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Chapter 11: Problem 3

Because cooking souffl茅s is incredibly difficult, the supply of souffl茅s in a small French town is controlled by two bakers, Gaston and Pierre. The demand for souffl茅s is given by \(P=30-2 Q\), and the marginal and average total cost of producing souffl茅s is \(\$ 6 .\) Because baking a souffl茅 requires a great deal of work and preparation, each morning Gaston and Pierre make a binding decision about how many souffl茅s to bake. a. Suppose that Pierre and Gaston agree to collude, evenly splitting the output a monopolist would make and charging the monopoly price. i. Derive the equation for the monopolist's marginal revenue curve. ii. Determine the profit-maximizing collective output for the cartel. iii. Determine the price Pierre and Gaston will be able to charge. iv. Determine profits for Pierre and Gaston individually, as well as for the cartel as a whole. b. Suppose that Pierre cheats on the cartel agreement by baking one extra souffl茅 each morning. i. What does the extra production do to the price of souffl茅s in the marketplace? ii. Calculate Pierre's profit. How much did he gain by cheating? iii Calculate Gaston's profit. How much did Pierre's cheating cost him? iv. How much potential profit does the group lose as a result of Pierre's cheating? c. Suppose that Gaston, fed up with Pierre's behavior, also begins baking one extra souffl茅 each morning. i. How does the extra production affect the price of souffl茅s in the marketplace? ii. Calculate Gaston's profit. How much did he gain by cheating? iii Calculate Pierre's profit. How much did Gaston's cheating cost him? iv. How much potential profit does the group lose as a result of Pierre's and Gaston's cheating? v. Demonstrate that it is in neither Pierre's nor Gaston's best interest to cheat further on their agreement.

Short Answer

Expert verified
Gaston and Pierre split the profit equally at $36 each. Cheating reduces their collective profit, with both earning $32 each if both cheat, resulting in a $8 group profit loss.

Step by step solution

01

Derive Marginal Revenue (MR)

To find the marginal revenue curve, we start with the demand equation: \(P = 30 - 2Q\). The total revenue (TR) can be derived by multiplying price \(P\) with quantity \(Q\): \[TR = P \times Q = (30 - 2Q)Q = 30Q - 2Q^2\]The marginal revenue \(MR\) is the derivative of \(TR\) with respect to \(Q\):\[MR = \frac{d(TR)}{dQ} = 30 - 4Q\]
02

Determine Profit-Maximizing Output

To find the profit-maximizing output for the cartel, set the marginal revenue equal to the marginal cost \(MC\). Since \(MC\) is equal to the average total cost \(ATC\) of $6, we have:\[30 - 4Q = 6\]Solving for \(Q\) gives:\[24 = 4Q\]\[Q = 6\]Thus, the collective output is 6 souffl茅s.
03

Determine Monopoly Price

Substitute the monopoly quantity \(Q = 6\) in the demand equation to find the price:\[P = 30 - 2 \times 6 = 30 - 12 = 18\]The monopoly price is $18.
04

Calculate Profits for Cartel and Individuals

To find total profit for the cartel, calculate total revenue and total cost:- Total revenue: \(TR = P \times Q = 18 \times 6 = 108\)- Total cost: \(TC = MC \times Q = 6 \times 6 = 36\)- Total profit: \(TR - TC = 108 - 36 = 72\)Each baker's output is \(Q = 3\), so each baker earns half the total profit: \(Profit = \frac{72}{2} = 36\). Each baker earns $36 in profit.
05

Pierre Cheats and Bakes One Extra Souffl茅

If Pierre bakes 1 more souffl茅, total quantity \(Q\) increases to 7. Calculate new price with \(Q = 7\):\[P = 30 - 2 \times 7 = 16\]New total revenue: \(TR = 16 \times 7 = 112\)Total cost for the cartel: \(TC = 6 \times 7 = 42\)New profit for cartel: \(TR - TC = 112 - 42 = 70\).Pierre's output is 4; his revenue: \(4 \times 16 = 64\).Pierre's cost: \(4 \times 6 = 24\).Pierre's profit: \(64 - 24 = 40\).Pierre gained \(4 by cheating (previous profit was \)36).
06

