Econ23950 homework 1
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Econ23950 homework 1
1. (15 points) The representative firm has a production function Y = zL, where z is the labor productivity. The representative household has a utility function u(c,h) = lnc + 2lnh. Let w denote the real wage rate.
(a) (1 point) Suppose that z = 100. What is the representative firm’s labor demand
function? What must the wage rate be at the equilibrium? At this wage rate, how much profit can this firm send to the representative household?
(b) (2 points) With this non-wage income, what is the household’s labor supply
curve?
(c) (2 points) Define and solve the competitive equilibrium.
(d) (5 points) We will introduce government into this economy. Part of govern- ment spending substitutes for private consumption. For examples, Head Start vs. private preschool, Medicare vs. private health insurance, and etc. Govern- ment spending, g, enters the utility function of the representative household as the following:
u(c,h;g) = ln(c + ηg) + 2ln(h)
Suppose that η = 0.4. This means that 40% of government spending is a direct substitute for private consumption. The parameter η reflects both substitutability and efficiency of publicly provided goods and services.
Suppose that its government collects a lump-sum tax to finance its spending. Define and calculate the competitive equilibrium. What is the fiscal multiplier? Does the multiplier increase or decrease with η? Explain your finding.
(e) (5 points) Suppose that its government collects a flat rate labor income tax
with a tax rate, τ to finance spending. Define and calculate the competitive equilibrium. Is the fiscal multiplier still positive in this case?
2. (15 points) In country RBC only labor is used in production and Y = zL, where z is the labor productivity. Economic booms and recessions are due to cyclical movements of labor productivity. For simplicity, suppose that z takes two values: either z = 95 or z = 105. The chance that z = 105 is 0.8 and the chance that z = 95 is 0.2. Roughly country RBC is in economic boom 80% of time and recession 20% of the time. The representative household in country RBC has a preference u(c,h) =^c + 14^h
(a) (6 points) Compute the competitive equilibriums in country RBC in booms and
in recessions. Is the drop of output bigger or smaller than the drop productivity when comparing a recession period with a boom period? Why? What about the magnitude of changes in employment and consumption?
(b) (3 points) The government in country RBC wants to use government spending to
reduce business cycle fluctuations. The government can collect a lump-sum tax to finance its spending. The objective of the government is to pick a spending level, g, during recessions so that the output in recessions is the same as the output in booms. The spending will be thrown in the ocean (wasted).
Show that g = 8.3355 during recessions will roughly achieve the goal.
(c) (3 points) The above stabilization policy leads to a waste of resources, hence may not be favored by households. Instead the government stabilizes production through wage subsidies financed by a lump-sum tax, which is equivalent to a pay- roll tax cut financed with debt. The representative household’s budget constraint becomes the following:
c = w(1 + s)(1 − h) − T.
Show that s = 13.78% and T = 5.0474 during recessions will achieve the goal of stable output as well.
(d) (3 points) Let (cb ,lb ) denote the equilibrium consumption and leisure during booms and (cr ,lr ) those during recessions. The welfare of the representative household is given by the following:
welfare = 0.8 · u(cb ,lb ) + 0.2 · u(cr ,lr )
Compute the welfare in country RBC without stabilization fiscal policy, that with increased government spending during recessions, and that with wage subsidy during recessions. What do you find? Were you surprised by this result? Why and why not?
2022-10-28