Econ303 Homework 6B 1 1. (15 points) The representative ﬁrm has production function Y = zN , where z is labor
productivity. The representative household has utility function u(c, l) = c + 18 l.
Let w denote the real wage rate.
(a) (2 points) Suppose that z = 200. 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 household?
(b) (3 points) Given this wage rate and non-wage income, what is the optimal time
allocation and consumption for the representative household?
(c) (1 point) At the equilibrium, how big is labor input and how much output is
(d) (3 points) Now the productivity drops. z = 190. At the new equilibrium how
big is labor input and how much output is produced?
(e) (1 points) Calculate the percentage decrease of productivity, labor input and
output? (% change = Af ter−Bef ore × 100%) Is the percentage decrease in output
bigger or smaller than the percentage decrease in productivity? Can you explain
(f) (3 points) Anticipating the productivity drop, government implements a stimulus
package by increasing the government spending from 0 to 8, which is ﬁnanced by
borrowing (lump-sum tax). With this stimulus package, how big is labor input
and consumption? How much output is produced?
(g) (1 point) Compared with the high productivity case, calculate the percentage
change of labor input, output, and consumption under lower productivity with
(h) (1 point) Compare the changes of labor input, consumption, and output with or
without stimulus package. What do you ﬁnd?
2. (5 points) The representative ﬁrm has production function Y = 30N . Part of government spending can be a perfect substitute for private consumption. Examples are
Head Start and private preschool, Medicare and private health insurance, and etc.
Government spending g enters the utility function of the representative household as
u(c, l; g) = ln(c + 0.5g) + 2.5 ln(l)
Base on the above utility function, 50% of government spending is a direct substitute
for private consumption. In this case, M RSl,c = 2.5(c+0.5g) .
Suppose that government spending g = 1.
(a) (2 points) Suppose that government collects lump-sum tax, where T = 1, to
completely ﬁnance its spending. What is household’s optimal time allocation?
How much is produced and how much is consumed? Calculate the welfare of
household as well. Econ303 Homework 6B 2 (b) (2 points) Suppose that government collects ﬂat rate labor income tax with tax
rate τ = 10%. If tax revenue is not enough to cover its spending, government will
borrow abroad. If tax revenue is more than its needs, government lends out to
foreigners. What is household’s optimal time allocation in this case? How much
is consumed and how much is produced? Calculate the welfare of household in
(c) (1 point) Does government need to borrow abroad? If so, how much? In this
case, is it still true that c + g = Y ?