A firm uses labour (workers, L) and capital (machinery, K) to produce output (quantity, Q). Assume the unitary cost of labour is 16 and that of capital is 4. 

a. Write the firm’s total cost function. What is the slope of the cost function? 

b. Draw the isocosts for the following levels of total cost: (1) TC = 160, (2) TC = 240, (3) TC = 320. Indicate the intercepts in all the cases. 

c. Assume the firm’s production function is given by the following formula . If the firm wants to find the optimal combination of labour and capital to produce a quantity Q = 2, how much labour and how much capital should it use? I.e. what are the optimal levels of L and K that allow the firm to produce Q = 2? 

Hint. Start from the production function, set the quantity to the desired one and rearrange to find an expression of K in term of L, or vice versa. Then, remember that the optimality condition requires MRTS equal price (of inputs) ratio. From this condition derive a second expression of K in terms of L, or vice versa. You will have a system with two equations and two variables, so you can use this to find the optimal levels of L and K.