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Lecture Notes Monopoly and Product Variety

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Unformatted text preview: Monopoly and Horizontal Product Variety A. Von Thünen Model and Land Rent Let us start with a spatial model. Consider the basic Heinrich von Thünen ­model1, where V denotes consumers’ WTP for the good, T the cost of transporting the good to the market and c denotes marginal production cost. Von Thünen wanted to explain the reach of a local market for wheat. Assume there is a market place surrounded by wheat fields. What is the maximum distance of wheat growing from the market? The distance will be where the net profit for the farmer equals zero. The farmer’s cost consist of growing cost plus the cost of hauling the harvested wheat to the market. The former is reflected by a constant (and distance ­independent) marginal cost c. The latter is depends on the distance from the market x and the freight cost per mile ­ton t. Overall transportation cost is thus T=tx. Note that t is the slope of the T ­curve. If V is the market’s WTP for wheat, c the marginal cost of growing wheat and T=tx the cost of hauling the wheat to the market, X1 will be the maximum distance from the market where farmer grow wheat; X1 is the maximum reach of the market. The farmer located at X1 will have a profit of zero. The market’s reach will increase, i.e., X1 will move to the right, if  ­ V increases  ­ t decreases or  ­ c decreases. and vice versa. 1 von Thünen, J.H. (1875). Der isolirte Staat in Beziehung auf Landwirthschaft und Nationalökonomie. 3rd ed., Berlin: Von Wiegandt. ` T=tx V A B c x M, x=0 X1 B. Monopoly and Horizontal Differentiation (1) Profit Maximizing rules for location, price, profit and number of outlets Our analysis of the monopolists horizontal differentiation is based on the von Thünen ­model. Assume a shop that is located on the middle of Main Street. The length of Main Street (z) is one mile, or simply 1. The shop is, therefore, located at milestone 0.5. Alternatively, we can interpret the length of Main Street as the size of the market, which now normalized to one. The shop sells a good with marginal production cost of c (not shown in the Figure), which is between zero and p, the price of the product. Consumers have a uniform WTP of V. However, since most consumer do not live at the shop they incur travel cost of tx in addition to p. Hence, consumer incur cost of p+tx. Like in the von Thünen ­model, t denotes the transportation cost per mile (and weight unit) and x denotes the distance from the shop (to either side). Similar to the von Thünen ­model, lowering the price will increase the markets reach. In the Figure above, lowering the price from p1 to p2 extends the shops market reach from x1 to x2 to either side. Likewise, A flatter tx curve and a higher V would increase the firm’s market reach as well. The maximum market reach is given where V=p+tx. Maximum price for one shop What is the maximum price if the firm wants to serve the entire market? In this case x=0.5 If x=0.5, the WTP of the marginal consumer, i.e., the one at the far edge, is V=p+t/2. That we can solve for the price and get p = V – t/2 Profit Function for one shop Assuming that there are N consumers on the market, profit for an N ­consumer market and one store, π ( N ,1) , will equal N(price–cost) minus set up cost F for the € store: π ( N ,1) = N (V − € t − c ) − F 2 Where do 2 stores locate? Since the stores do not want to compete with each other, each store needs to serve 1/2 of the market. That is, 1/4 to each side of each store. That is given at z=1/4 and z=3/4. Where do 3 stores locate? Similarly to the solution above, each store serves 1/3 of the market (1/6 to either side). The locations are thus 1/6, 3/6 and 5/6. € € € € € Where do n stores locate? n stores locate at 1/2n, 3/2n, 5/2n, … , (2n ­1)/2n. What is the maximum price for n stores? t p( N , n ) = V − 2n Now we also know the profit function for n stores: t π ( N , n ) = N (V − − c ) − nF 2n Note that we now subtracted nF for n stores. What is the profit maximizing number of stores? This can easily be solved by computing the profit function’s partial derivative with respect to n and setting this equal to zero. First write the profit function like: Ntn −1 π ( N , n ) = NV − − Nc − nF 2 ∂π Ntn −2 Nt Nt * = − F =0 ⇒ Nt = 2 Fn 2 ⇒ nπ = 2 =F ⇒ ∂n 2 2F 2n Example: If F=$50,000, N=5,000,000 and t=1 5, 000, 000 * nπ = = 50 ≈ 7 100, 000 C. Welfare and Horizontal Differentiation (1) Consumer Surplus, Producer Surplus, Welfare Loss Assuming marginal production cost of c=0, the respective surpluses are depicted in the Figure below. Consumer Surplus (CS) is the aggregated difference between consumers’ WTP (V) and the gross price paid (that is, p+tx). Producer Surplus (PS) is the aggregated difference between price (p) and marginal cost (c ), which is assumed to be zero. Therefore, PS= (PS1+PS2). The welfare loss (WL1+WL2) is the aggregated travel cost. Or in terms of product characteristics, welfare loss is the cost that stems from the fact that a product does not perfectly meet a consumer’s taste optimum. If we assume that the firm serves the entire market, and assuming all things being equal, the overall welfare loss is independent of the price level. A lower price would only shift surplus from the firm to consumers. CS WL1 WL2 PS 1 PS 2 p Now assume, the firm opens a second store. The store are now located at z=¼ and z=¾. The Figure below shows the welfare implications. C1 C2 WL1 WL3 WL4 WL2 P2 PS1 PS2 PS3 PS6 PS4 PS5 P1 PS7 PS8 PS9 PS10 Consumer Surplus With one store, CS was (C1+C2+WL3+WL4+PS1+PS2) with two stores it is only (C1+C2) That is, a large part of CS is lost:  (PS1+PS2) go to producers and become PS.  (WL3+WL4) become welfare loss. Producer Surplus Due the price increase from P1 to P2, there is a substantial increase in PS.  the firm gains (P1+P2+P3+P4+P5+P6) Welfare Loss With one store, WL was (WL1+PS3+PS4) and (WL2+PS5+PS6) with two stores it is (WL1+WL2+WL3+WL4) On the one hand, some WL has become PS (PS3+PS4+PS5+PS6) on the other hand, some CS has become welfare loss  the next effect on WL is thus [(WL3+WL4)  ­ (PS3+PS4+PS5+PS6)] and is negative in this example (negative means, the welfare loss has declined – a positive thing!) However, note that a comprehensive analysis of welfare losses should also consider the set up cost F, which is not contained in the Figure above. Therefore, we should also add one F, the set ­up cost of the second store, to the welfare loss. Conclusion: Welfare ­wise, more stores (or brands) are always bad for CS and always good for PS. The net effect on overall welfare is, however, not clear a priori. (2) Welfare ­optimizing number of stores/brands To find the welfare ­optimal number of stores we need to calculate all cost and minimize the cost function with respect to n. First, welfare cost is the area under the travel cost curve. Recall the price level does not influence this area. There will even be a welfare loss if p=0, as shown in the Figure below. ` € € € € ȹ 1 ȹȹ t ȹ ȹ t ȹ t The WL area of the left wedge is equal to 0.5 ȹ ȹȹ ȹ = 0.5ȹ 2 ȹ = 2 . ȹ 2 n Ⱥȹ 2 n Ⱥ ȹ 4 n Ⱥ 8 n Since each store has two of these areas we multiply by two and will get a loss of t . Since the base (i.e., 1/2n) only reflects a fraction of all N consumers, we need 4n2 € Nt to multiply by N to obtain $ of transportation cost: 2 . Finally we need t multiply 4n Nt by n to account for the number of stores and get . This is the area 4n (WL1+WL2+WL3+WL5) in the Figure above. € If we further consider nF set ­up costs we obtain the cost function for n ­shops as € Nt C ( N , n ) = + nF 4n Minimizing the cost function with respect to n yields ∂C ( N , n ) Nt Nt = − 2 + F =0 ⇒ = F ∂n 4n 4n2 We can further simplify to Nt Nt Nt = 4 Fn 2 ⇒ n 2 = ⇒ n= 4F 4F * Note that welfare optimizing n, (let us call this nWL ), is smaller than the profit ­ * * * maximizing nπ . Thus nπ > nWL . € € € € Example: Employing the same example as above with F=$50,000, N=5,000,000 and t= yields 5, 000, 000 * * nWL = = 25 = 5 [Recall, nπ ≈ 7 .] 200, 000 This indicates that, although some variety is not, there is a limit to the welfare ­ € optimizing variety. That limit is determined by the strength of consumers’ preferences (t), set ­up costs of new brands (F) and by the size of the market (N). D. Horizontal Differentiation and Price Discrimination Ideally, the monopolist would like to convert all remaining consumer surplus into producer surplus. Since we assumed all consumers to have identical reservation prices of V but different transportation cost T, the monopolist can charge a uniform delivery price and absorb the transportation cost himself. Charging each consumer V and is discriminatory pricing because, even if consumers pay the same price, V does not reflect the true cost of delivering the product to different locations. Everything being equal, uniform delivery prices leave the marginal consumer, i.e., the one with V=p+t/2n, neither worse nor better ­off. However, all other consumers (i.e., the infra ­marginal consumers) are worse ­off. Let s compare the price ­discriminatory firm with the no ­price discrimination case. Since the delivering firm incurs transportation cost of (t/2n) it will make a profit as long as V > c + t/2n. Since c<p, this condition is weaker than the one for the non ­discriminatory firm (V>p+t/2n), which underpins the quantity implications of price discrimination: quantity produced (or number of customers served) is higher in the presence of price discrimination. The price ­discriminatory firm considers the following profit ­relevant factors: Revenue: NV (it charges all N consumer their reservation price V) Production Cost: cN Transportation Cost: (tN/4n) (as shown in the welfare part above) Set ­Up Cost: nF € That yields the profit function: tN π ( N , n ) = NV − cN − − nF 4n Calculating the partial derivative with respect to n yields results that are identical tN with the welfare maximizing non ­discriminatory firm (see above): n * = . Dis 4F That is, price ­discriminatory firms will choose the socially efficient degree of € product variety. They reach remote customers by price reductions rather than by additional varieties. ...
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