Question (Mathematical Economics):
A firm’s production function is given by:
[
Q = 50L^{0.5}K^{0.5}
]
where ( Q ) is output, ( L ) is labor, and ( K ) is capital.
The price of output is ( P = 10 ), the wage rate is ( w = 5 ), and the rental rate of capital is ( r = 10 ).
Required:
- Derive the firm’s profit function in terms of ( L ) and ( K ).
- Determine the profit-maximizing levels of labor and capital.
- Calculate the maximum profit.
- Explain how the firm’s input choices would change if the wage rate increased to ( w = 8 ).