Question

Consider the following Cobb-Douglas production function for the bus transportation system in a particular city:
Q = Lβ1Fβ2Bβ3
Where L = labour input in worker hours F = fuel input in gallons
B = capital input in number of buses
Q = output measured in millions of bus miles
Suppose that the parameters (α, β1, β2 and β3) of this model were estimate using annual data for the past 25 years. The following results were obtained:
β1 = 0.45, β2 = 0.20 and β3 = 0.30
i. Determine the (i) labour, (ii) fuel, and (iii) capital-input production elasticities
(2 marks)

Answer 1

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

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)

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

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