Download Exam #1 (answers) - D. Mark Anderson

Exam #1 (answers)
ECNS 303
February 19, 2015
Name________________________
You may use a calculator and scratch paper for this exam, but nothing else!
1.) (10 points) The conventional approach for modeling the demand for addictive substances
posits an economic agent that maximizes the following utility function subject to a budget
constraint
U = f(C, X)
where C represents the consumption of an addictive substance and X represents the consumption
of a basket of other goods. This utility maximization problem produces the following demand
function
C = g(P, Y, Z)
where P is the price of the addictive substance, Y is income, and Z is a vector of variables
reflecting tastes.
What are the exogenous and endogenous variables in the demand function for the addictive
substance?
Endogenous variable: C
Exogenous variables: P, Y, and Z
2.) (20 points total) Consider a country that produces only three goods: ice axes, ice screws, and
crampons. Sales and price data for these three products for three different years are as follows:
# of pairs
price per
#ice axes
price per
#ice screws price per
of crampons pair of
Year sold
ice axe
sold
ice screw
sold
crampons
1990 50
$100
100
$25
40
$100
2000 80
$200
150
$50
80
$120
2010 100
$250
220
$60
100
$180
a.) (3 points) Calculate nominal GDP in 1990 and 2010.
Nominal GDP in 1990 = (50*100) + (100*25) + (40*100) = $11,500
Nominal GDP in 2010 = (100*250) + (220*60) + (100*180) = $56,200
b.) (4 points) Calculate real GDP in 2010 using 2000 as the base year.
Real GDP in 2010 (2000 base yr.) = (100*$200) + (220*$50) + (100*$120) = $43,000
c.) (4 points) Calculate the GDP deflator in 2010 (use 2000 as base year).
GDP deflator = Nominal GDP in 2010/Real GDP in 2010 = $56,200/$43,000 = 1.307
d.) (4 points) Calculate the CPI in 1990 using 2000 as the base year.
{($100*80) + ($25*150) + ($100*80)}/{($200*80) + ($50*150) + ($120*80)} = 19750/33100
= 0.597
3.) (20 points total) Assume we have an economy where production can be described by the
following function form
F(K,L) = [K2 + L2]1/2
where K is capital and L is labor.
a.) (7 points) Does this function exhibit the property of constant returns to scale? Make sure to
show your work! No work, no points.
Constant returns to scale requires that zY = F(zK, zL)
Here, we see that
F(zK, zL) = [(zK)2 + (zL)2]1/2 = [z2(K2 + L2)]1/2 = z(K2 + L2)1/2
Thus, it DOES exhibit the property of constant returns to scale.
b.) (6 points) Solve for the marginal product of labor and for the marginal product of capital.
MPL = L/(K2 + L2)1/2
MPK = K/(K2 + L2)1/2
c.) (7 points) Does this function exhibit the property of diminishing marginal returns to capital?
Make sure to show your work mathematically! No work, no points.
Via the quotient rule,
∂MPK/∂K = [(K2 + L2)1/2 – K2(K2 + L2)-1/2]/(K2 + L2) = L2/(K2 + L2)3/2 > 0
This function does not have diminishing marginal returns to capital. It has increasing marginal
returns to capital.
4.) (15 points) Suppose output in the economy is described by the following Cobb-Douglas
production function:
Y = K1/2L1/2
Also, suppose that 20% of output is saved each year, 5% of capital depreciates each year, and the
economy starts with 16 units of capital per worker.
How much capital per worker exists in this economy at the beginning of the second year? What
is the steady-state level of capital per worker?
First, we solve for the per-worker production function by dividing both sides of the above
equation by L
Y/L = (K/L)1/2
Which we can rewrite as
y = (k)1/2
Substituting in for k
y = (16)1/2 = 4 (16 units of capital per work produces 4 units of
output per worker)
Given that 20% of output is saved and invested each year, we know that
i = (0.2)(4) = 0.8
(0.8 units of output per worker is saved and
invested each year)
c = (0.8)(4) = 3.2
(3.2 units of output per worker is consumed
each year)
Given that 5% of capital stock depreciates each year, we know that
δk = (0.05)(16) = 0.8 (0.8 units of capital per worker depreciates
each year)
So, we can solve for the change in the capital stock from this first year to the second year
Δk = sf(k) – δk = 0.8 – 0.8 = 0
As a result, the amount of capital per worker that exists in this economy at the beginning of the
second year is
k stock at start of year 2 = (k stock at start of year 1) + Δk
= 16 + 0
= 16 We were already in the steady-state!!!
