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Chapter 17: Problem 4

Suppose all of the firms in Utopia obey the Pareto conditions for efficiency except General Widget (GW). That firm has a monopoly in production of widgets and is the only hirer of widget makers in the country. Suppose the production function for widgets is $$Q=2 L$$ (where \(L\) is the number of widget makers hired). If the demand for widgets is given by $$P=100-Q$$ and the supply curve of widget makers by $$w=20+2 L$$ how many widgets should GW produce to maximize profits? At that output, what will \(L\) \(w,\) and \(P\) be? How does this solution compare to that which would prevail if GW behaved in a competitive manner? Can you evaluate the gain to society of having GW be competitive?

Short Answer

Expert verified
Answer: At the optimal level for GW, the values are Q = 40, L = 20, P = 60, and w = 60. When comparing these values to a competitive market scenario, there is no change (Q = 40, L = 20, and w = P = 60). Thus, in this case, there is no gain to society by having the monopolist act competitively, as the welfare remains unchanged.

Step by step solution

01

Determine the profit function

To find the profit function, we first need to compute the Revenue (R) and the Cost (C) functions for the GW. Revenue is obtained by multiplying the price (P) by the quantity (Q), while the Cost function is the sum of the wages paid to widget makers multiplied by the number of widget makers (L). Revenue = P*Q Cost = w*L From the given equations, substitute the values of P and w to form the profit function: Profit (π) = R - C = (100-Q)*Q - (20+2L)*L As we know that Q = 2L, we can rewrite the profit function as: π = (100-2L)(2L) - (20+2L)*L
02

Maximize the profit function by finding the optimal level of output (Q)

To maximize the profit function, we are going to find the first derivative with respect to L and set it to zero. Taking the derivative with respect to L: $$\frac{d\pi}{dL} = -8L + 200 - 6L = 0$$ Solving for L: L = 20 Now, we will find the optimal level of output (Q) by substituting the value of L in Q = 2L: Q = 2 * 20 = 40
03

Determine the values of L, w, and P when the profits are maximized.

Now that we have determined the optimal output level (Q = 40), we will find the corresponding values for L, w, and P. We already know L = 20. So, using the given equations for P and w: P = 100 - Q = 100 - 40 = 60 w = 20 + 2L = 20 + 2(20) = 60 The values at the optimal output level are: L = 20, w = 60, and P = 60.
04

Compare these values with a competitive market scenario and evaluate the benefits to society.

In the case of a competitive market, the price would equal the supply curve (w) for widget makers, as firms would produce where marginal cost equals the price. P = w => 100 - Q = 20 + 2L Combining the equations, we get Q = 2L: 100 - 2L = 20 + 2L Solving for L, we get: L = 20 Thus, Q = 2 * 20 = 40. In a competitive market scenario, the values are: Q = 40, L = 20, and w = P = 60. Comparing these values, we see that both monopolistic and competitive market scenarios yield the same results in this particular case (Q = 40, L = 20, and w = P = 60). Since both scenarios provide the same outcome, there is no gain to society by having the monopolist act competitively. The welfare of the society remains unchanged.

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