Consider the following production function for three firms operating in the same industry using capital $($K$)$ and labor $($L$)$ as inputs.

Firm $$E Q$=$ K$^0\mathrm{.}7$ L$^0\mathrm{.}3$

Firm $$F Q$=$ K L$^2$

For each firm

$$

a$.$ Determine the firm$$s marginal product of capital function

$(2$ mark$)$

b$.$ Determine the firm$$s marginal product of labour function

$(2$ mark$)$

c$.$ Determine if the marginal product of capital function exhibits diminishing marginal returns

$(2$ marks$)$

d$.$ Determine if the marginal product of labour function exhibits diminishing marginal returns

$(2$ marks$)$

e$.$ Determine if the function exhibit increasing, constant or decreasing returns to scale.

$(2$ marks$)$

f$.$ Determine the function for the marginal rate of technical substitution

$(2$ marks$)$

g$.$ Determine if the production function has a diminishing marginal rate of technical substitution.

$(2$ marks$)$

h$.$ Use Lagrange Method to derive the conditional input demand functions for labour and capital by Minimizing cost subject to output

$(8$ marks$)$

i$.$ Use Lagrange Method to derive the conditional input demand functions for labour and capital by Maximizing output subject to cost constraint

$(8$ marks$)$

$$

j$.$ Choose the cost$/$output amount as well as the price of wages and the price of capital and illustrate the firm$$s output expansion path and the input demand for labour and capital for the firm chosen.

$(10$ marks $)$