Homework04Answer

Homework04Answer - Economics 150 Intermediate Microeconmics...

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Economics 150, Intermediate Microeconmics, Fall 2008 Homework 4: Answers Question 4.1 L = x 1 x 2 + λ ( p 1 x 1 + p 2 - I ) First order conditions: dL dx 1 = x 2 + λp 1 = 0 dL dx 2 = x 1 + λp 2 = 0 dL dx 1 = p 1 x 1 + p 2 x 2 - I = 0 Divide the first equation by the second to get: x 2 x 1 = p 1 p 2 Solve for x 2 : x 2 = p 1 x 1 p 2 Plug into the budget constraint: p 1 x 1 + p 2 p 1 x 1 p 2 = I Solve for x 1 : x 1 = I 2 p 1 Plug this into the budget constraint and solve for x 2 to get the demand for x 2 (he doesn’t ask for this, but it will be useful later): p 1 p 2 x 2 p 1 + p 2 x 2 = I x 2 = I 2 p 2 The income elasticity of demand is: dx 1 dI I x = 1 2 p 1 I I 2 p 1 = 1
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Question 4.2 Plug the demand functions for x 1 and x 2 into the utility function to get the indirect utility function: V ( p 1 , p 2 , I ) = I 2 p 1 I 2 p 2 = I 2 4 p 1 p 2 Question 4.3 Solve the expenditure minimization problem to get the compensated demand function: L = p 1 x 1 + p 2 x 2 - I + λ ( x 1 x 2 - U ) First order conditions: dL dx 1 = x 2 + λp 1 = 0 dL dx 2 = x 1 + λp 2 = 0 dL dx 1 = x 1 x 2 - U = 0 Divide the first equation by the second to get: x 2 x 1 = p 1 p 2 Solve for x 2 : x 2 = p 1 x 1 p 2 Plug into the utility constraint: x 1 p 1 x 1 p 2 = U Solve for x c 1 : x c 1 = s p 2 U p 1 2
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Question 4.4 CV = Z p 1 1 p 0 1 x c ( p 1 , p 2 , U ) dp 1 = Z 4 1 p 2 U p 1 1 2 dp 1 = 2 q p 1 p 2 U 4 1 = 2 q p 2 U 4 1 2 - 1 1 2 = 2 q p 2 U Now plug in the indirect utility function for U : CV = 2 s p 2 I 2 4 p 1 p 2 = I p 1 To get CV, evaluate at I = 10 and p 1 = 1: CV = 10 1 = 10 To get EV, evaluate at I = 10 and p 1 = 4: EV = 10 4 = 5 Finally, to get the change in consumer surplus, you want the area under the uncompensated demand curve: Δ CS = Z p 1 1 p 0 1 x ( p 1 , p 2 , I ) dp 1 = Z 4 1 I 2 p 1 dp 1 = I 2 ln p 1 4 1 = I 2 ln 4 3
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Evaluate at I = 10: Δ CS = 5 ln 4 7 CV, EV and ΔCS differ because they measure different concepts of welfare. Compensating variation measures how much you would have to compensate the consumer for the price change after it occurs to get the consumer back to their original utility. This is the area under the compensated demand curve at the original level of utility. Equivalent variation measures the change in wealth that would be equivalent to the price change if it were to occur (in other words, how much you would have to pay the consumer to get them to the new utility level). This is the area under the compensated demand curve at the new utility level. Consumer surplus measures the effect on wealth of a change in prices allowing for income effects. This is the area under the uncompensated demand curve. Looking at the below graph shows why CV > Δ CS > EV (in the case of a normal good), where CV = A + B + C , Δ CS = B + C , and EV = C : Question 4.5 L = 10 x 1 - x 2 1 + 10 x 2 + λ ( p 1 x 1 + p 2 - I ) 4
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First order conditions: dL dx 1 = 10 - 2 x
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