A.  Production Functions and Chapter 8

Suppose the aggregate production function is Y = F(K, L) = K L1-    where 0 <   < 1.

a.   Find the intensive form of the production function, y = f(k), where y = Y/L and k = K/L Answer:

From Y = F(K, L) = K L1-   , divide by sides by L to get

Y/L = K L-  = K /L = (K/L)

Since y = Y/L and k = K/L we have

y = k = f(k)

b.   Find the marginal product of labor and the marginal product of capital (MPK = ?Y/?K). Use

words to describe the MPL and MPK.

Answer:

MPL = ?Y/?L so from Y = K L1-  we get

MPL = (1 - )K L-  = (1 - )k

From the first equality above the MPL increases with an increase in K meaning workers’ productivity increases if they work with more capital, K.

On the other hand, the MPL falls as L increases, that is, ?MPL/?L < 0, so as labor increases the productivity of labor declines due to crowding.

MPK = ?Y/?K so from Y = K L1-  we get

MPK = K−1 L1-

Noticd that the MPK increases with an increase in L so capital gets more productive if the number of workers using that capital increases. That is, if a given piece of machinery is    used by more workers (perhaps a second shift) it produces more output.

On the other hand, the MPK falls as K increases, that is, ?MPK/?K < 0, so as capital increases the productivity of capital declines due to crowding of production.

c.   Find the marginal product of capital per worker (MPk = ?y/?k) and describe the MPk. Answer:

First use the intensive form of production from part a: y = f(k) = k .

Then MPk = dy/dk = k - 1 .

Since k = K/L an increase in the capital to labor ratio, an increase in k, reduces the MPk, that is, reduces the marginal productivity of capital per worker again due to crowding.    That is, as workers use more and more capital on average, the extra units of capital are   less and less productive.

d.   Show that MPK = MPk.

Answer: compare the results in parts d and e.

In part d we found:

MPK = K−1 L1-  = (K/L)−1   = k−1 where k = K/L

In part e we found:

MPk = k - 1

e.   Suppose  = 1/3. If in a given year population growth is 3% and the capital stock grows 6%,

use the growth trick to show that output grows 4% and output per worker grows 1%. Answer:

From Y = K1/3 L2/3, the growth trick gives gY = 1/3*gK + 2/3gL = 1/3*6%) + 2/3*3%+ 4%. Output per worker is Y/L so the growth trick yields g(Y/L) = gY – gL = 4% - 3% = 1%.

B.   Production Functions and Chapter 8

The aggregate production function is Y = K1/2 L1/2 . If in the Solow model the saving rate is 40%, depreciation is 8% per year and the population grows 2% per year, find output per worker,          capital per worker, and consumption per worker in the steady-state.

Answer:

In the steady state we know, sk = (6 + n)k. Using the givens in the problem we get, k = 16, y = 4 and c = (1 – s)*y = 2.4

C.  Problems and Applications from Chapter 8 in Mankiw

Chapter 8 Problem #1 parts b, c, d

Answer:

b.   To find the per-worker production function, divide Y = K1/3L2/3 by L:

 =

If we define y = Y/L, we can rewrite the above expression as:

y = K1/3/L1/3 = (K/L)1/3

Define k = K/L, and rewrite the above expression as, y = k1/3

c.   We know the following facts about countries A and B:

δ = depreciation rate = 0.20,

sa = saving rate of country A = 0. 1,

sb = saving rate of country B = 0.3, and

y = k1/3 is the per-worker production function derived in part (b) for countries A & B.

The growth of the capital stock k(dot) equals investment sf(k), minus the amount of depreciation 6k. That is, k(dot) = sf(k) – 6k. In steady state, the capital stock does not grow, so we can write   this as sf(k) = 6k.

To find the steady-state level of capital per worker, plug the per-worker production function into the steady-state investment condition, and solve for k* :

sk1/3 = 6k.

Rewriting this:

k2/3 = s/6

k = (s/6)3/2 .

To find the steady-state level of capital per worker k*, plug the saving rate for each country into the above formula:

Country A: k = (sa/6)3/2 = (0. 1/0.2)3/2 = 0.35.

Country B: k = (sb/6)3/2 = (0.3/0.2)3/2 = 1.84.

Now that we have found k* for each country, we can calculate the steady-state levels of income per worker for countries A and B because we know that y = k1/3 :

y*a = (0.35)1/3 = 0.71.

y*b = (1.84)1/3 = 1.22.

We know that out of each dollar of income, workers save a fraction s and consume a fraction (1 – s). That is, the consumption function is c = (1 – s)y. Since we know the steady-state levels of       income in the two countries, we find

Country A: c = (1 – sa)y = (1 – 0. 1)(0.71)

= 0.64.

Country B: c = (1 – sb)y = (1 – 0.3)(1.224)

= 0.86.

d.   If capital per worker is equal to 1 in both countries, we find the following values for income per worker and consumption per worker in each country:

Country A:  y = 1 and c = 0.9

Country B:  y = 1 and c = 0.7.

Chapter 8 Problem #2.

Answer:

a. The production function in the Solow growth model is Y = F(K, L), or expressed in terms of output per worker, y =f(k). If a war reduces the labor force through casualties but the capital stock is unchanged, then L falls and K remains constant so k = K/L rises. The production function tells us that total output falls because there are fewer workers. On the other hand, output per worker, Y/L, increases, since each worker has more capital.

b. The reduction in the labor force means that the capital stock per worker is higher after the war. Therefore, if the economy were in a steady-state prior to the war, then after the war the economy has a capital stock that is higher than the steady-state level. This is shown in Figure 8-2 as an increase in capital per worker from k* to k1 . As the economy returns to the steady-state, the capital stock per worker falls from k1 back to k*, so output per worker also falls.

In the transition to the new steady-state, the growth of output per worker is slower than normal. In the steady-state, we know that without technological change the growth rate of output per worker is zero. So, in this case, the growth rate of output per worker must be less than zero until the new  steady state is reached.

Chapter 8 Problem #3 parts a and b.

Answer:

a.   To find the per-worker production function, we divide Y = F(K, L) = K0.4L0.6 by L:

Y       F (K , L)      (  K   0.4

L =       L       = | L )|    .

Define y = Y/L, k = K/L, and y=f(k), so the per-worker production function can be expressed as y =f(k) = k0.4 .

b. If there is no population growth we have n = 0, so the equation of motion of the capital stock per worker is

k(dot) = sf(k) – 6k.

Steady-state capital per worker k* must satisfy the condition

sf(k*) = 6k*,

or

k*             s

=                .

f (k* )     6

In this example, we solve for steady-state capital per worker as follows: