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s - PA - tx' = s - pB - t(1 - 2)2 PAtti = PB + t(1 - 2)2 PA + ti' = PB + t(1 - 2x + 22) patti= pB+t - 2tx+ ti2 2tx = PB - PA +t C PB -PA+ 2t = QA(PA, PB) where QA(PA, PB) is the amount of quantity demanded Firm A realizes. This implies that Firm B enjoys the demand of QB(PA,PB) = 1-QA(PA,PB) = 1- =1 - PB - PA+ 2t QB(PA, PB) =PA -PB 2t + because we are working with a unit mass of consumers distributed across a unit interval from 0 to 1. (b) Using the demands derived in Part (a) we can set up the the Profit Maximization Problems (PMPs) as: CALCULUS PART: Firm A's maximization problem is max TA(PA, PB) = (PA - c) QA(PA, PB) PA20 max PA 20 TA(PA, PB) = (PA - c)i max TA(PA, PB) = (PA - C) (PB - PA PA20 2t + NIN Taking the derivative with respect to pa and setting equal to 0 On A(PA,PB) _ PB - PAIt _PA C OPA 2t 2t 2t = 0 (7) Firm B's maximization problem is max TB(PA, PB) = (PB - C) QB(PA; PB) PB 20 (1 - 2)max TB(PA, PB) = (PB - c) (1 - 2) PB 20 PB 20 max TB(PA; PB) = (PB - C) ( PA PB 2t + Taking the derivative with respect to pp and setting equal to 0 OTB(PA, PB) PA - PB +t PB OpB 2t 2t + 2t = 0 (8) where we now have two equations ((7) and (8)), and two choice variables (PA and pg) to solve for. CALCULUS PART FINISHED. YOUR CALCULATIONS START HERE. Using equations (7) and (8), please find the optimal price Firm A should set to capture their portion of the market (i.e. p*), and the optimal price Firm B should set to capture their portion of the market (i.e. p;). What happens when we set transportation costs to zero (i.e. t = 0)? What type of model does setting transportation costs equal to zero yield? (c) Find the optimal indifferent consumer (i.e. *), which is also the optimal demand Firm A will receive, and find (1 - *), which is the optimal demand Firm B will receive. (d) Find the equilibrium profits for Firm A and Firm B. What happens when we set t = 0? What does this condition resemble?4. (a) A Bertrand Model with Product Differentiation by Location Firms Consider two rms competing a la Bertrand {i.e. a price competition) where each rm has constant marginal costs to > 0]. Each rm is located at the end of a \"Linear City" interval where the interval is a line ranging from D to 1 inclusive {i.e. [[1, 1]). In this setting, the product differentiation between each rm is its location. Consumer Demand [ The Hotetting Demand Martel ] Consumers are distributed uniformly on this interval s.t. the distribution is a mass of con- sumers. We can imagine this distribution as being X N UNIF(D,1) where the expectation of X is % ( i.e. lE(X) = 12;" = Ll, ). Let t be the per-unit transportation cost for the squared distance traveled (d2). This implies the cost for each consumer to travel a certain amount of distance is tdQ. This expression can represent a multitude of costs to the consumer. It can be the consumer's value of time, gasoline, or perhaps some learning curve when adopting the product. In this setting we will consider it to be the consumer's \"traveling cost\" to each rm. We can think of the transportation cost for each consumer I for Firm A to be tie: and for Firm B to be t[1 3? since the consumer mass is distributed along the unit interval of X. Let s be the gross consumer surplus (i.e. its maximum willingness to pay for a good) where are going to assume that s is sufciently high for all consumers to purchase a product. The utility that consumer 1' gets from consuming at most one unit good is IL-=s(p+trt2) where p is the price of the good and tit2 is the travel cost. Notice that we can represent this utility for the consumer that purchases a good from Firm A and B respectively as Q3=s(p,4+txn) and U-E=S{Ps+t(1$)2) 1 because we are assuming this consumers are evenly distributed on the same unit interval they will be traveling. Use these two derived utility fimctions to nd the indierent consumer (i.e. it). This can be found by setting U? = Uf' and solving for a. This a: is if and is considered to be the demand that Firm A will capture. Note: I am giving you the answer to Part A. Please use these results in. the analy- ses fettowtng Part A. In order to determine the quantity demanded for each rm, we need to nd the consumer that is just indifferent between buying from Firm A and Firm B. = Let, :E be the location of the indierent consumer i s.t. the utility of consumer 'i is equal when buying from either rm at Us = v.3