拔高尼科爾森系列課件e and substitution effects

1、Chapter 5 E AND SUBSTITUTION EFFECTSContents Demand functionChanges in eChanges in a goods price The individuals demand curvesA maths development of response to price changeDemand elasticitiesConsumer surplusRevealed preference and the substitution effectDemand FunctionsDemand FunctionsThe optimal l

2、evels of x1,x2,xn can be expressed as functions of all prices and eThese can be expressed as n demand functions of the form:x1* = d1(p1,p2,pn,I)x2* = d2(p1,p2,pn,I)xn* = dn(p1,p2,pn,I)Demand FunctionsIf there are only two goods (x and y), we can simplify the notationx* = x(px,py,I)y* = y(px,py,I)Pri

3、ces and e are exogenousthe individual has no control over these parametersHomogeneityIf we were to double all prices and e, the optimal quantities demanded will not changethe budget constraint is unchangedxi* = di(p1,p2,pn,I) = di(tp1,tp2,tpn,tI)Individual demand functions are homogeneous of degree

4、zero in all prices and e HomogeneityWith a Cobb-Douglas utility functionutility = U(x,y) = x0.3y0.7 the demand functions areNote that a doubling of both prices and e would leave x* and y* unaffectedHomogeneityWith a CES utility functionutility = U(x,y) = x0.5 + y0.5 the demand functions areNote that

5、 a doubling of both prices and e would leave x* and y* unaffectedChanges in eChanges in eAn increase in e will cause the budget constraint out in a parallel fashionSince px/py does not change, the MRS will stay constant as the worker moves to higher levels of satisfactionIncrease in eIf both x and y

6、 increase as e rises, x and y are normal goodsQuantity of xQuantity of yCU3BU2AU1As e rises, the individual choosesto consume more x and y e consumption curve (ICC)Increase in eIf x decreases as e rises, x is an inferior goodQuantity of xQuantity of yCU3As e rises, the individual choosesto consume l

7、ess x and more yNote that the indifferencecurves do not have to be “oddly” shaped. Theassumption of a diminishing MRS is obeyed.BU2AU1Normal and Inferior GoodsA good xi for which xi/I 0 over some range of e is a normal good in that rangeA good xi for which xi/I 0the entire e effect is negativeif x i

8、s an inferior good, then x/I 0the entire e effect is positiveA Slutsky positionWe can demonstrate the position of a price effect using the Cobb-Douglas example studied earlierThe Marshallian demand function for good x wasThe Hicksian (compensated) demand function for good x wasA Slutsky positionThe

9、substitution effect is found by differentiating the compensated demand functionThe overall effect of a price change on the demand for x isA Slutsky positionCalculation of the e effect is easierInterestingly, the substitution and e effects are exactly the same sizeDemand ElasticitiesMarshallian Deman

10、d ElasticitiesMost of the commonly used demand elasticities are derived from the Marshallian demand function x(px,py,I)Price elasticity of demand (ex,px)Marshallian Demand Elasticities e elasticity of demand (ex,I)Cross-price elasticity of demand (ex,py)Price Elasticity of DemandThe own price elasti

11、city of demand is always negativethe only exception is Giffens paradoxThe size of the elasticity is importantif ex,px -1, demand is inelasticif ex,px = -1, demand is unit elasticPrice Elasticity and Total SpendingTotal spending on x is equal tototal spending =pxxUsing elasticity, we can determine ho

12、w total spending changes when the price of x changesPrice Elasticity and Total SpendingThe sign of this derivative depends on whether ex,px is greater or less than -1if ex,px -1, demand is inelastic and price and total spending move in the same direction （agricultural products), higher price ,higher

13、 spendingif ex,px -1, demand is elastic and price and total spending move in opposite directions, lower price ,high spendingCompensated Price ElasticitiesThe compensated own price elasticity of demand (exc,px) isThe compensated cross-price elasticity of demand (exc,py) isIf the compensated demand fu

14、nction isxc = xc(px,py,U) Compensated Price ElasticitiesThe relationship between Marshallian and compensated price elasticities can be shown using the Slutsky equationIf sx = pxx/I, thenCompensated Price ElasticitiesThe Slutsky equation shows that the compensated and pensated price elasticities will

15、 be similar ifthe share of e devoted to x is smallthe e elasticity of x is smallRelationships among Demand ElasticitesHomogeneityDemand functions are homogeneous of degree zero in all prices and eEulers theorem for homogenous functions shows thatAny proportional change in all prices and e will leave

16、 the quantity of x demanded unchangedEngel AggregationWe can see this by differentiating the budget constraint (I=Pxx+Pyy）with respect to e (treating prices as constant)Engels law :IF e elasticity of demand for food items is less than one, THEN e elasticity of demand for all nonfood items must be gr

