Macroeconomic Analysis

ECON 6022A

Fall 2011

Solutions to Problem Set 3

October 21, 2011

1 Golden rule saving rate in the Solow model

Consider a Solow economy which is identical to the one we studied in Lecture 5. The production function

is Cobb-Douglas, Y = A ·Kα ·N1−α. The saving rate is s, population growth rate n and depreciation rate

d.

1. Derive the steady-state level of capital-labor ratio, k∗, as a function of s, n and d.

2. Derive the steady-state level of consumption, c∗, as a function of k∗.

3. Derive the golden rule steady-state capital-labor ratio, k∗G, at which the maximum of steady state

level of consumption is achieved.

4. Derive the golden rule saving rate, sG so that k∗G is the steady state level of capital-labor ratio in

this economy (and the maximum of steady state level of consumption is achieved).

Solution:

1. The production function in per worker term is y = A · kα. In steady-state we have

∆k = s ·A · kα − (n+ d) · k = 0.

Thus, the steady-state level of capital-labor ratio

k∗ = (sA

n+ d)

11−α .

2. The steady-state level of consumption,

c∗ = A · k∗α − (n+ d) · k∗

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3. To achieve the maximum of steady-state consumption, we derive the first order condition,

αA · k∗(α−1) = n+ d.

Therefore, the golden rule steady-state capital-labor ratio,

k∗G = (αA

n+ d)

11−α .

4. We can choose saving rate s, such that

(sA

n+ d)

11−α = (

αA

n+ g)

11−α .

Thus, the golden rule saving rate is

sG = α.

2 Shocking the Solow Model

In Lecture 5, we show a case where there is an one-time increase in the saving rate. Suppose that the saving

rate is constant over time and the depreciate rate is d1. There is an one-time decrease in the depreciation

rate at t = 0. The new depreciation rate is d2 and d1 > d2. Before the shock, the economy is on its steady

state.

1. Show how the economy converges to a new steady-state in a Solow Diagram.

2. (Optional) Can you plot the trajectories for output per worker, y(t), aggregate output, ln(Y (t)),

before and after time t = 0.

Solution:

1. See Figure 1.

2. See Figure 2 for y(t) and Figure 3 for ln(Y (t)).

3 Growth accounting

The production function is Yt = A·Kαt ·N1−α

t , where α = 0.3. The growth rates of the inputs are ∆KK = 3%

and ∆NN = 1%.

1. Suppose A does not grow. Find the growth rate of Y .

2. Suppose ∆AA = 1.4%. Find the growth rate of Y and y, where y = Y

N

3. (optional) Is the ratio KY constant in this economy, given the assumption in subquestion 2? How

about the ratio of aggregate consumption (or investment) to output, CY (or I

Y )?

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Figure 1:

Figure 2:

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Figure 3:

4. (optional) Let B = A1

1−α and the production function can be rewritten as Yt = Kαt · (Bt · Nt)1−α.

We define another variable, capital per effective unit of worker, k = KB·N . Find the growth rates of

k, capital per worker k and B.

5. (optional) Is this economy at the steady state? (Hint: think of a proper definition of steady state.)

Solution:

1. ∆YY = α · ∆K

K + (1 − α) · ∆NN = 0.3 × 3% + 0.7 × 1% = 1.6%.

2. ∆YY = ∆A

A + α · ∆KK + (1 − α) · ∆N

N = 1.4% + 0.3 × 3% + 0.7 × 1% = 3%.

∆yy = ∆Y

Y − ∆NN = 3% − 1% = 2%.

3. As K and Y grow at the same rate, 3%, KY is constant.

Since ∆K = I − d ·K = Y − C − d ·K, we get ∆KK · KY = 1 − C

Y − d · KY . Given KY is constant, C

Y is

also constant. Since IY = Y−C

Y = 1 − CY , the investment to output ratio is also constant.

4. Growth rate of B, ∆BB = 1

1−α · ∆AA = 1

0.7 × 1.4% = 2%.

Growth rate of k, ∆kk

= ∆KK − ∆B

B − ∆NN = 3% − 2% − 1% = 0.

Since k = k ·B, growth rate of k, ∆kk = ∆k

k+ ∆B

B = 0 + 2% = 2%.

5. This economy is at the steady state in the sense that the capital per effective unit of worker, k, and

the output per effective unit of worker, y = kα, are all constant. This situation is called “balanced

growth ” . The key feature is that the capital-output ratio is constant in this economy. In the version

of Solow model without productivity growth (the one in Lecture 5), the capital per worker is constant

in steady state, and the constant capital-output ratio is also implied. It is fairly easy to see.

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