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Problem Set #7 Suggested Answers to Problems 7.1; 7.3, 7.5 and 7.9 7.1 Imagine a market for X composed of four individuals: Mr. Pauper (P), Ms. Broke (B), Mr Average (A), and Ms. Rich (R). All four have the same demand function for X: It is a function of income (I), PX, and the price of an important substitute (Y) for X: X IPY 2 PX a. What is the market demand function for X? If PX = PY = 1, IP = IB = 16, IA = 25 and IR = 100, what is the total market demand for X? What is eX,PX? eX, PY? eX, I? X = [IP(PY)] .5/[2(PX)] + [IBPY ] .5/[2(PX)] +[ IAPY] .5/[2(PX)] +[ IRPY] .5/[2(PX)] = X = [16(1)] .5/[2(1)] + [16(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[100(1)] .5/[2(1)] = =2 + 2 + 2.5 + 5 = 11.5 Let IAgg denote IP.5+ IB.5+ IA.5 + IR.5. Then eX,PX =[X/Px][PX/X] = -[IAgg (1).5]/[2PX2] { PX/ X} = -1 (recalling the definition of X) by similar reasoning eX, PY = [X/Py][Py/X] = (1/2)[IAgg (PY) -.5] /[2PX] { PY/ X} = 1/2 The income elasticity cannot be computed without knowing the distribution of income changes. b. If PX doubled, what would the new level of X demanded? If Mr. Pauper lost his job and his income fell 50 percent, how would that affect the market demand for X? What if Ms. Rich’s income were to drop 50%? If the government imposed a 100 percent tax on Y, how would the demand for X be affected? - Double PX to 2 and quantity demanded becomes [IAgg (1) .5] /[2(2)], or half the previous quantity demanded (5.75) - If IP = 8 then X’ = [16(1)] .5/[2(1)] + [8(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[100(1)] .5/[2(1)] = X’ = 2 + 2+ 2.5 + 5 = 10.91 - If IR = 50 then X’ = [16(1)] .5/[2(1)] + [16(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[50(1)] .5/[2(1)] = X’ = 2 + 2+ 2.5 + 2.52 = 10.03 -If Py + t = 2, then X’ = [16(2)] .5/[2(1)] + [16(2)] .5/[2(1)] +[25(2)] .5/[2(1)] +[50(2)] .5/[2(1)] = X’ =11.5(2).5 = 16.26 c. If IP = IB = IA = IR =25, what would be the total demand for X? How does that figure compare with your answer to (a)? Answer (b) for these new income levels and PX = PY = 1. X’ = 4[25(1)] .5/[2(1)] = 10. - Double PX to 2 and quantity demanded becomes [IAgg (1) .5] /[2(2)], or half the previous quantity demanded (5) - If IP falls by half (to 12.5) then X’ = 3[25(1)] .5/[2(1)] + [12.5(1)] .5/[2(1)] = X’ = 7.5 +1.77 = 9.57 - If IR falls by half, we repeat the above. -If Py + t = 2, then X’ = 4[25(2)] .5/[2(1)] = 102 = 14.1. d. If Ms. Rich found Z a necessary complement to X, her demand function for X might be described by the function IPY 2PX PZ What is the new market demand function for X? If PX = PY = PZ= 1 and income levels are those described by (a), what is the demand for X? What is eX,PX? eX, PY? eX, I? eX, PZ? What is the new level of demand for X if the price of Z rises to 2? Notice that Ms. Rich is the only one whose demand for X drops. X Market demand X = [IP(PY)] .5/[2(PX)] + [IBPY ] .5/[2(PX)] +[ IAPY] .5/[2(PX)] +[ IRPY]/[2(PX) (PZ)] At current prices X = [16(1)] .5/[2(1)] + [16(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[100(1)] /[2(1)(1)] = 2 + 2 + 2.5 + 50 = 56.5 eX,PX =[X/Px][PX/X]. Here [X/Px] = -1[IP(PY)] .5/[2(PX)2] + -1[IB(PY)] .5/[2(PX)2]+ -1[IA(PY)] .5/[2(PX)2]+-1IRPY/[2(PX)2PZ] and, with current parameters = -1[16(1)] .5/[2(1)] +-1 [16(1)] .5/[2(1)] +-1 [25(1)] .5/[2(1)] +-1 [100(1)] ./[2(1)] =-1(56.5) = -X/PX. Thus, eX,PX = [X/Px][PX/X] = -1 eX, PY = [X/Py][Py/X] = Here X/Py = .5[IP.5(PY)- .5] /[2PX] + .5[IB.5(PY)- .5] /[2PX] +.5[IA.5(PY)- .5] /[2PX] + IR /[2PX PZ] = .5X + .5IR /[2PX PZ] Thus eX, PY = {.5X + IR /[2PX PZ]} [Py/X] =[.5(56.5) + 100/2]1/56.5 = .5 + .884 = 1.38 -Income elasticity Again, income