ECON 5110 Solutions to Problem Set #2

1. Not graded. Undeterminded coeffi cients will not work. Need to use a different solution technique.

2. Consider the production function Y = AK +BL, where A and B are positive constants. Also assume

constant population growth at rate n and constant depreciation at rate . [Barro and Sala-i-Martin,

2004]

(a) Write output per person as a function of capital per person. What is the marginal product of k?

What is the average product of k?

Solution. Output per person is y = Ak +B. The marginal product is A. The average product

is A+B/k.

(b) Write down the fundamental dynamic equation for the Solow model.

Solution. The fundamental dynamic equation is

k(t) = sy(t) (n+ )k(t) = s(Ak(t) +B) (n+ )k(t).

(c) Under what conditions does this model have a steady state with no growth of k, and under what

conditions does the model display endogenous growth?

Solution. The actual investment line is s(Ak(t)+B). The break-even investment line is (n+)k(t).

If (n+ ) > sA, then the model has a steady-state with no growth. If (n+ ) sA, then there

is endogenous growth but the growth rate declines over time.

(d) In the case of endogenous growth, how does the growth rate of the capital stock behave over time

(that is, does it increase or decrease)? What about the growth rates of output and consumption

per capita?

Solution. The growth rate of capital is given by

k =k(t)

k(t)= sA+

sB

k(t) (n+ ).

With endogenous growth, the growth rate of capital decreases over time. The relevant derivative

iskt

= sBk(t)2k(t) = sBk/k(t) < 0.

Output and consumption per capita grow at the same rate, y = c because c(t) = (1 s)y(t).

The growth rates of output and capital are related according to

y =y(t)

y(t)= A

k(t)

k(t)

k(t)

y(t)= A

k(t)

y(t)k.

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(e) If s = 0.4, A = 1, B = 2, = 0.08, and n = 0.02, what is the long-run growth rate of this

economy? What if B = 5? Explain the differences.

Solution. For B = 2, we have

k = 0.3 + 0.8/k(t).

For B = 5, we have

k = 0.3 + 2/k(t).

This suggests that as k(t), the growth rate approaches k = 0.3. The marginal products of

labor (B) has an effect on the growth rate of the economy in the short run, but not the long run.

3. This problem is illustrates the crucial role played by the assumption of constant returns in new growth

theories. The framework is one of a simple neoclassical model with a constant saving rate. The

production function for firm i is

yi(t) = A(t)ki(t)

where 0 < < 1, A(t) = 1NNi=1 ki(t), and N is the number of firms. Suppose that s is the constant

saving rate, n is the constant population growth rate, and is the rate of depreciation of physical

capital.

(a) Find the differential equation for k when all firms are identical.

Solution. The differential equation is

k = sk(t)+ (n+ )k(t).

The growth rate of capital is

k = sk(t)+1 (n+ ).

(b) Represent graphically the solutions to the model for the cases where the production function

exhibits (i) diminishing returns to scale, + < 1, (ii) constant returns to scale, + = 1, and

(iii) increasing returns to scale, + > 1. What is meant by the "knife-edge" property of the

AK model? Explain.

Solution. The "knife-edge" property refers to the idea that long-run growth is only possible if

+ = 1. Any deviation from this condition sends the economy to either zero or infinite growth.

(c) Examine the effect on the long-run growth rate of a change in the saving rate for each of the three

cases.

Solution. A change in the saving rate will have no effect on long-run growth in either the

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decreasing or increasing returns case (i.e., + < 1 or + > 1). However, with constant

returns to scale, a change in the saving rate will direct impact on the long-run growth rate.

(d) Consider the effect of a once-off shock. Suppose an earthquake destroys half the capital stock of

the economy. Examine what happens in each of the three cases, both in the short run and long

run.

Solution. Three cases:

i. Decreasing Returns (+ < 1). The is the standard Neoclassical case and hence the standard

results apply. A one-time reduction in the capital stock per worker causes a higher growth

rate temporarily as k(t) converges back to the pre-earthquake steady state. Hence, the level

of income returns back to its original level and all effects are temporary.

ii. Constant Returns ( + = 1). This is the AK model. Since k is constant at s n ,

a one-time reduction in the capital stock does not affect the long-run growth rate of the

economy. However, since the growth rate is unaffected, this does imply that as a result of

the earthquake the capital stock and level of income are permanently lower than they would

have been otherwise.

iii. Increasing Returns ( + > 1). The reduction in the capital stock for this case implies

that the growth rate will be reduced immediately after the shock. If the economy was on

a path toward infinite growth a suffi ciently large shock could push it toward a zero-growth,

zero-capital path; otherwise it would continue toward infinite growth. If the economy was

heading toward zero growth, the shock would only speed the process up. In sum, there are

likely to be no permanent effects.

4. Rework the Green Solow model assuming AK growth. Specifically, how does this effect Figure 4 and

the test for conditional emission convergence?

Solution. With AK growth, the key dynamic equation is

k =k

k= s(1 ) ( + n+ gB).

Assuming s(1 ) > ( + n+ gB), the Green Solow model does not have a steady state or transition

dynamics. Capital per effective worker, k, grows continually at rate k. This implies that output per

worker, Y/L, grows at rate

Y/L = k + gB = s(1 ) ( + n).

Emissions will grow at rate

gE = Y gA.

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Figure 4 shows that if gE < 0 and Y/L > 0 (i.e., growth is sustainable), then emissions will grow

initially and then fall continuously if k(0) < k(T ). This generates the EK curve because initially

the marginal product of capital is high relative to the growth in abatement. New emissions exceed

abatement and the EKC curves up. As the marginal product of capital falls, eventually abatement

exceeds new emissions and the EKC slopes down. With AK growth, there are no transition dynamics.

If gE > 0, then emissions continually rise. If gE < 0, then emissions continually fall. If Y = gA,

then emissions are flat.

There is no income convergence in the AK growth model. As a result, there will be no convergence in

emissions and 1 = 0 from the estimating equation.

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