Question

Stolper-Samuelson Suppose that the economy produces two goods, 1 and 2 and that the production functions of sector 1 and 2 are respectively given by Q = K^0.3L^0.7 and Q = K^0.6L^0.4. Answer the following questions.

1. (a) Determine which of the goods is capital intensive.

2. (b) Derive the corresponding unit cost functions.

3. (c) If the prices of goods 1 and 2 are 1 and 1, make use of the unit cost functions to derive the wage and rental rates, assuming that both goods are produced. Show roughly the two unit cost schedules in a suitable diagram.

4. (d) If the price of good 1 increases to 2 while that of good 2 remains constant, derive the new wage and rental rates. By comparing your results in (c) and (d), argue that the Stolper-Samuelson theorem holds.

Answer 1

Solution:

Production function for good 1: Q = K^0.3L^0.7

Production function for good 2: Q = K^0.6L^0.4

a) Observing the production functions for the two gods, it is easy to notice that capital intensity is higher for good 2 (0.6 > 0.3). This is because the power to factor K, i.e., capital, tells the share of capital in production of 1 unit of output of a good (and correspondingly, the power to factor L, i.e., labor, tells the share of labor in production of 1 unit of output of a good).

b) Deriving unit cost functions:

The total cost, C = w*L + r*K, where w is wage rate and r is rental rate.

The above cost function is function of factors; in order to derive unit cost, we mean to derive this cost function as a function of the units produced, i.e, Q.

The cost minimizing condition is MPL/MPK = w/r, where MPL is marginal product of labor, calculated as $\partial Q/\partial L$ and MPK is the marginal product of capital, calculated as $\partial Q/\partial K$

Then, for good 1:

MPL = $\partial Q/\partial L$ = 0.7*(L^0.7-1)*K^0.3 = 0.7*(K/L)^0.3

MPK = $\partial Q/\partial K$ = 0.3*(K^0.3-1)*L^0.7 = 0.3*(L/K)^0.7

So, MPL/MPK = (7/3)*(K/L)

And the cost minimizing condition gives us (7/3)*(K/L) = w/r implying K = 3wL/7r

Substituting this value for K in the production function we get: Q = (3wL/7r)^0.3L^0.7 = (3w/7r)^0.3*L

So, L = Q*(7r/3w)^0.3 and thus, K = (3w/7r)*(Q*(7r/3w)^0.3) = Q*(3w/7r)^0.7

C = w*(Q*(7r/3w)^0.3) + r*(Q*(3w/7r)^0.7)

C = 1.29Q*w^0.7r^0.3 + 0.55Q*w^0.7r^0.3

C = 1.84Q*w^0.7r^0.3

Similarly, for good 2:

MPL = $\partial Q/\partial L$ = 0.4*(L^0.4-1)*K^0.6 = 0.4*(K/L)^0.6

MPK = $\partial Q/\partial K$ = 0.6*(K^0.6-1)*L^0.4 = 0.6*(L/K)^0.4

So, MPL/MPK = (4/6)*(K/L)

And the cost minimizing condition gives us (4/6)*(K/L) = w/r implying K = 6wL/4r

Substituting this value for K in the production function we get: Q = (6wL/4r)^0.6L^0.4 = (6w/4r)^0.6*L

So, L = Q*(4r/6w)^0.6 and thus, K = (6w/4r)*(Q*(4r/6w)^0.3) = Q*(6w/4r)^0.7

C = w*(Q*(4r/6w)^0.6) + r*(Q*(6w/4r)^0.4)

C = 0.78Q*w^0.4r^0.6 + 1.18Q*w^0.4r^0.6

C = 1.96Q*w^0.4r^0.6

Denoting cost of good 1 by C1, and cost of good 2 by C2, and by divind the cost function by Q (number of units) we have found the unit cost functions as (note that for per unit case, you could also divide by Q the production function in the beginning, and generate k (=K/Q) and l(=L/Q) as the per unit factor requirements, and proceed, answer would still be the same; verify!) :

c1 = 1.84(w)^0.7(r)^0.3

c2 = 1.96(w)^0.4(r)^0.6

c) Given p1 = p2 = 1

We know that at equilibrium, unit cost function should equal per unit price for a particular good.

So, for good 1, c1 = p1

gives 1.84(w)^0.7(r)^0.3 = 1

and for good 2, c2 = p2 gives 1.96(w)^0.4(r)^0.6 = 1

Now, that we have two equations in two unknowns, we can find unique solution for (r, w)

Using the two equations above, 1.84(w)^0.7(r)^0.3 = 1.96(w)^0.4(r)^0.6

w^0.3 = 1.065*r^0.3

w = 1.23*r (approx)

So, substituting this in any 1 of the above equations, 1.84*(1.23r)^0.7r^0.3 = 1

Solving for r, r = 0.47

Then, w = 1.23*0.47 = 0.58

d) Now, p1 = 2 and p2 = 1

Again as in part c), for good 1, 1.84(w)^0.7(r)^0.3 = 2 or 0.92(w)^0.7(r)^0.3 = 1

and for good 2, 1.96(w)^0.4(r)^0.6 = 1

Solving the two equations,

0.92(w)^0.7(r)^0.3 = 1.96(w)^0.4(r)^0.6

w^0.3 = 2.13*r^0.3

w = 12.44*r (approx)

So, substituting this in any 1 of the above equations, 0.92*(12.44r)^0.7r^0.3 = 1

Solving for r, r = 0.19

Then, w = 12.44*0.19 = 2.32

As we have already established in part (a) that good 2 is capital intensive, in same way (or correspondingly) we note that good 1 is labor intensive. Stolper-Samuelson theorem states that if the relative price of labor intensive good increases, then labor-owners shall be better off and capital-owners shall be worse off and vice-versa.

Here, in our question, as the relative price of good 1 which is labor intensive increases (p1/p2 = 2 which was initially 1, so increased), we expect the labor to be better of, and capital be worse off. Clearly, we have established that wage rate has increased tremendously (in absolute terms and also, higher than prices, so that real wage rate also increases) while the rental rate has fallen (so returns to capital have fallen in absolute terms, and with higher prices, real rental rate has also gone down). Thus, Stolper-Samuelson theorem holds here.