Problem set 2 solutions Prof. Justin Marion Econ 100M Winter 2012 1. I+S effects Recognize that the utility function U =min{2x 1,4x 2 } represents perfect complements, and that the goods will be consumed in fixed proportion. The utility maximizing solution will always be where 2x 1 = 4x 2, or x 1 = 2x 2. So this consumer will always want twice as much good 1 as good 2 (think of the example with 2 lumps of sugar with each cup of coffee). a. The budget line: 5x 1 +x 2 = 50. Substitute for x 1 using x 1 = 2x 2. 5(2x 2 )+x 2 = 50 So x 2 = 50/11 4.5 and x 1 = 2x 2 = 9. b. New budget line: 2x 1 +x 2 = 50. Again substituting for x 1 : 2(2x 2 )+x 2 = 50. x 2 = 10 and x 1 = 20. c. m = 2(9)+1(4.5) = 22.5 Find bundle at new prices p 1, and compensated income: Substitute x 1 = 2x 2 into 2x 1 +x 2 = 22.5: x 2 = 4.5 and x 1 = 2(x 2 ) = 9. d. What you found in parts (a) and (c) was x 1 (p 1,,m) and x 1 (p 1,,m ) which allows you to find the substitution effect, which tells you how the consumer would change consumption of good 1 in response to a change in price, holding constant purchasing power. The substitution effect is 0, as x 1 (p 1,,m) = x 1 (p 1,,m ) = 9. You can find the income effect using what you found in parts (b) and (c): x n 1 = x 1(p 1,,m) x 1 (p 1,,m ). In other words, represent the gain in purchasing power with the difference between m and m. Holding constant prices at there new level p 1,, how does consumption change? x n 1 = 20 9 = 11. The entire response in consumption is due to the income effect. There is no scope for substitution since these preferences are represented by perfect complements there is no way for the consumer to substitute toward good 1 holding constant utility or purchasing power. e. CV = e(p 1,,u 0 ) e(p 1,,u 0 ) or in words how much less does it cost to achieve u 0 at the lower price p 1 1. First, find u 0. This is min{2(9),4(4.5)} = 18. How much does it cost to achieve a utility level of 18 at the new prices? This is easy to find with the perfect complements utility function. It s just the m = 22.5 you found in part (c). So CV = 22.5 50 = 27.5. This notion is captured in figure 1, where the dotted line represents the budget line that compensates for the price drop to leave utility unchanged. Letu 1 betheutilitylevelyoucanachieveafterthepricechange. EV = e(p 1,,u 0 ) e(p 1,,u 1 ) 1
or in words what change in income would be equivalent to the change in price? The first term is easy this is m = 50. The second term is trickier. From your answer to (b), we see that u 1 =min{2(20),4(10)} = 40. What is the least costly way to achieve utility of 40? This is where x 1 = 20 and x 2 = 10. All other bundles on the u(x1,x2) = 40 indifference curve involve greater expenditures. Purchasing this bundle at the original prices would require 5(20) + 1(10) = $110. So e(p 1,,u 1 ) = 110 and the EV = 50 110 = 60. This is illustrated graphically in Figure 2, where the dotted line is the budget line that allows you to afford u 1 at the original prices. 