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Chapter 14: Problem 11

In an important recent working paper, M. Fabinger and E. G. Weyl characterize tractable monopoly problems. \(^{18}\) A "tractable" problem satisfies three conditions. First, it must be possible to move back and forth between explicit expressions for inverse and direct demand (invertibility). Second, inverse demand-which can also be interpreted as average revenuemust have the same functional form as marginal revenue, and average cost must have the same functional form as marginal cost (form preservation). Third, the monopolist's first-order condition must be a linear equation (linearity), if not immediately after differentiation, then at least after suitable substitution. The authors show that the broadest possible class of tractable problems has the following functional form for inverse demand and average cost: \\[ \begin{aligned} P(Q) &=a_{0}+a_{1} Q^{-s} \\ A C(Q) &=c_{0}+c_{1} Q^{-s} \end{aligned} \\] where \(a_{0}, a_{1}, c_{0}, c_{1},\) and \(s\) are non-negative constants. a. Solve for the monopoly equilibrium quantity and price given these functional forms. What substitution \(x=f(Q)\) do you need to make the first- order condition linear in \(x ?\) b. Derive the solution in the special case with constant average and marginal cost. c. If one is willing to relax tractability a bit to allow the monopoly's first-order condition to be a quadratic equation (at least after suitable substitution), the authors show that the broadest class of tractable problems then involves the following functional forms: \\[ \begin{aligned} P(Q) &=a_{0}+a_{1} Q^{-s}+a_{2} Q^{s} \\ A C(Q) &=c_{0}+c_{1} Q^{-s}+c_{2} Q^{s} \end{aligned} \\] Solve for the monopoly equilibrium quantity and price. What substitution \(x=f(Q)\) is needed to make the first-order condition quadratic in \(x ?\) d. While slightly complicated, the functional forms in part (c) have the advantage of being flexible enough to allow for U-shaped average cost curves such as drawn in Figure 14.2 in addition to constant, increasing, and decreasing. Demonstrate this by graphing this average cost curve for well- chosen values of \(c_{0}, c_{1}, c_{2}\) to illustrate the various cases. The flexible functional forms in part (c) also allow for realistic demand shapes, for example, one that closely fits the U.S. income distribution (which implicitly takes income to proxy for consumers' willingness to pay). These realistic demand shapes can be used in calibrations to address important policy questions. For example, the text mentioned that, in theory, the welfare effects of monopoly price discrimination can go either way, either being higher or lower than under uniform pricing. Calibrations involving the demand curves from part (c) invariably show that welfare is higher under price discrimination.

Short Answer

Expert verified
(4 points) 2. In the case with constant average and marginal cost, what is set to be equal to marginal cost? (1 point) 3. How does the flexible functional forms help improve policy questions or calibrations? (2 points)

Step by step solution

01

Calculate Total Revenue (TR) and Total Cost (TC)

To find the monopoly equilibrium quantity and price for the given functional forms, first we need to compute total revenue and total cost. Total revenue is given by TR = P(Q) * Q, and total cost is given by TC = AC(Q) * Q.
02

Compute Marginal Revenue (MR) and Marginal Cost (MC)

Next, we need to find the marginal revenue and marginal cost. To find marginal revenue, differentiate the total revenue with respect to quantity Q: MR = d(TR)/dQ. To find marginal cost, differentiate the total cost with respect to quantity Q: MC = d(TC)/dQ.
03

Set MR = MC to find equilibrium quantity

To find the monopoly equilibrium quantity, set marginal revenue equal to marginal cost: MR = MC. Solve the resulting equation for the quantity Q.
04

Find the equilibrium price

Plug the equilibrium quantity found in step 3 back into the inverse demand equation to find the equilibrium price: P(Q) where Q is obtained from step 3.
05

Find the substitution x=f(Q) to make the first-order condition linear in x

To find the substitution x=f(Q) that makes the first-order condition linear in x, examine the resulting equation from setting MR = MC in step 3 and look for a suitable substitution. #b. Deriving the solution with constant average and marginal cost#
06

Set constant average cost

To derive the solution in the case with constant average and marginal cost, set AC(Q) = c0 (a constant). This means that c1 = 0
07

Compute the equilibrium quantity and price

Follow steps 3 and 4 from part a with the constant average cost to find the monopoly equilibrium quantity and price. #c. Finding the monopoly equilibrium quantity and price for the functional forms when the first-order condition is allowed to be a quadratic equation#
08

Update the given functional forms of inverse demand and average cost

In this part, use the updated functional forms given for inverse demand and average cost: \\ \[ \begin{aligned} P(Q) &=a_{0}+a_{1} Q^{-s}+a_{2} Q^{s} \\\ A C(Q) &=c_{0}+c_{1} Q^{-s}+c_{2} Q^{s} \end{aligned} \\]
09

Compute marginal revenue and marginal cost

Differentiate the total revenue and total cost functions as in part a to find the updated marginal revenue and marginal cost.
10

