Accelerated Macro Spring 2015
Solutions to HW #2
1
Money demand and the velocity of money
ABC Ch. 7 NP #2
Money demand in an economy in which no interest is paid on money is
Md
P
= 500 + 0.2Y − 1000i.
a. Suppose that P = 100, Y = 1000, and i = 0.10. Find real money demand, nominal money demand,
and the velocity of money.
Solution: Real money demand is
Nominal money demand is
Md
P P
Md
P
= 500 + 0.2Y − 1000i = 500 + 0.2(1000) − 1000(0.10) = 600
= 600(100) = 60, 000
d
Velocity of money is V = P Y /M = 100(1000)/60, 000 = 5/3
b. The price level doubles from P = 100 to P = 200. Find real money demand, nominal money
demand, and the velocity of money.
Solution: Neither income Y nor the interest rate i have changed, so real money demand is
unchanged.
Nominal money demand is
Md
P P
= 600(200) = 120, 000
Velocity of money is V = P Y /M d = Y /(M d /P ), and as neither income Y nor real money demand
Md
P have changed, the velocity of money is unchanged.
c. Starting from the values of the variables given in Part (a) and assuming that the money demand
function written holds, determine how velocity is affected by an increase in real income, by an
increase in the nominal interest rate, and by an increase in the price level.
Solution: V =
Y
M d /P
=
Y
500+0.2Y −1000i
Y
Y
• An increase in real income: With i = 0.10, V = 500+0.2Y −1000(0.10)
= 400+0.2Y
.
dV
10000
dY = (Y +2000)2 > 0, so when Y increases, the velocity of money increases.
1000
• An increase in the real interest rate: With Y = 1000, V = 500+0.2(1000)−1000i
=
dV
100
=
>
0,
so
when
i
increases,
the
velocity
of
money
increases.
di
(7−10x)2
1000
700−1000i .
• An increase in the price level: There is no effect on velocity, since we can write velocity as a
function just of Y and i. Nominal money demand changes proportionally with the price level,
so that real money demand, and hence velocity, is unchanged.
2
Log-linearization and elasticities
Log-linearize the consumption Euler equation you derived in HW #1 Question 5.b.
1− 1
1− 1
(for the utility form U (ct , ct+1 ) =
ct σ
1
1− σ
+β
ct+1σ
1
1− σ
)
After log-linearization, one of the percentage deviation terms should have a coefficient. What is the
interpretation of this coefficient as an elasticity?
Solution: Taking natural logs of our Euler equation, ct = ct+1 (β(1 + rt ))−σ , yields
ln(ct ) = ln(ct+1 ) − σln(β) − σln(1 + rt ) ⇐⇒
ln(c∗t ) +
∆ct
ct
= ln(c∗t+1 ) +
∆ct+1
ct+1
∆ct
ct
t
= σ ∆1+r
1+rt
−
∆ct+1
ct+1
t
− σln(β) − σln(1 + rt∗ ) − σ ∆1+r
⇐⇒
1+rt
σ is interpreted as the elasticity of intertemporal substitution, which measures how responsive consumption growth is to changes in interest rates.
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3
The Solow model: shocks to the steady state
ABC Ch. 6 AP #1
According to the Solow model, how would each of the following affect consumption per worker in the
long run (i.e., in the steady state)? Explain.
a. The destruction of a portion of the nation’s capital stock in a war.
Solution: “The destruction of some of a country’s capital stock in a war would have no effect
on the steady state, because there has been no change in s, f (), n, or δ. Instead, k is reduced
temporarily, but equilibrium forces eventually drive k to the same steady state value as before.”
b. A permanent increase in the rate of immigration (which raises the overall population growth rate).
Solution: Immigration raises n, which lowers steady state k, leading to lower steady state output
and thus consumption per worker. (See Figure 6.3.)
c. A permanent increase in energy prices.
