This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 ( 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 marketclearing 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 twogood 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 marketclearing 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 (...
View
Full
Document
This note was uploaded on 01/14/2012 for the course ECON 201 taught by Professor Witte during the Spring '08 term at Northwestern.
 Spring '08
 Witte
 Macroeconomics, Utility

Click to edit the document details