2. The Slutsky equation a. TheLagrangian is L = x 1 + x 2 λ[p 1 x 1 + x 2 m]. The consumer s first-orderconditions are: x 1 : x 1/2 1 /2 = λp 1 x 2 : x 1/2 2 /2 = λ λ: p 1 x 1 + x 2 = m Combine the first two equations: x 2 = ( p 1 ) 2 x 1 (1) (notice that this is just the condition that the slope of the indifference curve is equal to the ratio of relative prices). Substitute this into the budget constraint: p 1 x 1 +p2(( p 1 ) 2 x 1 ) = m Solving for x 1 : Use this in (1) to obtain x 1 (p 1,,m) = x 2 (p 1,,m) = These are the regular Marshallian demand curves. b. Price elasticity is x i p i p i x i. Price elasticity for good 1: m p 1 (p 1 + ) mp 1 (p 1 + ) (2) (3) ǫ = 2p 1 + (p 1 + ) (4) where we have substituted for x 1 using (2). Income elasticity is similarly x i p i p i x i. Income elasticity for good 1: ǫ m = mp 1(p 1 + ) m where again we have substituted for x 1 using (2). p 1 (p 1 + ) = 1 (5) c. The expenditure minimization problem is: min p 1 x 1 + x 2 subject to the constraint that x 1 + x 2 = u 0. Thelagrangian for this problem is given by L = p 1 x 1 + x 2 λ[ x 1 + x 2 u 0 ] 2
curve: The first-order conditions are x 1 : p 1 = λx 1/2 1 /2 x 2 : = λx 1/2 2 /2 λ: x 1 + x 2 = u 0 Divide the first of these equations by the second and you get: x 2 = ( p 1 ) 2 x 1. Substitute this into the constraint and solve for x 1 to get the Hicksian (compensated) demand h 1 (p1,p2,u 0 ) = u 2 2 (p 1 + ) 2. (6) Using this in x 2 = ( p 1 ) 2 x 1, you can solve for the Hicksian demand for x 2 : h 2 (p1,p2,u 0 ) = u 2 1 (p 1 + ) 2. (7) d. e(p 1,,u 0 ) = p 1 h 1 (p1,p2,u 0 ) + h 2 (p1,p2,u 0 ). Using the Hicksian demands you just solved for, this becomes e(p 1,,u 0 ) = u2 p 1 p 1 +. We can verify that x 1 (p 1,,e(p 1,,u 0 )) = h 1 (p1,p2,u 0 ) by substituting for m in equation (2) using e(p 1,,u 0 ): e. The Slutsky equation is x 1 / p 1 = h 1 x x 1 1 p 1 m The term on the left-hand side tells you the slope of the regular demand curve. The first term on the right-hand side tells you the slope of the Hicks demand curve. The last term is negative if demand is normal, so regular demand is more price responsive than Hicks demand. Therefore the Hicks demand curve will be steeper. f. x 1 / p 1 = m(2p 1 + 1 (p 1+ ) 2 h 1 / p 1 = 2 u2 0 p2 2 (p 1 + ) 3 p x 1 / m = ( 2 p 1 (p 1 + ) Substitute these into the Slutsky equation (8): m(2p 1 + 1 (p = 2 u2 1+ ) 2 0 p2 2 (p 1 + ) x 3 1 ( p 1 (p 1 + ) Now use the fact that x 1 = m p 1 (p 1 + ) and that u 0 = x 1 + x 2, which after a bit of work simplifies tou 2 0 = m(p 1+ ) p 1. Once you substitute these into the above expression, the right-hand side reduces to m(2p 1 + 1 (p. 1+ ) 2 3. Labor supply and welfare a. F +10L = 20000 where full income is $ 20,000. b. U(F,L) = 2lnL+lnF (8) 3
The usual two conditions hold at the optimum. The budget constraint must be satisfied and the marginal rate of substitution is equal to relative prices. This last condition is MRS = MU L /MU F = w/p F 2F/L = 10 F = 5L. Use this in the equation for the budget line to find L = 1333.33. F = 5(1333.33) = 6666.67. c. See figure 3. d. In figure 4, I plotted the original (before welfare program) bundle A. For this person, the welfare program leads to a decline in effective wages, which leads to a substitution effect from point A to point B. This is an increase in leisure (a reduction in work). The program also has an income effect from B to C. While the effective wage is lower, which would normally have an adverse income effect, here the person receives money from the government which leads to an improvement in income. As long as leisure is normal, this leads to more leisure (less work). Since both the income and substitution effects go in the same direction, we can conclude that the welfare program will tend to discourage work. Figure 1: X2 U1 U0 X1 4
Figure 2: X2 U1 U0 X1 Figure 3: F Slope = -10 Slope = -5 12000 6000 800 2000 L 5
Figure 4: F A B 2000 C L 6