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*,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,Chapter Eight,Slutsky Equation,2024/9/22,1,Effects of a Price Change,What happens when a commoditys price decreases?,Substitution effect,: the commodity is relatively cheaper, so consumers substitute it for now relatively more expensive other commodities.,Income effect,: the consumers budget of $y can purchase more than before, as if the consumers income rose, with consequent income effects on quantities demanded.,2024/9/22,2,Effects of a Price Change,x,2,x,1,Original choice,Consumers budget is $y.,2024/9/22,3,Effects of a Price Change,x,1,Lower price for commodity 1,pivots the constraint outwards.,Consumers budget is $y.,x,2,2024/9/22,4,Effects of a Price Change,x,1,Lower price for commodity 1,pivots the constraint outwards.,Consumers budget is $y.,x,2,Now only $y are needed to buy the original bundle at the new prices, as if the consumers income has increased by $y - $y.,2024/9/22,5,Effects of a Price Change,Changes to quantities demanded due to this extra income are the income effect of the price change.,Slutsky discovered that changes to demand from a price change are always the sum of a pure substitution effect and an income effect.,2024/9/22,6,Real Income Changes,Slutsky asserted that if, at the new prices,less income is needed to buy the original bundle then “real income” is increased,more income is needed to buy the original bundle then “real income” is decreased,2024/9/22,7,Real Income Changes,x,1,x,2,Original budget constraint and choice,2024/9/22,8,Real Income Changes,x,1,x,2,Original budget constraint and choice,New budget constraint,2024/9/22,9,Real Income Changes,x,1,x,2,Original budget constraint and choice,New budget constraint,; real income has risen,2024/9/22,10,Real Income Changes,x,1,x,2,Original budget constraint and choice,2024/9/22,11,Real Income Changes,x,1,x,2,Original budget constraint and choice,New budget constraint,2024/9/22,12,Real Income Changes,x,1,x,2,Original budget constraint and choice,New budget constraint,; real income has fallen,2024/9/22,13,Pure Substitution Effect,Slutsky isolated the change in demand due only to the change in relative prices by asking “What is the change in demand when the consumers income is adjusted so that, at the new prices, she can only just buy the original bundle?”,2024/9/22,14,Pure Substitution Effect Only,x,2,x,1,x,2,x,1,2024/9/22,15,Pure Substitution Effect Only,x,2,x,1,x,2,x,1,2024/9/22,16,Pure Substitution Effect Only,x,2,x,1,x,2,x,1,2024/9/22,17,Pure Substitution Effect Only,x,2,x,1,x,2,x,2,x,1,x,1,2024/9/22,18,Pure Substitution Effect Only,x,2,x,1,x,2,x,2,x,1,x,1,2024/9/22,19,Pure Substitution Effect Only,x,2,x,1,x,2,x,2,x,1,x,1,Lower p,1,makes good 1 relativelycheaper and causes a substitutionfrom good 2 to good 1.,2024/9/22,20,Pure Substitution Effect Only,x,2,x,1,x,2,x,2,x,1,x,1,Lower p,1,makes good 1 relativelycheaper and causes a substitutionfrom good 2 to good 1.,(x,1,x,2,),(x,1,x,2,),is the,pure substitution effect,.,2024/9/22,21,And Now The Income Effect,x,2,x,1,x,2,x,2,x,1,x,1,(,x,1,x,2,),2024/9/22,22,And Now The Income Effect,x,2,x,1,x,2,x,2,x,1,x,1,(,x,1,x,2,),The,income effect,is,(x,1,x,2,),(x,1,x,2,),.,2024/9/22,23,The Overall Change in Demand,x,2,x,1,x,2,x,2,x,1,x,1,(,x,1,x,2,),The change to demand due to lower p,1,is the sum of the income and substitution effects,(x,1,x,2,),(x,1,x,2,),.,2024/9/22,24,Slutskys Effects for Normal Goods,Most goods are normal (i.e. demand increases with income).,The substitution and income effects reinforce each other when a normal goods own price changes.,2024/9/22,25,Slutskys Effects for Normal Goods,x,2,x,1,x,2,x,2,x,1,x,1,(,x,1,x,2,),Good 1 is normal becausehigher income increasesdemand,2024/9/22,26,x,2,x,1,x,2,x,2,x,1,x,1,(,x,1,x,2,),Good 1 is normal becausehigher income increasesdemand, so the income and substitution effects reinforce each other.,Slutskys Effects for Normal Goods,2024/9/22,27,Since both the substitution and income effects increase demand when own-price falls, a normal goods ordinary demand curve slopes down.,The Law of Downward-Sloping Demand therefore always applies to normal goods.,Slutskys Effects for Normal Goods,2024/9/22,28,Slutskys Effects for Income-Inferior Goods,Some goods are income-inferior (i.e. demand is reduced by higher income).,The substitution and income effects oppose each other when