Calculate Gaston's Profit After Pierre Cheats

Gaston's output remains 3. His revenue: \(3 \times 16 = 48\).Gaston's cost: \(3 \times 6 = 18\).Gaston's profit: \(48 - 18 = 30\).Gaston lost \(6 due to Pierre's cheating (previous profit was \)36).
07

Calculate Potential Group Loss Due to Cheating

Initial cartel profit was \(72. New cartel profit after Pierre cheats is \)70.Group loss due to cheating: \(72 - 70 = 2\). Cartel loses $2 in potential profit due to Pierre's cheating.
08

Both Cheat and Bake One Extra Souffl茅

Total quantity \(Q\) increases to 8 with both cheating. New price:\[P = 30 - 2 \times 8 = 14\]New total revenue: \(TR = 8 \times 14 = 112\).Total cost for the cartel: \(TC = 6 \times 8 = 48\).New profit for cartel: \(TR - TC = 112 - 48 = 64\).
09

Gaston's Profit After He Also Cheats

With Gaston cheating, both bakers have output of 4. Gaston's revenue: \(4 \times 14 = 56\).Gaston's cost: \(4 \times 6 = 24\).Gaston's profit: \(56 - 24 = 32\).Gaston gained $2 from cheating alone.
10

Pierre's Profit After Gaston Also Cheats

Pierre's output is 4. His revenue: \(4 \times 14 = 56\).His cost: \(4 \times 6 = 24\).Pierre's profit: \(56 - 24 = 32\).Pierre lost \(8 due to Gaston's cheating (previous profit was \)40).
11

Potential Group Loss from Both Cheating

Initial cartel profit was \(72. New cartel profit after both cheat is \)64.Total group loss from both cheating is \(72 - 64 = 8\).The group collectively loses $8 in potential profit from both cheating.
12

Evaluate Incentive to Cheat Further

If either baker cheats further on their agreement, it continues to drive the price down, resulting in reduced profits for both. Neither has an incentive to produce more, as profits decline with increased competition and lower market prices. It is not in their best interest to cheat further as it reduces overall profits.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cartel Behavior
In economics, a cartel is an agreement between competing firms to control prices or output in a certain market. This is often done to avoid the competitive pressures of a free market. By colluding, these firms can act almost as a monopoly by collectively deciding on quantities to produce and prices to charge.
  • Cartels usually form in markets where only a few firms exist, such as the case with Gaston and Pierre.
  • The primary objective of a cartel is to maximize joint profits as if the collective firms were a single monopolistic entity.
  • This is achieved through the coordination of supply and pricing, which reduces market competition and can elevate prices.
However, cartels are typically illegal in many countries because they stifle competition and can lead to higher prices for consumers.
Monopoly Pricing
Monopoly pricing refers to the strategy of a single firm, or cartel in this case, setting prices to maximize profits without concern for competitive pricing. Unlike in a competitive market where supply and demand dictate pricing, a monopolist sets a price that consumers are willing to pay while maximizing profit.

For Pierre and Gaston, they calculate the monopoly price where their marginal revenue equals their marginal cost. This leads to maximizing their profit based on demand dynamics. This calculated monopoly price reflects the highest price they can charge while still selling their desired quantity.
  • The monopoly price is typically higher than in a competitive market, as seen in the price they set at $18 for the souffl茅s.
  • By acting as a single decision-maker, they can set a price that captures maximum consumer surplus.
However, the challenge for monopolies is finding the balance between pricing and the amount consumers are willing to buy at that price.
Marginal Revenue and Cost
Marginal Revenue (MR) is the additional revenue gained from selling one more unit. Marginal Cost (MC), on the other hand, is the cost of producing one additional unit. In economic theory, firms maximize profit when their marginal revenue equals their marginal cost.

The calculation starts by differentiating the total revenue to get the MR. For Gaston and Pierre, their equilibrium occurs where MR equals MC.
  • MR: This is derived from the demand equation and helps set the pace for how much they can increase production.
  • MC: In the example, the cost of producing each additional souffl茅 is constant at $6.
  • To maximize profit, they set their production such that MR = MC, which maximizes their profit potential.
Understanding these concepts helps businesses in deciding output levels and in setting prices for maximum profitability.
Collusion and Cheating
Collusion occurs when two or more parties join forces to gain an advantage, often through means that would be considered illegal or against competitive norms. Gaston and Pierre start by colluding, but the dynamics of cheating become evident as one attempts to outdo the other.

