A paint manufacturing company has a production function, Q = K+√?L???. For this production function MPK = 1 and MPL = 1/(2√?L???) The firm faces a price of labor w that equals $1 per unit and a price of capital services r that equals $50 per unit.
a) Verify that the firm's cost-minimizing input combination to produce Q = 10 involves no use of capital.
b) What must the price of capital fall to in order for the firm to use a positive amount of capital, keeping Q at 10 and w at 1?
c) What must Q increase to for the firm to use a positive amount of capital, keeping w at 1 and r at 50?