly work in this case? Hint: The proportional tax on earnings means that, if
the wage is w, then the after-tax wage is (1 Ã¢Ë†â€™ Ãâ€ž )w. Solve ShellyÃ¢â‚¬â„¢s problem again with her
after-tax wage.
3
Labor supply curve for different preferences
Derive the labor supply functions that are associated withe the three utility functions given
below. Assume that the individualÃ¢â‚¬â„¢s time endowment is T = 16, non-labor income V = 32
and the price of consumption is one. In each case, plot the labor supply function.
Question 3.1 U (C, L) = 0.2 ln(C) + 0.8 ln(L)
Question 3.2 U (C, L) = 0.4C + 0.6L HINT : This utility function is linear and it violates
the conditions we typically impose on the utility function. In particular, this utility function
does not feature the diminishing marginal utility of consumption and leisure (why?). As a
results, there will not be interior solution.
Question 3.3 U (C, L) = C + 0.8 ln L
2