elasticity cannot be determined without knowing the distribution of an income change Finally suppose that the PZ increases to 2. Then new market demand becomes X = [16(1)] .5/[2(1)] + [16(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[100(1)] /[2(1)(2)] =2 + 2 + 2.5 + 25 = 31.5 7.3 Tom, Dick and Harry constitute the entire market for scrod. Tom’s demand curve is given by Q1 – 100 – 2P for P< 50. For P>50, Q1 = 0. Dick’s demand curve is given by Q2 – 160 – 4P for P<40. For P>40, Q2 = 0. Harry’s demand curve is given by Q3 – 150 – 5P for P<30. For P>30, Q3 = 0. Using this information, answer the following: a. How much scrod is demanded by each person at P = 50? At P = 35? At P = 25? At P = 10? At P =0? Price 50 35 25 10 0 Tom (1) 0 30 50 80 100 Dick (2) 0 20 60 120 160 Harry (3) 0 0 25 100 150 Market 0 50 135 300 410 b. What is the total market demand for scrod at each fo the prices specified in part (a)? See the rightmost column in the table above. c. Graph each individual’s demand curve Tom 60 Dick 60 50 50 50 40 40 40 30 30 30 20 20 20 10 10 10 0 0 0 200 0 0 400 Harry 60 200 400 0 200 400 d. Use the individual demand curves and the results of part (b) to construct the total market demand curve for scrod. Summing horizontally, Market 60 50 40 30 20 10 0 0 200 400 P 7.5 For this linear demand, show that the price elasticity of demand at any given point (say point E) is given by minus the ratio of distance X to distance Y in the figure. How might you apply this result to a nonlinear demand curve? Our task is to show that [X/P] [P/X] = -X/Y D D Y Notice first that X is the distance from the origin to P*, or P*. Distance Y is the distance from the intercept to P*, or Po – P*. E P* Y0 Now, a linear demand curve may be written as X Q* Q = a – bP, b>0. Inverting, P* = a/b – Q*/b. Thus, When Q* = 0, Po Q = a/b. At Q*, P* = a/b – Q*/b. Differencing, Po – P* Thus X/Y = = Q*/b = P*/[Q* a/b - [a/b - Q*/b] /b] = b[P*/Q*], where b = Q/P 7.9 In Example 7.2 we showed that with 2 goods the price elasticity of demand of a compensated demand curve is given by esX PX = -(1-sx) where sx is the share of income spent on good X and us the substitution elasticity. Use this result together with the elasticity interpretation of the Slutsky equation to show that: a. if =1 (the Cobb-Douglas case), eX PX + eY PY = -2 b. >1 implies eX PX + eY PY < -2 and <1 implies eX PX + eY PY > -2. These results can easily be generalized to cases of more than two goods. Both a and b are answered similarly. The Slutsky equation implies for X and Y that eX PX = esX PX + sx eX I and eY PY = esY PY + sY eY I Adding the two Slutsky equations together eX PX + eY PY = esX PX + esY PY - sx eX I - sY eY I Now, by Engel’s law, sx eX I + sY eY I = 1. Thus eX PX + eY PY = esX PX + esY PY - 1 Finally, for either X or Y compensated demand may be written esX PX = -(1-sx) or esY PY = -(1-sY). Inserting eX PX + eY PY = -(1-sx) -(1-sY) =--1 - 1 (The latter expression since the sum of shares equals 1) Thus = 1 implies the sum of own price elasticities equals -2, and <1 implies that the sum of own price elasticities is <-2. Finally, it is useful to consider more explicitly the definition of the elasticity of substitution parameter, . In the utility context = [(Y/X)/(Y/X)] / [(Py/Px)/(Py/Px)] That is, the substitution elasticity measures the percentage change in relative input use induced by a one percent change in relative prices. Thus, the expression esX PX = -(1-sx) = implies that the compensated elasticity of demand for a good is a the extent to which consumers shift away from X as the relative price of X increases, () adjusted by the prominence of X in the consumption bundle.