Set MR = MC and find the substitution x=f(Q) to make the first-order condition quadratic in x

Set marginal revenue equal to marginal cost as in part a. The resulting equation should be a quadratic equation. Find the suitable substitution x=f(Q) that makes the equation quadratic in x.
11

Solve the quadratic equation for x

Once the quadratic equation is obtained using the substitution x=f(Q), solve it for x.
12

Find the equilibrium quantity and price

Return to the terms of Q by substituting x=f(Q) back into the equation. Solve for Q and use this result to find the equilibrium price by plugging it into the inverse demand equation. #d. Demonstrating the flexibility of functional forms in part (c)#
13

Graph the average cost curve

Demonstrate the different shapes of the average cost curve for well-chosen values of c0, c1, c2 by graphing the average cost curve AC(Q).
14

Calibration and welfare analysis

Discuss the implications of using the flexible functional forms in part (c) for calibrations and policy questions, especially regarding the welfare effects of monopoly price discrimination.

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Key Concepts

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

Inverse Demand
In economic terms, inverse demand is a way of expressing the relationship between the price of a good and the quantity that consumers want to buy. It essentially tells us how much consumers are willing to pay for a specific quantity of a good. Instead of simply saying 'at this price, consumers will buy that many units,' inverse demand flips the equation around: 'to sell this many units, the price must be set at that level.'

For a monopolist, the inverse demand curve is crucial because it highlights how they can adjust prices to influence demand. In the textbook exercise, the inverse demand function is given by the expression \( P(Q) = a_{0} + a_{1}Q^{-s} \), where \( P(Q) \) is the price at a quantity \( Q \), and \( a_{0}, a_{1}, \) and \( s \) are constants. Monopolists use it to figure out the price they should charge to maximize profits without setting it so high that they lose customers.
Average Cost
Moving onto average cost, which in the context of a monopoly, represents the total cost of production divided by the quantity of output produced. It provides invaluable insight into how costs behave in relation to output. Typically, there might be economies of scale at work, meaning that as a firm produces more, the average cost per unit decreases.

In our exercise, the average cost function has a similar form to the inverse demand function, described as \( AC(Q) = c_{0} + c_{1}Q^{-s} \). If the average cost remains the same as output increases, we are likely dealing with constant returns to scale. However, if the average cost goes down with increased output, it's a case of increasing returns to scale. Conversely, decreasing returns to scale occur when the average cost rises as output grows. Understanding these relationships helps the monopolist determine the optimum production level for maximum profitability.
Marginal Revenue
The term marginal revenue (MR) is about the additional revenue a firm earns by selling one more unit of a good or service. It's a key concept in deciding the level of output that maximizes profit for any firm, including a monopolist. To calculate MR, we differentiate the total revenue (TR) function with respect to Q.

In monopoly settings, because the firm must lower the price to sell more units, MR can differ significantly from the price, which would not be the case in a competitive market. In the solution steps, the computation of MR requires differentiation of TR, which involves applying calculus to the inverse demand function. This is instrumental for determining the optimum quantity to produce to maximize profits.
Marginal Cost
On the other side of the coin, we have marginal cost (MC). This is the cost of producing one more unit of a good. For an efficient producer, it's critical to understand where MC stands in relation to MR. The equilibrium for production and price-setting is found where MR equals MC — this is the profit-maximizing output for the monopoly.

Just like MR, the MC is found by differentiation. Only here we differentiate the total cost (TC) function. In a perfect world with no other market frictions, if MC were to exceed MR at a certain output level, the monopolist should decrease production since each additional unit would cut into profits.
First-Order Condition
The first-order condition for profit maximization is essentially the 'golden rule' for monopolists. It's a mathematical way of stating 'at this point, you can't make more profit by producing more or less.' It is achieved when MR equals MC, which is the rule enforced in the textbook solution.

However, solving these equations isn't always straightforward. Depending on the functions for demand and cost, mathematicians often need to perform a substitution to linearize the equation, making it easier to solve. In our exercise, the substitution \( x = f(Q) \) transforms the first-order condition into a linear or quadratic equation. This simplification allows us to find the monopoly equilibrium quantity and price, which is where the monopolist will operate to maximize profits.

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Most popular questions from this chapter

Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of \(\$ 10\) per unit. Monopolized marginal costs increase to \(\$ 12\) per unit because \(\$ 2\) per unit must be paid to lobbyists to retain the widget producers' favored position. Suppose the market demand for widgets is given by \\[ Q_{D}=1,000-50 P \\] a. Calculate the perfectly competitive and monopoly outputs and prices. b. Calculate the total loss of consumer surplus from monopolization of widget production. c. Graph your results and explain how they differ from the usual analysis.