17、eater than oneCournot AggregationThe size of the cross-price effect of a change in the price of x on the quantity of y consumed is restricted because of the budget constraintWe can demonstrate this by differentiating the budget constraint with respect to pxCournot Aggregation調(diào)價(jià)的作用（此商品的支出）決定于被調(diào)價(jià)商品在人們

18、生活開支中的作用Demand ElasticitiesThe Cobb-Douglas utility function isU(x,y) = xy(+=1,=1)The demand functions for x and y areDemand ElasticitiesCalculating the elasticities, we getDemand ElasticitiesWe can also showhomogeneityEngel aggregationCournot aggregationDemand ElasticitiesWe can also use the Slutsk

19、y equation to derive the compensated price elasticityThe compensated price elasticity depends on how important other goods (y) are in the utility functionDemand ElasticitiesThe CES utility function (with = 2, = 0.5) isU(x,y) = x0.5 + y0.5The demand functions for x and y areDemand ElasticitiesWe will

20、 use the “share elasticity” to derive the own price elasticityIn this case,Demand ElasticitiesThus, the share elasticity is given byTherefore, if we let px = pyDemand ElasticitiesThe CES utility function (with = 0.5, = -1) isU(x,y) = -x -1 - y -1The share of good x isDemand ElasticitiesThus, the sha

21、re elasticity is given byAgain, if we let px = pyMore substitution, more own price elasticityDemand ElasticitiesDemand ElasticitiesFactors That Influence the Elasticity of DemandCloseness of Substitutes. The closer the substitutes, the more elastic the demand.Proportion of e Spent on the Good The gr

22、eater the proportion of e spent on a good, the more elastic the demand.Time Elapsed（時(shí)滯）Since Price ChangeThe longer the time, the more elastic the demand.Short-run demandLong-run demandFactors That Influence the Elasticity of DemandConsumer SurplusConsumer SurplusAn important problem in welfare econ

23、omics is to devise a monetary measure of the gains and losses that individuals experience when prices changeConsumer WelfareOne way to evaluate the welfare cost of a price increase (from px0 to px1) would be to compare the expenditures required to achieve U0 under these two situationsexpenditure at

24、px0 = E0 = E(px0,py,U0)expenditure at px1 = E1 = E(px1,py,U0)Consumer WelfareIn order to compensate for the price rise, this person would require a compensating variation (CV) ofCV = E(px1,py,U0) - E(px0,py,U0)Consumer WelfareQuantity of xQuantity of yU1ASuppose the consumer is maximizing utility at

25、 point A.U2BIf the price of good x rises, the consumer will maximize utility at point B.The consumers utility falls from U1 to U2expenditure at px0 = E0 = E(px0,py,U0)Consumer WelfareQuantity of xQuantity of yU1AU2BCV is the amount that the individual would need to be compensatedThe consumer could b

26、e compensated so that he can afford to remain on U1Cexpenditure at px1 = E1 = E(px1,py,U0)expenditure at px0 = E0 = E(px0,py,U0)Consumer WelfareThe derivative of the expenditure function with respect to px is the compensated demand function (Shephards lemma)Consumer WelfareThe amount of CV required

27、can be found by integrating across a sequence of small increments to price from px0 to px1this integral is the area to the left of the compensated demand curve between px0 and px1welfare lossConsumer WelfareQuantity of xpxxc(pxU0)px1x1px0 x0When the price rises from px0 to px1,the consumer suffers a

28、 loss in welfareThe Consumer Surplus ConceptAnother way to look at this issue is to ask how much the person would be willing to pay for the right to consume all of this good that he wanted at the market price of px rather than doing without the good complete.The Consumer Surplus ConceptThe area belo

29、w the compensated demand curve and above the market price is called consumer surplusthe extra benefit the person receives by being able to make market transactions at the prevailing market priceConsumer WelfareQuantity of xpxxc(.U0)px1x1When the price rises from px0 to px1, the actual market reactio

30、n will be to move from A to Cxc(.U1)x(px)ACpx0 x0The consumers utility falls from U0 to U1Consumer WelfareQuantity of xpxxc(.U0)px1x1Is the consumers loss in welfare best described by area px1BApx0 (CV) using xc(.U0) as base or by area px1CDpx0 (EV) using xc(.U1) as base?xc(.U1)ABCDpx0 x0Is U0 or U1

31、 the appropriate utility target?Consumer WelfareQuantity of xpxxc(.U0)px1x1We can use the Marshallian demand curve as a compromisexc(.U1)x(px)ABCDpx0 x0The area px1CApx0 falls between the sizes of the welfare losses defined by xc(.U0) and xc(.U1)Consumer SurplusWe will define consumer surplus as the

32、 area below the Marshallian demand curve and above priceshows what an individual would pay for the right to make voluntary transactions at this pricechanges in consumer surplus measure the welfare effects of price changesWelfare Loss from a Price IncreaseSuppose that the compensated demand function

