A monopolist uses one input X, which she purchases at the fixed price p=5 in order to produce output q. Her demand and production functions are: P=85-3q and q= 2×1/2 respectively. Derive the equilibrium output and equilibrium profit

To find the equilibrium output, we need to set the marginal cost equal to the marginal revenue.

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The monopolist’s profit-maximizing condition is (MR = MC).

Given that the demand function is (P = 85 – 3q) and the production function is (q = 2\sqrt{x}), we can derive the marginal revenue (MR) and marginal cost (MC) as follows:

1. Marginal Revenue (MR):
   [MR = \frac{d(TR)}{dq}]
   where (TR) is the total revenue. [TR = P \cdot q]
   [MR = \frac{d(85q – 3q^2)}{dq}]
2. Marginal Cost (MC):
   [MC = \frac{d(TC)}{dq}]
   where (TC) is the total cost. The cost function depends on the price of input (X), which is purchased at a fixed price (p = 5), so (TC = 5x).

Now, set (MR) equal to (MC) and solve for (q) to find the equilibrium output. Once you find (q), you can find the equilibrium price ((P)) using the demand function.

To find equilibrium profit, calculate total revenue ((TR)) and total cost ((TC)) at the equilibrium output, then subtract (TC) from (TR).

Remember to check if the found output is feasible given the production function.