  • Cheating within a cartel occurs when a member produces more than agreed upon, trying to unjustly gain more profit.
  • The short-term gain allows the cheater to capture a larger share of the market, but it comes at the cost of reducing overall cartel profits.
  • In Gaston and Pierre's case, when one cheats, it disturbs the balance, reducing prices further due to increased supply.
Cheating eventually leads to instability in cartels as trust diminishes and each party pursues individual rather than collective profit. This cycle can dismantle the entire cartel structure, returning the market to a competitive state.

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Most popular questions from this chapter

Two farmers, Tito and Helen, supply a chain of islands with kale. The inverse demand for kale in the islands is given by \(P=60-0.5 Q\) where \(Q\) is the combined output of Tito \(\left(q_{T}\right)\) and Helen \(\left(q_{H}\right)\). measured in 10 -pound bunches. Tito grows kale at a constant marginal and average cost of \(\$ 12\) per bunch; Helen grows kale at a constant marginal and average cost of \(\$ 10\) per bunch. a. Suppose this market is a Stackelberg oligopoly and Tito is the first-mover. How much will he and Helen produce? What will the market price of kale be? How much profit will each farmer earn? b. Now suppose that Helen is the first-mover in this Stackelberg oligopoly. How much will each farmer produce? What will the market price of kale be? How much profit will each farmer earn? c. Quantify the value to Helen of being the first-mover in this Stackelberg game.

Suppose that three grocery stores sell Bubba's Gourmet Red Beans and Rice. Bullseye Market is able to acquire, stock, and market them for \(\$ 2.00\) per package. OKMart can acquire, stock, and market them for \(\$ 1.98\) per package. SamsMart can acquire, stock, and market them for \(\$ 1.96\) per package. a. If the three competitors are located in close proximity to one another, so the cost of going to a different store to purchase red beans and rice is negligible, and if the market for prepackaged gourmet red beans and rice is characterized by Bertrand competition, what will the prevailing market price be? b. Where will customers buy their red beans and rice? Bullseye Market, OKMart, or SamsMart? What does your answer suggest about the potential rewards to small improvements in efficiency via cost-cutting? c. Suppose that each day, equal numbers of customers begin their shopping at each of the three stores. If the cost of going to a different store to purchase red beans and rice is 2 cents, is the Bertrand result likely to hold in this case? Where will customers purchase red beans and rice? Where will they not purchase them?

There are only three big tobacco companies, but they produce dozens of brands of cigarettes. Compare and contrast Bertrand competition with undifferentiated and differentiated products to explain why the big three tobacco companies devote many resources to support so many different brands instead of each producing just a single type of generic cigarette. Do you think supporting all these different brands is good for society, or bad?

One big question economics ponders is how to produce the greatest material well-being using the fewest resources. Compare and contrast perfect competition and monopolistic competition in achieving that end.

The market for nutmeg is controlled by two small island economies, Penang and Grenada. The market demand for bottled nutmeg is given by \(P=100-q_{P}-q_{G}\) where, \(q_{P}\) is the quantity Penang produces and \(q_{G}\) is the quantity Grenada produces. Both Grenada and Penang produce nutmeg at a constant marginal and average cost of $$ 20$$ per bottle. a. Verify that the reaction function for Grenada is given by $$q_{G}=40-0.5 q_{P}$$ then verify that the reaction function for Penang is given by $$q_{P}=40-0.5 q_{G}$$. b. Find the Coumot equilibrium quantity for each island. Then solve for the market price of nutmeg and for each firm's profit. c. Suppose that Grenada transforms the nature of competition to Stackelberg competition by announcing its production targets publicly in an attempt to seize a first-mover advantage. i. Grenada must first decide how much to produce, and to do this, it needs to know the demand conditions it faces. Substitute Penang's reaction function into the market demand curve to find the demand faced by Grenada. ii. Based on your answer to the problem above, find the marginal revenue curve faced by Grenada. iii Equate marginal revenue with marginal cost to find Grenada's output. iv. Plug Grenada's output into Penang's reaction function to determine Penang's output. v. Plug the combined output of Grenada and Penang into the market demand curve to determine the price. How do the industry quantity and price compare to those under Cournot competition? vi. Determine profits in Grenada and Penang. How do the profits of each compare to profits under Cournot competition? Is there an advantage to being the first-mover?
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