an income-inferior goods own price changes.,2024/9/22,29,Slutskys Effects for Income-Inferior Goods,x,2,x,1,x,2,x,1,2024/9/22,30,Slutskys Effects for Income-Inferior Goods,x,2,x,1,x,2,x,1,2024/9/22,31,Slutskys Effects for Income-Inferior Goods,x,2,x,1,x,2,x,1,2024/9/22,32,Slutskys Effects for Income-Inferior Goods,x,2,x,1,x,2,x,2,x,1,x,1,2024/9/22,33,Slutskys Effects for Income-Inferior Goods,x,2,x,1,x,2,x,2,x,1,x,1,The pure substitution effect is as fora normal good. But, .,2024/9/22,34,Slutskys Effects for Income-Inferior Goods,x,2,x,1,x,2,x,2,x,1,x,1,(,x,1,x,2,),The pure substitution effect is as for a normal good. But, the income effect is in the opposite direction.,2024/9/22,35,Slutskys Effects for Income-Inferior Goods,x,2,x,1,x,2,x,2,x,1,x,1,(,x,1,x,2,),The pure substitution effect is as for a normal good. But, the income effect is in the opposite direction. Good 1 is income-inferior because an,increase to income causes demand to fall.,2024/9/22,36,Slutskys Effects for Income-Inferior Goods,x,2,x,1,x,2,x,2,x,1,x,1,(,x,1,x,2,),The overall changes to demand arethe sums of the substitution and income effects.,2024/9/22,37,Giffen Goods,In rare cases of extreme income-inferiority, the income effect may be larger in size than the substitution effect, causing quantity demanded to fall as own-price rises.,Such goods are Giffen goods.,2024/9/22,38,Slutskys Effects for Giffen Goods,x,2,x,1,x,2,x,1,A decrease in p,1,causes quantity demanded of good 1 to fall.,2024/9/22,39,Slutskys Effects for Giffen Goods,x,2,x,1,x,2,x,1,x,1,x,2,A decrease in p,1,causes quantity demanded of good 1 to fall.,2024/9/22,40,Slutskys Effects for Giffen Goods,x,2,x,1,x,2,x,2,x,1,x,1,x,1,x,2,Substitution effect,Income effect,A decrease in p,1,causes quantity demanded of good 1 to fall.,2024/9/22,41,Slutskys Effects for Giffen Goods,Slutskys decomposition of the effect of a price change into a pure substitution effect and an income effect thus explains why the Law of Downward-Sloping Demand is violated for extremely income-inferior goods.,2024/9/22,42,Income & Substitution Effects: Summary,The Substitution effect is always,NEGATIVE,P up, Q Down,The Income effect is,normally,positive,P Up,Real Income Down,Therefore Q down,So a rise in Price =,Sub effect - Q down,Inc Effect - Q Down,Overall - Q Down,2024/9/22,43,Income & Substitution Effects: Summary,The Substitution effect is always,NEGATIVE,P up, Q Down,BUT,income effect can be NEGATIVE,Inferior Good,P Up - Real Income Down,But now Q,UP,So a rise in Price =,Sub effect - Q down,Inc effect - Q,UP,Overall Which effect is stronger?,2024/9/22,44,Income & Substitution Effects: Summary,The Substitution effect is always,NEGATIVE,P up, Q Down,BUT income effect can be NEGATIVE,Inferior Good, P Up - Real Income Down Q,UP,So P up,Sub effect - D down,Inc Effect - Q,UP,Overall Which effect is stronger?,Sub effect Inc Effect = Inferior good,Q down,Inc effect Sub Effect =,Giffen,good,Q UP,2024/9/22,45,Algebra and calculation,Original,choice,Final,choice,pivot,shift,x,1,x,2,Substitution,effect,Income,effect,m/p,2,m,/p,1,The substitution effect is always negative,2024/9/22,46,This equation is called Slustsky identity.,When x,1,is a normal good,(-),(-),(-),When x,1,is an inferior,good,(?),(-),(+),The total change in demand,2024/9/22,47,Income effect,= total effect,pivot,shift,Original,choice,Final,choice,Original,budget,line,Final budget line,Substitution effect = total effect,2024/9/22,48,Substitution effect =,total effect,Final budget line,Original,budget,line,Quasilinear preferences,2024/9/22,49,Units of Good,x,1,Units of good,x,2,f,B,1,X*,1,h,B,2,X,1,g,x,s,1,Income effect,of the price rise,Overall,B,2a,I,6,I,5,I,4,I,3,I,2,I,1,normal good case,(x*,1,x*,2,),(x,1,x,2,),(x,s,1,x,s,2,),Income and substitution effects: geometric,2024/9/22,50,E.g,U= x,2,1,x,2,2,200= 2 p,1,+ 4 p,2,P,1,=1,SE, IE =?,2024/9/22,51,Price change and demand curves,The Marshallian Demand Curve,Hicksian Demand Curve,and Slutsky Demand Curve,2024/9/22,52,x,x,y,p,x,In this part of the diagram we have drawn the choice between x on the horizontal axis and y on the vertical axis. Soon we will draw an indifference curve in here,Down below we have drawn the relationship between x and its price P,x.,This is effectively the space in which we draw the demand curve.,We start with the following diagram,2024/9/22,53,x,y,p,x,Next we draw in the indifference curves showing the consumers tastes