Econ 101AGoldman/Wang/ChanFall 96Answer Key for Midterm 11. (a) The marginal rate of substitution (MRS) is the negative of the slope of the indi®erencecurve at some point. For two goods, the MRS re°ects how much the individual mustreceive of one good to be just as well o® after losing another good.(b) The Slutsky income compensation is the amount of money given or taken away thatmakes the original bundle consumed just a®ordable under the new prices.(c) The expenditure function shows the minimum amount of money needed to attain a givenutility for a particular set of prices.(d) If the demand for a good increases when income increases (or demand decreases whenincome decreases), then that good is a normal good for that range of income.(e) If A and B are any two situations or bundles, an individual has complete preferencesif she can specify (1) A is preferred to B, (2) B is preferred to A, or (3) A and B areequally attractive.2. (a) UNCERTAINThe statement is false for normal goods but true for inferior goods. The change inconsumer surplus is, by de¯nition, equal to the change in expenditure that is necessaryto keep utility constant. Hence, it is measured by the area bounded by the Hicksiandemand curve at the initial utility level, and the original and new price levels. Fornormal goods, since the income e®ect is positive, the Hicksian demand curve is steeperthan the Marshallian demand curve, thereforeRp2p1dxdpxis bigger(not smaller) than thechange in consumer suplus. The opposite is true for inferior goods.(b) FALSEBoth A and C are a®ordable at budget line BL1, and the consumer chooses A; thereforeARC. However, choosing C at BL2does not violate WARP, because A is not a®ordablethen.(c) UNCERTAINThe statement is true for interior solutions, but is not necessarily true for corner solu-tions. For example, if x = 0 and y > 0, it is possible thatUxpx<Uypy.(d) TRUESince both x, y are strictly positive, at the original optimal bundle,Uxpx=Uypy. Supposethe income goes up by ¢I. If the consumer spends some of ¢I in good X, then Uxwillgo down because Uxx< 0, and Uywill go up because Uxy> 0. Now,Uxpx<Uypy; thereforethe consumer will spend some of the rest of ¢I on good Y. As a result, the consumptionof both X and Y will increase.3. (a) Note that a decrease in prices implies an increase in purchasing power that a®ects bothgoods. The total e®ect on demand depends on both the substitution and the incomee®ects. Therefore, the income e®ect might be revealed by examining the sign of thechange in total demand and the sign of the substitution e®ect.1First consider the e®ect of the price decrease on software. The Slutsky decompositiontells us@s@ps=@s@ps¯¯¯¯U constant| {z }Substitution E®ect¡ s ¢@s@I| {z }Income E®ect:The own-price substitution e®ect will always be negative. The income e®ect will benegative if software is an inferior good and will be positive if software is a normalgood. But for a su±ciently large substitution e®ect, Prof. Goldman will consume moresoftware when software prices decrease even if software is an inferior good. Consequently,Prof. Goldman's increase in software consumption after a price decrease does not tell uswhether software is normal or inferior.Now consider the e®ect pshas on diet coca cola. The Slutsky decomposition tells us@d@ps=@d@ps¯¯¯¯U constant| {z }Substitution E®ect¡ d ¢@d@I| {z }Income e®ect:Note that@d@psjU constant> 0 when Prof. Goldman only consumes two goods and hasmonotonic preferences. Why? We know that the own-price substitution e®ect willalways be negative. Consequently, when psdecreases Prof. Goldman will consume moresoftware than before the price change. If@d@psjU constant< 0 then the price decrease alsoimplies Prof. Goldman consumes more diet coca-cola. But if Prof. Goldman consumesmore of both goods, monotonic preferences imply that he is better o®. Prof. Goldmancannot be better o® because the substitution e®ect holds utility constant. Therefore,@d@psjU constant> 0.If the cross-price substitution e®ect is negative then we must have@d@I> 0 for@d@ps> 0.Therefore, Prof. Goldman's increased consumption of diet coca-cola implies that dietcoca-cola is a normal good.(b) Prof. Goldman's original utility level isU¤= V (ps; pd; I) = V (9; 1; 39) = ¡³p9 +p1´236= ¡49:His increase in consumer surplus can be calculated by using the expenditure function¢CS = E(9; 1; 36) ¡E(4; 1; 36) = ¡7U¤=634:Alternatively, we can calculate the change in consumer surplus by integrating the hick-sian demand for software.4.max U(C; L) = CL + L2subject to C + wL = wTL = CL + L2+ ¸(wT ¡C ¡ wL)and2@L@C= l ¡¸;@L@L= C + 2L ¡¸w:If L > 0 and C > 0, then@L@C=@L@L= 0. This impliesC = L(w ¡ 2)Subtituting into the budget constraint givesL =wT2(w ¡ 1); C =wT (w ¡ 2)2(w ¡ 1); T ¡L =T (w ¡ 2)2(w ¡ 1):T-L, the number of hours that steve will work, is always non-negative. That will be the casei® w ¸ 2. There is no interior solution when w < 2.It is obvious that L will not be zero.If L = T and C = 0, then@L@C· 0 and@L@L= 0.@L@L= 0 ) ¸ = C +2Lwand@L@C= L ¡C ¡2Lw= T (1 ¡2w):Therefore@L@C· 0 i® w · 2. There is no corner solution when w ¸ 2.(a) Steve will not work i® w · 2.(b) Steve's labor supply is given byT ¡L =T(w ¡2)2(w ¡ 1)when w ¸ 2:(c) The after tax wage rate is equal to (1 ¡t100)w and the labor supply will be equal toT ¡L = 0 when (1 ¡t100)w < 2=T ((1 ¡t100)w ¡ 2)2((1 ¡t100)w ¡ 1)when (1 ¡t100)w ¸ 2Therefore, the tax revenue,t100w(T ¡L) is equal tot100wT ((1 ¡t100)w ¡ 2)2((1 ¡t100)w ¡ 1)when (1 ¡t100)w ¸ 2(d) No. From part c, we know that Steve will work less when the tax is increased. If the taxis too high, the e®ect of the reduction in labor supply will more than o®set the e®ect ofthe increase in tax rate. In particular, if t is so high such that (1 ¡t100)w < 2, Steve willnot work at all, and the tax revenue will drop to