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2. [Theory] In this question, well calculate CV and EV for a simple fall in prices. WRITE YOUR FINAL ANSWERS IN THE TABLE BELOW. THE TA WILL NOT GIVE MARKS IF THE FINAL NUMERICAL ANSWER IS NOT ON THE TABLE Part Item Value (2 decimal places) C. d. e. cV Xev Yev lev EV nd Consider the following situation. There are two goods, X and Y. Good X costs p per unit, a good Y is our numeraire and costs 1 per unit. Let the quantity consumed of good X be x, the quantity consumed of good Y be y, and I be income (that is, total spending). Utility is given by U(x,y)s хгу. Total spending (income) is 1 = px + y Lets plot this on a graph where we have y on the y-axis and x on the x-axis (just like in the lecture If we let u represent the level of utility, the formuia fo ). r the corresponding indifference curves can be written as Indifference curve for utility level u Slope of indifference curve: Similarly, from our income expression we can derive the budget line: Budget line: y = 1-px dy 2u Slope of budget line: p dy dx
Originally, at time O, the price of x is po 100 and income I 300. Using a subscript o to values at this time, the consumer chooses a bundle (xo.Yo) such that xo 2, and yo is therefore Uo 22 x 100 400. At time 1, the price of x falls to p1- 50. Income is unchanged: I-300. The consumer chooses a bundle (xi.Vi) such that 1,600. x1 4, and y1 100 (yes, the same as before). Utility is therefore U- ). Utility is therefore U The situation is described by the table and graph below. Description Symbol Before (O) After (1) 50 100 2 Price of x 4. 100 100 00 300 Quantity of x Quantity of yy 400 1,600 Utility Income 300 200 100 Lets find the Compensating Variation for the change in prices. For that, well need to keep the NEW PRICE (p1) and the OLD INDIFFERENCE CURVE (Utility Uo What were finding is illustrated in the diagram below. We want to find the income level, lev, such that at the new, lower price the consumer stays on the original indifference curve. The bundle the consumer chooses is (xevYe) and the compensating variation is equal to her original income, 300, minus lev. CV lo-lev 300-l
300 200 IC 100 We have three unknowns (xevYov,lev) and three equations. The first equation is our budget constraint: total spending must equal lev. The second equation says we must stay on the original indifference curve: U(xov,Vev) Uo. The third equation says that at the optimal consumption bundle, the slopes of the indifference curve and the budget line must be equal to each other. CV Equation 1: piv + ycv CV Equation 2: U(xcv, yeu)-x; oe,-240 CV Equation 3: rcv (1 mark) Solve CV Equation 3 for xev. Show your work and write your answer down correct to two decimal places. a. (1 mark) Solve CV Equation 2 for yo. Use your value for Xev from a. Show your work and write your answer down correct to two decimal places. b. (1 mark) Solve CV Equation 1 for lev. Use your values from parts a. and b. Show your work and write your answer down correct to two decimal places: c. d. (1 mark) Calculate the compensating variation using CV lo-lev. Use your value for le from part c. Show your work and write your answer down correct to two decimal places. Now lets calculate the equivalent variation (EV). For that, well need to keep the OLD PRICE (Po and the NEW INDIFFERENCE CURVE (Utility-U1).
such that at the original price the consumer achieves the new, higher bundle the consumer chooses is (Xev,yev) and the compensating variati her original income, 300. EV = lev-lo = lev-300. ere finding is illustrated in the diagram below. We want to find the income level, lev level of utility U1. The on is equal to lev minus 5001 ev U(x, y)-U 400 EV 300 200 evi yet 100 4
have three unknowns (Xev, Yev,ley) and three equations. The first equation is our budget constraint: total spending must e indifference curve: U(xevyev)-Un The third equation says that at the optimal consumption qual lev. The second equation says we must stay on the new bundle, the slopes of the indifference curve and the budget line must be equal to each other. EV Equation 1: Poxev + Yev le EV Equation 2: U(ev yev)xavyev EV Equation 3:- 2tu (1 mark) Solve EV Equation 3 for xev. Show your work and write your answer down correct to two decimal places. e. (1 mark) Solve EV Equation 2 for yev. Use your value for Xev from e. Show your work and write your answer down correct to two decimal places. f. (1 mark) Solve EV Equation 1 for lev. Use your values from parts e. and f. Show your work and write your answer down correct to two decimal places. g. h. (1 mark) Calculate the equivalent variation using EV lev-lo. Use your value for lev from part g. Show your work and write your answer down correct to two decimal places.
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Date-_ (a) CV Equatim 3 3 2 (Hua-50 ,-400 50 c샤 = 400 (2.5143) 40 6.349 63 002 c시Date -/-/ CV Equation 50 (2.5198)+63.002 -ICN 135.99 +63-002 188 442 る00-188 .992 CV 111.008 ろ 3 2 (1600 u160 2 (IL00 lo O 3 32Date--/--1. → (3120 ye-1600 6o 0 3 124 2 0:014 158 82 G EV equation 100 (3.GH) + 158.824 ะ 1ey 313-니 +158-824 Eley > у-1 6: 229-300

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