for x and y.,Then we draw in the budget constraint and find the initial equilibrium,x,0,y,0,2024/9/22,54,x,y,p,x,x,0,y,0,Recall the slope of the budget constraint is:,2024/9/22,55,x,y,p,x,x,0,y,0,From the initial equilibrium we can find the first point on the demand curve,Projecting x,0,into the diagram below, we map the demand for x at p,x,0,x,0,p,x,0,2024/9/22,56,x,y,p,x,x,0,y,0,x,0,p,x,0,Next consider a rise in the price of x, to p,x,1,. This causes the budget constraint to swing in as -p,x,1,/p,y,0,is greater,To find the demand for x at the new price we locate the new equilibrium quantity of x demanded.,x,1,x,1,p,x,1,Then we drop a line down from this point to the lower diagram.,This shows us the new level of demand at p,x,1,2024/9/22,57,We are now in a position to draw the ordinary Demand Curve,x,y,p,x,x,0,y,0,x,0,p,x,0,x,1,x,1,p,x,1,First we highlight the the p,x,and x combinations we have found in the lower diagram.,D,x,And then connect them with a line.,This is the Marshallian demand curve for x,2024/9/22,58,x,y,p,x,x,0,y,0,x,0,p,x,0,x,1,x,1,p,x,1,D,x,Our next exercise involves giving the consumer enough income so that they can reach their original level of utility U,2,U,2,So we take the new budget constraint.,And gradually increase the agents income, moving the budget constraint out.,.,until we reach the indifference curve U,2,U,1,2024/9/22,59,x,y,p,x,x,0,y,0,x,0,p,x,0,x,1,x,1,p,x,1,D,x,The new point of tangency tells us the demand for x when the consumer had been compensated so they can still achieve utility level U,2,but the relative price of x and y has risen to p,x,1,/p,y,0,.,U,1,2024/9/22,60,This is called the Hicksian demand for x and we will label it x,H,x,y,p,x,x,0,y,0,x,0,p,x,0,x,1,x,1,p,x,1,D,x,The level of demand for x represents the pure substitution effect of the increase in the price of x,x,H,U,2,U,1,2024/9/22,61,x,y,p,x,x,0,y,0,x,0,p,x,0,x,1,x,1,p,x,1,D,x,x,H,x,H,We derive the Hicksian Demand curve by projecting the demand for x downwards into the demand curve diagram,Notice this is the,compensated,demand for x when the price is p,x,1,To get the Hicksian demand curve we connect the new point to the original demand x,0,p,x,0,U,2,U,1,2024/9/22,62,x,y,p,x,y,0,x,0,p,x,0,x,1,p,x,1,D,x,x,H,Notice that the Hicksian Demand Curve is steeper than the Marshallian demand curve, when the good is a normal good,We label the curve H,x,H,x,U,2,U,1,2024/9/22,63,Notice that an alternative compensation scheme would be to give the consumer enough income to buy their original bundle of goods, (x,0,y,o,),x,y,p,x,y,0,x,0,p,x,0,x,1,p,x,1,D,x,x,H,x,0,In this case the budget constraint has to moved out even further until it goes through the point (x,0,y,o,),U,2,2024/9/22,64,x,y,p,x,y,0,x,0,p,x,0,x,1,p,x,1,D,x,x,H,x,0,But now the consumer doesnt have to consume (x,0,y,o,),So they will choose a new equilibrium point . On a higher indifference curve,U,3,U,2,U,1,2024/9/22,65,x,y,p,x,y,0,x,0,p,x,0,x,1,p,x,1,D,x,x,H,x,0,U,3,U,2,Once again we find the demand for x at this new higher level of income by dropping a line down from the new equilibrium point to the x axis. We call this x,s,. It is the Slutsky demand,.,x,s,H,x,x,s,Once again this,income,compensated demand is measured at the price p,x,1,This diagram is going to get quite messy now .,2024/9/22,66,x,y,p,x,y,0,x,0,p,x,0,x,1,p,x,1,D,x,x,H,x,0,U,3,U,2,H,x,x,s,Finally, once again we can draw the,Slutsky,compensated demand curve through this new point (x,s,p,x,1,)and the original (x,0,p,x,0,),S,x,M,x,The new demand curve S,x,is steeper than either the Marshallian or the Hicksian curve when the good is normal,2024/9/22,67,M,H,S,p,x,x,We can derive three demand curves on the basis of our indifference curve analysis.,Summary,2024/9/22,68,M,H,S,p,x,x,1.,The normal Marshallian Demand Curve, holding income fixed,2024/9/22,69,M,H,S,p,x,x,2.,The Hicksian compensated demand curve where agents are given sufficient income to maintain them on their original utility curve: holding utility fixed.,2024/9/22,70,M,H,S,p,x,x,3.,The Slutsky income compensated demand curve where agents have sufficient income to purchase their original bundle: holding purchasing power fixed,2024/9/22,71,M,H,S,p,x,x,Finally, for a,normal,good the Marshallian demand curve is flatter than the Hicksian, which in turn is flatter than the Slutsky demand curve.,THREE DEMAND CURVES,2024/9/22,72,
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