Answer to Question #233145 in Microeconomics for Clement Paul

Question

The Cobb- Douglas production function is given by the form Y = A La Kb

Where, Y = Output (mt/day), L = Labour (hours/mt) K = Capita (Rs/mt). By taking the natural logarithm of each term in the function, that is LnY = LnA+aLnL+bLnK. Using a standard multiple linear regression model the parameters of the function were obtained as follows;

Ln Y = 4.45 +0.5LnL + 0.4LnK

(2.2) (0.01) (0.02)

The standard error for each estimated parameter is shown in parentheses below the estimate.

Construct equation for Cobb- Douglas production function and briefly explain the properties of this production function (4 marks)

Measure marginal product of labour and average product of capital when L=10 and K=5 (3 marks)

Is this production function exhibit the law of diminishing returns in the short run? (3 marks)

Answer 1

Ln Y = 4.45 +0.5LnL + 0.4LnK

Differentiating both sides w.r.t. labour we get:

$\frac{1}{Y}$$\times$$\frac{dY}{DL}=$$\frac{0.5}{L}$

$\frac{dY}{dL}=$$\frac{0.5Y}{L}$

Elasticity of labour $=$$\frac{dY}{DL}$$\times$$\frac{L}{Y}$

E(L)=$\frac{0.5Y}{L}$ $\times$ $\frac{L}{Y}$

E(L)=0.5

Elasticity of capital =$\frac{dY}{dK}\times\frac{K}{Y}$

Differentiating output equation w.r.t to capital we get

$\frac{1}{Y}\times\frac{dY}{dK}=\frac{0.4}{K}$

Putting in elasticity equation we get;

E(K)=$\frac{0.4Y}{K}\times\frac{K}{Y}$

E(K)=0.4

Therefore, cobb douglas function is given by

Q = 4.45L^0.5K^0.4

properties of Cobb-Douglas production function

1. Homogeneous of Degree One i.e Q=AL^αK^β. α + β= 1
2. Average and marginal products of L and K, i.e., AP_L, AP_K, MP_L, and MP_K would all be the functions of L-K or K-L ratio.
3. MRTS_L.K = $\frac{MPL}{MPK}$  = function of $\frac{L}{K}$ ratio

marginal product of labour

MPL is calculated by partially differentiating the cobb douglas function w.r.t. to L

MPL= 2.225 L^-0.5K^0.4

But L=10 and K= 5

Therefore MPL = 1.354

average product of capital

APK= $\frac{Q}{K}$ = AL^αK^β-1

APK= 4.45L^0.5K^-0.6

But L=10 and K= 5

APK = 4.45 * 10^0.5*5^-0.6

APK = 5.36

This function exhibits the law of diminishing returns in the short run since the variables of labor and capital are varied in isolation.