Answer to Question #221303 in Microeconomics for Senal

Question

Given the following data on output and inputs for ten production period

Production period Output (Q) Capital (K) Labour (L)

1 225 10 20

2 240 12 22

3 278 10 26

4 212 14 18

5 199 12 16

6 297 16 24

7 242 16 20

8 155 10 14

9 215 8 20

10 160 8 14

1. Estimate the parameters (A, α and β) of a Cobb-Douglas production function using the least squares regression method.

2. Use estimated parameters to determine (a) returns to scale and (b) factor intensity

3. Determine elasticity of labour and elasticity of capital

4. Measure marginal product of labour and capital for the input combination (L=20 and k=30)

5. Construct the equation for isoquant and graph the isoquant assuming output is 100 units and L = 2,4,6,8,10,12,14.16 and 18.

Answer 1

1 -

Linear regression refers to the linear approach to modeling the relationship between a scalar response and one or more explanatory variables. the case of one explanatory variable is called simple regression.

the data is as follows,

the Cobb Douglas production is;

$Q=AK^\alpha\times{L^\Beta}$

it can be written as

$LogQ=LogA+\alpha{LogK}+\beta{LogL}$

$LogQ=A'+\alpha{LogA}+\beta{LogL}$

The regression output using OLS (excel):

The regression equation using OLS is ;

$LogQ=LogA+\alpha{LogK}+\beta{LogL}$

$LogQ=0.964+0.11\times{LogK}+0.98\times{LogL}$

Here $A'=LogA=0.964$

so,$A=9.2$

thus, The Cobb Douglas production function is

$Q=9.2\times{K^{0.11}}\times{L^{0.98}}$

2 - a

RETURNS TO SCALE

$\alpha+\beta=0.11+0.98=1.09>1$

since ${\alpha+\beta}>1$, there are increasing return to scale

2 - b

FACTOR INTENSITY

$\alpha=0.11$

$\beta=0.98$

factor intensity = $\frac{0.11}{0.98}$

factor intensity = 0.1122

3 -

MARGINAL PRODUCT OF LABOUR (MPL)

= $\frac{\delta Q}{\delta L}$

$= 9.2\times{0.98}\times{K^{0.11}}\times{L^{0.98-1}}$

$=9.016\times{K^{0.11}}\times{L^{-0.02}}$

MARGINAL PRODUCT OF CAPITAL (MPK)

= $\frac{\delta Q}{\delta K}$

$=9.2\times{0.11}\times{K^{0.11-1}}\times{L^{0.98}}$

$=1.012\times{K^{-0.89}}\times{L^{0.98}}$

At L=20 ,K=30

MPL $=9.016\times{30^{0.11}}\times{20^{0.98}}$

$=12.34457$

MPK $=1.012\times{30^{0.89}}\times{20^{0.98}}$

$=0.092264$

4 - Isoquant is the locus of different possible combination of inputs for a fixed level of output.

For this question, where increasing units of labor employment is given for a fixed level of output (given 100), the level of capital employment is assumed to be decreasing. So, the possible combinations of labor and capital can be as follows.

Here, the curve AB is an isoquant for a fixed level of output i.e. 100 units.

Since, this diagram is completely based on assumed levels of capital employed, the shape of the curve is much like a downward sloping straight line.