Question

There are 2 pages in this assignment. There are 3 questions with parts. The weight of each part is indicated in the margin. Please answer the questions carefully. When asked to draw a graph, make sure you label all the curves and the axes. Sloppy work will not be graded or will lose points. Read carefully Chapter 3 as well as your class notes. (30pts) 1. Consider a perfectly competitive economy with K amount of capital and L amount of labour. All factors of production are fixed and fully employed. Technology is described by a Cobb-Douglas production function given by: Y = F(R.D) = (R) (6) a) For each of the following cases, indicate whether the production function exhibits increasing, decreasing or constant returns to scale. Demonstrate your result (by introducing a "multiple :" of the factors of production) and explain. i) a=0.4, b=0.5 ii) a=0.3, b = 0.7 iii) a = 0.6, b = 0.6 (6) b) Suppose that a=0.3, b =0.7. If factors are paid their marginal products, what is the share of income received by capital and labour, respectively? Show your result and explain. (18) c) Suppose that a natural disaster (for ex., an earthquake) destroys 30% of this economy's capital stock but that the labour force remains unchanged. (Assume again that a -0.3, b -0.7). (i) What is the percentage change in total output? Explain and draw a graph to illustrate (ii) What is the percentage change in the real rental rate? Explain and draw a graph to illustrate. (iii) What is the percentage change in the real wage rate. Explain and draw a graph to illustrate, In each case indicate the direction of change increase or decrease) and derive the percentage change. Show your work. Explain in words. Draw a graph to illustrate the change in each case. (HINT: express the new levels of K and L in terms of the old (i.e., initial) values. For example, Knew = 0.7 KOLD.)

Answer 1

a).

Let’s assume the production function be “Y = F(K, L) = K^a*L^b”, if we increase both the factor with proportion “z” then the we have.

=> F(zK, zL) = (zK)^a*(zL)^b = z^a+b*(K)^a*(L)^b = z^a+b*Y, => F(zK, zL) = z^a+b*Y. as we increase both the factor with proportion “z” the output also increases by “z^a+b”, => the production is homogeneous of degree “a+b”. Now, if “a+b > 1”, => the production function exhibits IRS, if “a+b < 1”, => the production function exhibits DRS and , if “a+b = 1”, => the production function exhibits CRS.

Now, here if “a=0.4, b=0.5”, => a+b = 0.9 < 1, => the production function exhibits DRS.

If “a=0.3, b=0.7”, => a+b = 1, => the production function exhibits CRS.

Finally, if “a=0.6, b=0.6”, => a+b = 1.2 > 1, => the production function exhibits IRS.

b).

The production function is given by, => Y = K^a*L^b, where factors are paid their marginal products. So, the marginal product of both the inputs are given by.

=> MP_L = dY/dL = b*L^b-1*K^a, and MP_K = dY/dK = a*K^a-1*L^b.

So, the labor share of income is given by, => MP_L*L/Y = [b*L^b-1*K^a]*L/Y = [b*L^b*K^a]/ L^b*K^a.

=> MP_L*L/Y = b = labor share of income.

Similarly, the capital share of income is given by, => MP_K*K/Y = [a*K^a-1*L^b]*K/Y.

=> MP_K*K/Y = [a*K^a*L^b]/ L^b*K^a = a = capital share of income.

Here we have given that the “a=0.3” and “b=0.7”, => the labor share and the capital share of income are “0.7” and “0.3” respectively.

c).

The production function is given by, => Y = K^a*L^b, => LogY = a*LogK + b*LogL.

=> dLogY = a*dLogK + b*dLogL, => g_Y = a*g_K + b*g_L, here “labor” is fixed as before, => “g_L=0” and capital destroy by 30%, => “g_K = 30% = 0.3”, => the above relationship can written as f

=> g_Y = a*g_K + b*g_L, => g_Y = a*(-0.3) + b*0 = 0.3*(-0.3) = (-0.09) = (-9%), => output also decreases by “9%”. The following fig shows the total output initially and new after the change in capital.

Y = F(K1, L) Y = F(K2, L)

ii).

The real rental rate is given by, => R/P = MPK, => MPK = a*K^a-1*L^b, => R/P = a*K^a-1*L^b.

=> gr = (a-1)*g_K + b*g_L, => gr = (0.3-1)*(-0.3) = 0.21 > 0, => gr = 21%. So, as the capital decreases by 30% the real rental rate will increase by 21%. The following fig shows the increase in real rental rate.

R/P = MPK K2

iii).

W/P = MPL1 MPL2

The real wage rate is given by, => W/P = MPL, => MPL = b*K^a*L^b-1, => W/P = b*K^a*L^b-1.

=> gw = a*g_K + (b-1)*g_L, => gw = 0.3*(-0.3) + 0 = (-0.09) < 0, => gw = (-9)%. So, as the capital decreases by 30% the real wage rate will also decrease by 9%. The above fig shows the decrease in real wage rate.