Solution: The rise in energy prices reduces the productivity of capital per worker (i.e., reduces
Total Factor Productivity, A). This causes sf (k) to shift downward, resulting in a decline in
steady state k. Steady state consumption per worker falls beacuse the decline in productivity
reduces output for any fixed factor level, but the steady state k is also reduced, further reducing
output and consumption. (See Figure 6.4.)
d. A temporary rise in the saving rate.
Solution: “A temporary rise in s has no effect on the steady state equilibrium.”
e. A permanent increase in the fraction of the population in the labor force (the population growth
rate in unchanged).
Solution: “The increase in the size of the labor force does not affect the growth rate of the labor
force, so there is no impact on the steady state capital-labor ratio. However, because a larger
fraction of the population is working, output and consumption per person increases.”
4
Per-worker production function
Based on ABC Ch. 6 NP #5
An economy has the per-worker production function yt = 3kt0.5 where yt is output per worker and kt is
the capital-labor ratio. The depreciation rate is 0.1, and the population growth rate is 0.05. Savings is
St = 0.3Yt , where St is total national savings and Yt is total output.
a. What are the steady state values of the capital-labor ratio, output per worker, and consumption
per worker?
Solution: sf (k) = (n + δ)k ⇐⇒ 0.3(3k 0.5 ) = (0.05 + 0.1)k ⇐⇒ 6 = k 0.5 ⇐⇒ k ∗ = 36
y ∗ = 3(k ∗ )0.5 = 3(6) = 18
c∗ = y ∗ − (n + δ)k ∗ = 18 − (0.15)36 = 12.6
The rest of the problem shows the effects of changes in the three fundemental determinants of
long-run standards of living.
b. Repeat Part (a) for a savings rate of 0.4 instead of 0.3. Explain why the steady state capital-labor
ratio is changing the way it is.
Solution: sf (k) = (n + δ)k ⇐⇒ 0.4(3k 0.5 ) = (0.05 + 0.1)k ⇐⇒ 8 = k 0.5 ⇐⇒ k ∗ = 64
y ∗ = 3(k ∗ )0.5 = 3(8) = 24
c∗ = y ∗ − (n + δ)k ∗ = 24 − (0.15)64 = 14.4
When the savings rate rises, more capital is accumulated, so K/N ↑; Y /N is increasing in K/N ,
so Y /N ↑; C/N is increasing linearly in Y /N , so C/N ↑.
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c. Repeat Part (a) for a population growth rate of 0.08 (with a saving rate of 0.3). Explain why the
steady state capital-labor ratio is changing the way it is.
Solution: sf (k) = (n + δ)k ⇐⇒ 0.3(3k 0.5 ) = (0.08 + 0.1)k ⇐⇒ 5 = k 0.5 ⇐⇒ k ∗ = 25
y ∗ = 3(k ∗ )0.5 = 3(5) = 15
c∗ = y ∗ − (n + δ)k ∗ = 15 − (0.18)25 = 10.5
When the population growth rate rises, a larger workforce dilutes the capital stock per worker, so
K/N ↓; Y /N is increasing in K/N , so Y /N ↓; C/N is increasing linearly in Y /N , so C/N ↓ .
d. Repeat Part (a) for a per-worker production function yt = 4kt0.5 . Assume the saving rate and population rate are at their original values. Explain why the steady state capital-labor ratio is changing
the way it is.
Solution: sf (k) = (n + δ)k ⇐⇒ 0.3(4k 0.5 ) = (0.05 + 0.1)k ⇐⇒ 8 = k 0.5 ⇐⇒ k ∗ = 64
y ∗ = 4(k ∗ )0.5 = 4(8) = 32
c∗ = y ∗ − (n + δ)k ∗ = 32 − (0.15)64 = 22.4
When productivity rises, more output is produced from the given factor levels. Savings and consumption are linear function of Y /N , so C/N ↑ and S ↑. The increase in savings will increase the
capital stock per worker, so K/N ↑.
Can you back out the original Cobb-Douglas production function from the given per-worker production
function yt ?
Solution: yt = 3kt0.5 ⇐⇒
Yt
Nt
t 0.5
= 3( K
⇐⇒ Yt = 3Kt0.5 Nt0.5
Nt )
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