The taxation of monopoly can sometimes produce results different from those that arise in the competitive case. This problem looks at some of those cases. Most of these can be analyzed by using the inverse elasticity rule (Equation \(14.1)\) a. Consider first an ad valorem tax on the price of a monopoly's good. This tax reduces the net price received by the monopoly from \(P\) to \(P(1-t),\) where \(t\) is the proportional tax rate. Show that, with a linear demand curve and constant marginal cost, the imposition of such a tax causes price to increase by less than the full extent of the tax. b. Suppose that the demand curve in part (a) were a constant elasticity curve. Show that the price would now increase by precisely the full extent of the tax. Explain the difference between these two cases. c. Describe a case where the imposition of an ad valorem \(\operatorname{tax}\) on a monopoly would cause the price to increase by more than the tax. d. A specific tax is a fixed amount per unit of output. If the tax rate is \(\tau\) per unit, total tax collections are \(\tau Q .\) Show that the imposition of a specific tax on a monopoly will reduce output more (and increase price more) than will the imposition of an ad valorem tax that collects the same tax revenue.

Suppose a monopoly can produce any level of output it wishes at a constant marginal (and average) cost of \(\$ 5\) per unit. Assume the monopoly sells its goods in two different markets separated by some distance. The demand curve in the first market is given by \\[ Q_{1}=55-P_{1} \\] and the demand curve in the second market is given by \\[ Q_{2}=70-2 P_{2} \\] a. If the monopolist can maintain the separation between the two markets, what level of output should be produced in each market, and what price will prevail in each market? What are total profits in this situation? b. How would your answer change if it costs demanders only \(\$ 4\) to transport goods between the two markets? What would be the monopolist's new profit level in this situation? c. How would your answer change if transportation costs were zero and then the firm was forced to follow a single-price policy? d. Now assume the two different markets 1 and 2 are just two individual consumers. Suppose the firm could adopt a linear two-part tariff under which marginal prices charged to the two consumers must be equal but their lump-sum entry fees might vary. What pricing policy should the firm follow?

Suppose a monopolist produces alkaline batteries that may have various useful lifetimes \((X) .\) Suppose also that consumers' (inverse) demand depends on batteries' lifetimes and quantity (Q) purchased according to the function \\[ P(Q, X)=g(X \cdot Q) \\] where \(g^{\prime} < 0 .\) That is, consumers care only about the product of quantity times lifetime: They are willing to pay equally for many short-lived batteries or few long-lived ones. Assume also that battery costs are given by \\[ C(Q, X)=C(X) Q \\] where \(C^{\prime}(X )> 0 .\) Show that, in this case, the monopoly will opt for the same level of \(X\) as does a competitive industry even though levels of output and prices may differ. Explain your result. Hint: Treat \(X Q\) as a composite commodity.

Some firms employ the marketing strategy of posting a low price for the good, but then tack on hidden fees or high prices for add-ons that can add up to an "all-in" price that is exorbitant compared to the posted price. A television ad may blare that a perpetually sharp knife sells for \(\$ 20,\) leaving the additional \(\$ 10\) handling charge-or worse, that the \(\$ 20\) is just one of three installments-for the small print. A laser printer printing photo-quality color prints may seem like a bargain at \(\$ 300\) if one doesn't consider that the five toner cartridges must be replaced each year at \(\$ 100\) each. If consumers understand and account for these additional expenses, we are firmly in a neoclassical model, which can be analyzed using standard methods. Behavioral economists worry about the possibility that unsophisticated consumers may underestimate or even ignore these shrouded prices and firms do their best to keep it that way. This question introduces a model of shrouded prices and analyzes their efficiency consequences. a. Consumers' demand for a good whose price they perceive to be \(P\) is given by \(Q=10-P .\) A monopolist produces the good at constant average and marginal cost equal to \(\$ 6 .\) Compute the monopoly price, quantity, profit, consumer surplus, and welfare (the sum of consumer surplus and profit) assuming the perceived is the same as the actual price, so there is no shrouding. b. Now assume that while the perceived price is still \(P\), the actual price charged by the monopolist is \(P+s,\) where \(s\) is the shrouded part, which goes unrecognized by consumers. Compute the monopoly price, quantity, and profit assuming the same demand and cost as in part (a). What amount of shrouding does the firm prefer? c. Compute the consumer surplus (CS) associated with the outcome in (b). This requires some care because consumers spend more than they expect to. Letting \(P_{s}\) and \(Q_{s}\) be the equilibrium price and quantity charged by the monopoly with shrouded prices, \\[ C S=\int_{0}^{Q} P(Q) d Q-P_{s} Q_{s} \\] This equals gross consumer surplus (the area under inverse demand up to the quantity sold) less actual rather than perceived expenditures. d. Compute welfare. Find the welfare-maximizing level of shrouding. Explain why this is positive rather than zero. e. Return to the case of no shrouding in part (a) but now assume the government offers a subsidy s. Show that the welfare-maximizing subsidy equals welfare-maximizing level of shrouding found in part (d). Are the distributional consequences (surplus going to consumers, firm, and government) the same in the two cases? Use the connection between shrouding and a subsidy to argue informally that any amount of shrouding will be inefficient in a perfectly competitive market.
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