Question

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

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

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

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

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.

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.

Key concepts

Pareto Efficiency

Pareto efficiency, named after the Italian economist Vilfredo Pareto, is a state of resource allocation where it is impossible to make any one individual better off without making at least one individual worse off. Imagine a world where everyone receives the maximum possible benefit from resources, and any change to benefit one party would disadvantage another.

In economic systems, achieving Pareto efficiency is often a goal because it indicates that resources are being utilized in the best possible economic manner. However, real-world systems, like monopolies, can obstruct this optimal allocation.

In our example, General Widget (GW) holds a monopoly over the production of widgets. This means it is the only producer and faces no competition. This lack of competition can affect the Pareto efficiency of the market. Unlike a competitive market, a monopoly might not allocate resources in the most efficient way because it can set prices to maximize its profits, disregarding the overall resource utility.

It's essential to examine monopolies in terms of their impact on efficiency, as these entities often wield significant economic power that can lead to sub-optimal allocation of resources.

Production Function

A production function describes the relationship between inputs and outputs in a production process. For GW, the simplified production function is given by the equation \(Q = 2L\). This equation suggests that the output of widgets \(Q\) is directly proportional to the number of widget makers \(L\) hired.

Key takeaways:

• The relationship is constant, implying that hiring more workers will linearly increase the output.
• This function helps in determining how efficiently inputs (widget makers) are converted into outputs (widgets).
• Understanding this function is vital for profit calculation and generation, as it indicates how labor input scales to produce greater quantities of widgets.

In monopolistic situations, such understanding empowers the firm to adjust inputs efficiently to control production and maximize profits.

Profit Maximization

Profit maximization for firms is the process of determining the best output and input levels in order to enhance profitability. It involves finding the quantity of production where the difference between revenue and costs is greatest.

For GW, the profit function derived from its economic activities was \(\pi = (100-2L)(2L) - (20+2L)L\). Maximizing profits required determining where the derivative of this profit function with respect to \(L\) equalled zero, resulting in \(L = 20\).

Steps to Profit Maximization:

• Determine revenue and cost from production: Revenue is price times quantity, and cost relates to the cost of labor.
• Derive the profit function from revenue minus cost.
• Set the derivative of the profit function with respect to labor (or output) to zero to find the optimal level of labor \(L\) or output \(Q\).

At the calculated optimum values, profits are maximized, maximizing the monopolist firm's control over pricing and output.

Competitive Market

A competitive market is characterized by many buyers and sellers, where firms have little control over the market price of goods. Here, prices adjust to balance supply and demand, fostering efficient resource allocation. In contrast to monopolistic markets, competitive settings seek to ensure that prices reflect the true cost and value of goods and services.

In our scenario, comparing competitive and monopolistic market outputs, both provide the same results for widget production (\(Q = 40\), \(L = 20\), and \(P = w = 60\)), which is rare. Generally, monopolies can lead to inefficient pricing and reduced consumer welfare.

Benefits of Competitive Markets:

• They typically encourage better resource allocation due to pricing signals.
• They reduce inefficiencies caused by monopolistic pricing power and output controls.
• Consumers benefit from lower prices and greater product choices.

In this case, even though there was no societal gain from GW acting competitively, competitive markets are considered more favorable under typical economic conditions.