Suppose a monopolist faces production function Q = 3KL and demand

function Q = 12 − P. Derive the input demand (as a function of Q, r
and w) for capital and labor.

To derive the input demand for capital and labor, we need to use the production function and the demand function.

Starting with the production function: Q = 3KL, we can solve for capital (K) in terms of labor (L). Rearrange the equation to isolate K:
K = Q / (3L)

Next, we use the demand function: Q = 12 - P, to solve for P in terms of Q:
P = 12 - Q

Now, we have expressions for capital (K) and price (P) in terms of labor (L) and quantity (Q).

To determine the demand for capital (K), we need to find the conditions that maximize the monopolist's profit. The profit-maximizing condition is where the marginal cost (MC) of production equals the marginal revenue (MR) of sales.

The marginal cost of production can be calculated by taking the derivative of the production function with respect to labor (L), and then multiplying it by the cost of labor (w):
MC = (dQ/dL) * w

Now, let's find the derivative of the production function:
dQ/dL = 3K

Substituting K = Q / (3L), we get:
dQ/dL = 3(Q / (3L)) = Q / L

Now, multiply the derivative by the cost of labor (w) to get the marginal cost of production:
MC = (Q / L) * w

On the other hand, the marginal revenue (MR) can be calculated by taking the derivative of the demand function with respect to quantity (Q):
MR = dP/dQ

To find dP/dQ, differentiate the demand function:
dP/dQ = -1

Now, we have the expression for marginal revenue:
MR = -1

The monopolist's profit-maximizing condition is MC = MR. Substituting the expressions for marginal cost (MC) and marginal revenue (MR) obtained above, we have:
(Q / L) * w = -1

Simplifying the equation, we get:
Q / L = -1 / w

Now, we can solve for the input demand for labor (L):
L = -Q / (w * L)

Finally, we can substitute the expression for labor (L) into the production function equation to find the input demand for capital (K):
K = Q / (3L)

Substituting the expression for L, we get:
K = Q / (3 * (-Q / (w * L)))

Simplifying, we have:
K = -w / 3

Therefore, the input demand for capital (K) is -w / 3, and the input demand for labor (L) is -Q / (w * L).