5.) (10 points) Suppose the country of Andersonland is described by the Solow model with
population growth and technological progress. In the country of Andersonland, the growth rate
of output is .05 per year, the depreciation rate is .10 per year, and the capital-output ratio is 4.
Suppose Andersonland is currently in a steady state. Calculate the savings rate in this steady
state. (Hint: recall that the growth rate of output is equivalent to the sum of the population and
technological progress growth rates)
In the steady state, we know that
Δk = sf(k) – (δ+n+g)k = 0
Solving for the savings rate, we obtain the following
sf(k) = (δ+n+g)k
 s = [(δ+n+g)k]/f(k)
because δ=.10, n+g = .05, and k/y = 4, we can make the following substitutions
s = [(.1 + .05)k]/y = (.1 + .05)(4) = .60
 s = .6 (or a savings rate of 60%)
6.) (15 points total) Suppose the following national income per capita equation describes the
economy
y=c+i
a.) (7 points) In a model that includes population growth and technological progress, solve for
the condition that describes the Golden Rule.
We know that the Golden Rule steady-state is the particular steady-state where consumption is
maximized. We also know that our steady-state level of consumption is described by the
following modification of our national income equation
c* = f(k*) – (δ + n + g)k*
To solve for the level of c* that maximizes the above equation, we take the derivative of c* with
respect to k*
dc*/dk* = df(k*)/dk* - (δ + n + g) = 0
=>
MPK = δ + n + g
b.) (8 points) Suppose you are a policymaker that is in charge of setting a savings rate that
maximizes steady-state consumption per worker. Show graphically the savings rate you would
choose.
n, g, δ
(δ + n + g)k*
f(k*)
c*Golden Rule
sf(k*)
k*Golden Rule
Choose the savings rate, s, above that maximizes the distance between f(k*) and (δ + n + g)k*
k*
7.) (15 points total) Note: this problem is difficult, so allocate your time appropriately
before trying to answer this one.
Consider how unemployment would affect the Solow growth model. Suppose that output is
produced according to the production function
Y = Kα[(1 – u)L]1-α
where K is capital, L is the labor force, and u is the natural rate of unemployment. The national
saving rate is s, the labor force grows at rate n, and capital depreciates at rate δ.
a.) (10 points) Express output per worker (y = Y/L) as a function of capital per worker (k = K/L)
and the natural rate of unemployment. Also, solve for the steady-state values of k and y.
To find output per worker we divide total output by the number of workers:
Y/L = {Kα[(1 – u)L]1-α}/L
y = (K/L)α(1-u)1-α
y = kα(1-u)1-α
Notice that unemployment reduces the amount of output per worker for any given
capital-labor ratio because some of the workers are not producing anything.
Our steady-state level equation looks like what we are used to:
sy = (δ+n)k
Plugging in for y:
skα(1-u)1-α = (δ+n)k
k* = (1-u)(s/(δ+n))1/(1-α)
Unemployment lowers the marginal product of capital per worker and, hence, acts like a
negative technological shock that reduces the amount of capital the economy can maintain in
steady state.
Finally, to get steady-state output per worker, plug the steady-state level of capital per
worker into the production function:
y* = ((1-u*)(s/(δ+n))1/(1-α))α(1-u*)1-α
= (1-u*)(s/(δ+n))α/(1-α)
Unemployment lowers steady-state output for two reasons: for a given k, unemployment
lowers y, and unemployment also lowers the steady-state value k*.
b.) (5 points) Suppose that some change in government policy reduces the natural rate of
unemployment. Describe how this change affects output both immediately and over time. You
may use a graph (although it is not necessary for this problem) to support your answer.
As soon as unemployment falls from u1 to u2, output jumps up from its initial steady-state value
of y*(u1). The economy has the same amount of capital (since it takes time to adjust the capital
stock), but this capital is combined with more workers. At that moment the economy is out of
steady-state: it has less capital than it wants to match the increased number of workers in the
economy. The economy begins its transition by accumulating more capital, raising output even
further than the original jump. Eventually the capital stock and output converge to their new,
higher steady-state levels.