Problem-Set-6-Solutions

Problem-Set-6-Solutions - Problem Set 6 Solutions...

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Problem Set 6 Solutions — Competitive Equilibrium Problem 1 Competitive Equilibrium ( a ) Consumer 1: We know an interior allocation is a solution to consumer 1’s utility maxi- mization problem if the slope of her indifference curve is the same as the slope of the budget constraint. By equating MRS 1 to the price ratio, we get MRS 1 = 1 x 1 1 y 1 = y 1 x 1 = p x p y . Hence, y 1 = p x p y x 1 . (If the above equation did not have a solution — as will be the case in Problem 3 — then we would need to check for corner solutions.) Substituting this into the budget constraint, we get p x x 1 + p x x 1 = 4 p x + 2 p y . Therefore, the demand for good x by consumer 1 is x * 1 ( p x , p y ) = 4 p x + 2 p y 2 p x = 2 + p y p x . Since y 1 = p x p y x 1 , the demand for good y by consumer 1 is y * 1 ( p x , p y ) = 2 p x p y + 1 . Consumer 2: Again, check for interior solutions by equating MRS 2 to the price ratio: MRS 2 = 2 1 x 2 3 1 y 2 = 2 y 2 3 x 2 = p x p y . Hence, y 2 = 3 2 p x p y x 2 . Substituting this into the budget constraint, we get p x x 2 + 3 2 p x x 2 = 5 p x + 5 p y . Therefore, consumer 2’s demand of good x is given by x * 2 ( p x , p y ) = 5 p x + 5 p y 5 2 p x = 2 + 2 p y p x . Since y 2 = 3 2 p x p y x 2 , we then have y * 2 ( p x , p y ) = 3 p x p y + 3 . 1
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( b ) We already know that these demand functions will satisfy Walras’ law since this result is a theorem that holds for any demand functions. However, to verify directly that it holds for this problem, we simply plug in the above formulas for the demand functions to get p x h x * 1 ( p x , p y ) + x * 2 ( p x , p y ) - e x 1 - e x 2 i + p y h y * 1 ( p x , p y ) + y * 2 ( p x , p y ) - e y 1 - e y 2 i = p x h 2 + p y p x + 2 + 2 p y p x - 4 - 5 i + p y h 2 p x p y + 1 + 3 p x p y + 3 - 2 - 5 i = h 3 p y - 5 p x i + h 5 p x - 3 p y i = 0 ( c ) To calculate the market-clearing prices, first consider market for good x : x * 1 ( p x , p y ) + x * 2 ( p x , p y ) = e x 1 + e x 2 2 + p y p x + 2 + 2 p y p x = 9 3 p y p x = 5 p x p y = 3 5 . Therefore, when the ratio of prices of good x to good y is p x p y = 3 5 , the market for good x clears. By Walras’ law, in a two-good economy, when one market clears, so does the other. Therefore a price ratio of 3 5 also ensures the market for good y clears. To verify the above claim, let’s check what prices will clear the market for good y . 2 p x p y + 1 + 3 p x p y + 3 = 7 . Clearly, this also boils down to p x p y = 3 5 . Thus, the market-clearing price ratio is p x p y = 3 5 . In other words, any prices p x and p y that satisfy p x p y = 3 5 will clear both markets. ( d ) There are multiple competitive equilibrium (CE) prices. Let’s use ( p x , p y ) = ( 3 5 , 1). We substitute ( p x , p y ) = ( 3 5 , 1) into the demand functions for consumers 1 and 2 to solve the equilibrium consumption bundles for consumers 1 and 2: x * 1 ( 3 5 , 1) = 2 + 5 3 = 11 3 x * 2 ( 3 5 , 1) = 2 + 2 · 5 3 = 16 3 y * 1 ( 3 5 , 1) = 2 · 3 5 + 1 = 11 5 y * 2 ( 3 5 , 1) = 3 · 3 5 + 3 = 24 5 Thus, the equilibrium allocation is ( x * 1 , y * 1 ) = (3 2 3 , 2 1 5 ) and ( x * 2 , y * 2 ) = (5 1 3 , 4 4 5 ).
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