33、for x is given byThe welfare cost of a price increase from px = 1 to px = 4 is given byWelfare Loss from a Price IncreaseIf we assume that V = 2 and py = 2,CV = 222(4)0.5 222(1)0.5 = 8If we assume that the utility level (V) falls to 1 after the price increase (and used this level to calculate welfar

34、e loss), CV = 122(4)0.5 122(1)0.5 = 4Welfare Loss from Price IncreaseSuppose that we use the Marshallian demand function insteadThe welfare loss from a price increase from px = 1 to px = 4 is given byWelfare Loss from a Price IncreaseIf e (I) is equal to 8, loss = 4 ln(4) - 4 ln(1) = 4 ln(4) = 4(1.3

35、9) = 5.55this computed loss from the Marshallian demand function is a compromise between the two amounts computed using the compensated demand functionsRevealed PreferenceRevealed Preference and the Substitution EffectThe theory of revealed preference was proposed by Paul Samuelson in the late 1940s

36、The theory defines a principle of rationality based on observed behavior and then uses it to approximate an individuals utility functionRevealed Preference and the Substitution EffectConsider two bundles of goods: A and BIf the individual can afford to purchase either bundle but chooses A, we say th

37、at A had been revealed preferred to BUnder any other e arrangement, B can never be revealed preferred to ARevealed Preference and the Substitution EffectQuantity of xQuantity of yAI1Suppose that, when the budget constraint isgiven by I1, A is chosenBI3A must still be preferred to B when eis I3 (beca

38、use both A and B are available)I2If B is chosen, the budget constraint must be similar to that given by I2 where A is not availableNegativity of the Substitution EffectSuppose that an individual is indifferent between two bundles: C and DLet pxC,pyC be the prices at which bundle C is chosenLet pxD,p

39、yD be the prices at which bundle D is chosenNegativity of the Substitution EffectSince the individual is indifferent between C and DWhen C is chosen, D must cost at least as much as CpxCxC + pyCyC pxCxD + pyCyD When D is chosen, C must cost at least as much as DpxDxD + pyDyD pxDxC + pyDyCNegativity

40、of the Substitution EffectRearranging, we getpxC(xC - xD) + pyC(yC -yD) 0pxD(xD - xC) + pyD(yD -yC) 0Adding these together, we get(pxC pxD)(xC - xD) + (pyC pyD)(yC - yD) 0Negativity of the Substitution EffectSuppose that only the price of x changes (pyC = pyD)(pxC pxD)(xC - xD) 0This implies that pr

41、ice and quantity move in opposite direction when utility is held constantthe substitution effect is negativeMathematical GeneralizationIf, at prices pi0 bundle xi0 is chosen instead of bundle xi1 (and bundle xi1 is affordable), thenBundle 0 has been “revealed preferred” to bundle 1Mathematical Gener

42、alizationConsequently, at prices that prevail when bundle 1 is chosen (pi1), thenBundle 0 must be more expensive than bundle 1Strong Axiom of Revealed PreferenceIf commodity bundle 0 is revealed preferred to bundle 1, and if bundle 1 is revealed preferred to bundle 2, and if bundle 2 is revealed pre

43、ferred to bundle 3,and if bundle K-1 is revealed preferred to bundle K, then bundle K cannot be revealed preferred to bundle 0Contents Demand functionChanges in eChanges in a goods price The individuals demand curvesA maths development of response to price changeDemand elasticitiesConsumer surplusRe

44、vealed preference and the substitution effectImportant Points to Note:Proportional changes in all prices and e do not shift the individuals budget constraint and therefore do not alter the quantities of goods chosendemand functions are homogeneous of degree zero in all prices and eImportant Points t

45、o Note:When purchasing power changes ( e changes but prices remain the same), budget constraints shiftfor normal goods, an increase in e means that more is purchasedfor inferior goods, an increase in e means that less is purchasedImportant Points to Note:A fall in the price of a good causes substitu

46、tion and e effectsfor a normal good, both effects cause more of the good to be purchasedfor inferior goods, substitution and e effects work in opposite directionsno unambiguous prediction is possibleImportant Points to Note:A rise in the price of a good also causes e and substitution effectsfor norm

47、al goods, less will be demandedfor inferior goods, the net result is ambiguousImportant Points to Note:The Marshallian demand curve summarizes the total quantity of a good demanded at each possible pricechanges in price prompt movements along the curvechanges in e, prices of other goods, or preferen

48、ces may cause the demand curve to shiftImportant Points to Note:Compensated demand curves illustrate movements along a given indifference curve for alternative pricesthey are constructed by holding utility constant and exhibit only the substitution effects from a price changetheir slope is unambiguously negative (or zer