Question

Consider the following Cobb-Douglas production function for the bus transportation system in a city:

 Q = L^β1F^β2B^β3 , Where L = labour input in worker hours  F = fuel input in gallons  B = capital input in a number of buses  Q = output measured in millions of bus miles  

Suppose that the parameters (α, β1, β2 and β3) of this model were estimated using annual data for the past 25 years. The following results were obtained:  

β1 = 0.45, β2 = 0.20 and β3 = 0.30

a.  Determine the labour, fuel, and capital-input production elasticities

b.  Suppose that labour input (worker hours) is increased by 2 percent next year (with the other inputs held constant), determine the approximate percentage change in output.

Answer 1

a)

$$Q=L^{\beta_1}F^{\beta_2}B^{\beta_3}$$

i)

$$\frac {\delta Q}{\delta L}=\beta_1L^{\beta_1-1}F^{\beta_2}B^{\beta_3}$$

$$\frac {\delta Q}{\delta L}=0.45L^{-0.55}F^{0.2}B^{0.3}$$

ii)

$$\frac {\delta Q}{\delta F}=\beta_2L^{\beta_1}F^{\beta_2-1}B^{\beta_3}$$

$$\frac {\delta Q}{\delta F}=0.2L^{0.45}F^{-0.8}B^{0.3}$$

iii)

$$\frac {\delta Q}{\delta B}=\beta_3L^{\beta_1}F^{\beta_2}B^{\beta_3-1}$$

$$\frac {\delta Q}{\delta B}=0.3L^{0.45}F^{0.2}B^{-0.7}$$

b) percentage change in $Q=0.2(\%$ change in L)

it is given that percentage change in labor is $2\%$

Hence,

$\%change$ in $Q=0.45\times2=0.90\%$