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2. Consider a consumer located at point $x$ with the utility function of $U_{x}={?p_{A}???x?a??p_{B}???x?(L?b)??if she buys fromAif she buys fromB?$ where $p_{A}$ (resp. $p_{B}$ ) is the price paid for purchasing from firm A (resp. B), $x?$ $[0,L]$ denotes the location of each consumer, $L>0$ is the length of a linear street, $?$ is transportation cost per unit of distance when a consumer has to pay for shopping at a (resp. b) where A (resp. B) store is located, given that the total number of consumers in the economy is $L$. (a) $(20%)$ Find the demand functions faced by firm A and firm B, respectively. (b) $(20%)$ Find the Bertrand-Nash equilibrium prices and firm profits. (c) $(10%)$ Comments on your results.

a) The demand functions faced by firm A and firm B can be found by taking the derivative of the utility function with respect to the price. For firm A

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