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1、Chapter 5 第五章INCOME AND SUBSTITUTION EFFECTS收入和替代效应110:16Demand Functions 需求函数 The optimal levels of x1,x2,xn can be expressed as functions of all prices and income x1,x2,xn的最优水平可以表示为所有价格和收入的函数 These can be expressed as n demand functions of the form 这可以表示为 n个需求函数的形式2x1* = d1(p1,p2, pn,I)x2* = d2(p1
2、,p2, pn,I)xn* = dn(p1,p2, pn,I)10:16Demand Functions 需求函数 If there are only two goods (x and y), we can simplify the notation 如果只有两种商品（ x和 y ），可以 简化记法x* = x(px,py,I)y* = y(px,py,I) Prices and income are exogenous 价格和收入是外生的 the individual has no control over them 个人不能 控制310:16Homogeneity齐次性 If we wer
3、e to double all prices and income, the optimal quantities demanded will not change 将所有的价格和收入翻倍，最优的需求数量将不会改变 the budget constraint is unchanged 预算约束没有变化xi* = di(p1,p2,pn,I) = di(tp1,tp2,tpn,tI) Individual demand functions are homogeneous of degree zero in all prices and income 个人需求函数对所有价格和收入是 零次齐次 的4
4、10:16Homogeneity齐次性 With a Cobb-Douglas utility function 由柯布 -道格拉斯效用函数utility = U(x,y) = x0.3y0.7the demand functions are 需求函数为：5 Note that a doubling of both prices and income would leave x* and y* unaffected. 注意，将价格和收入翻倍不会影响 x*和 y* xpx I3.0* ypy I7.0* 10:16Homogeneity齐次性 With a CES utility functio
5、n CES效用函数utility = U(x,y) = x0.5 + y0.5the demand functions are需求函数为：6 Note that a doubling of both prices and income would leave x* and y* unaffected. 注意，将价格和收入翻倍不会影响 x*和 y* xyx pppx I/11*yxy pppy I/11*10:16Changes in Income 收入变化 An increase in income will cause the budget constraint out in a paral
6、lel fashion收入增加会使预算约束向外平行移动 Since px/py does not change, the MRS will stay constant at the utility-maximizing point 由于 px/py并没有改变，在效用最大化处的 MRS （收入提高前后）保持不变710:16Increase in Income收入增加 If both x and y increase as income rises, x and y are normal goods 如果随着收入增加， x和 y都增加，则 x和 y是 正常 商品8Quantity of xQuan
7、tity of yCU3BU2AU1As income rises, the individual choosesto consume more x and y 随着收入增加，个人选择消费更多的 x和 y10:16Increase in Income收入增加 If x decreases as income rises, x is an inferior good如果随着收入增加， x减少，则 x是劣等商品9Quantity of xQuantity of yCU3As income rises, the individual choosesto consume less x and more
8、 y随着收入增加，个人选择消费较少的 x和较多的 yNote that the indifference curves do not have to be “oddly” shaped. The assumption of a diminishing MRS is obeyed.注意，无差异曲线并不一定是奇怪的形状，仍旧满足 MRS递减的假设BU2AU110:16Normal and Inferior Goods正常商品和劣等商品 A good xi for which xi/I 0 over some range of income is a normal good in that rang
9、e如果在某收入范围内，商品 xi满足xi/I 0，那么 在这一范围内 xi为 正常 商品 A good xi for which xi/I 0 如果 x是正常商品，那么 x/I 0 the entire income effect is negative 总的收入效应是负的 if x is an inferior good, then x/I -1, demand is inelastic如果 ex,px -1，需求是缺乏弹性的 if ex,px = -1, demand is unit elastic如果 ex,px = -1，需求为单位弹性5110:18Price Elasticity a
10、nd Total Spending 价格弹性和总花费 Total spending on x is equal to对 x的总花费等于total spending =pxx Using elasticity, we can determine how total spending changes when the price of x changes 通过弹性，我们可以决定总花费如何随着 x的价格变化而变化521)( , xpxxxxx exxpxppxp10:18Price Elasticity and Total Spending价格弹性和总花费 The sign of this deri
11、vative depends on whether ex,px is greater or less than -1这一导数的符号取决于 ex,px大于还是小于 -1 if ex,px -1, demand is inelastic and price and total spending move in the same direction如果 ex,px -1，需求是缺乏弹性的，价格和总花费按同一方向运动 if ex,px -1, demand is elastic and price and total spending move in opposite directions如果 ex,
12、px -1，需求是有弹性的，价格和总花费按不同方向运动531)( , xpxxxxx exxpxppxp10:18Homogeneity齐次性 Demand functions are homogeneous of degree zero in all prices and income需求函数对所有价格和收入是零次齐次的 Eulers theorem for homogenous functions shows that 齐次函数的欧拉定理表明54II xpxppxpyyxx 010:18Homogeneity齐次 Dividing by x, we get 除以 x，得55I,0 xpxp
13、x eee yx Any proportional change in all prices and income will leave the quantity of x demanded unchanged 所有价格和收入的按比例变化会使得 x的需求数量保持不变10:1856Engel Aggregation恩格尔加总 By differentiating the budget constraint with respect to income (treating prices as constant) , we can get 通过预算约束对收入微分（把价格视为常数），可以得出II yp
14、xpyx1IIIIIIII ,1 yyxxyx esesyyypxxxp 10:18 The weighted average on income elasticities for all goods that a person buys must be one 一个人所买的所有商品的收入弹性的加权平均一定为 1Engel Aggregation 恩格尔加总 Engels law suggests that the income elasticity of demand for food items is less than one 恩格尔法则表明对食物需求的收入弹性是小于 1的 this i
15、mplies that the income elasticity of demand for all nonfood items must be greater than one这意味着对所有非食物商品需求的收入弹性一定是大于 1的5710:18Cournot Aggregation古诺加总 The size of the cross-price effect of a change in the price of xon the quantity of y consumed is restricted because of the budget constraint因为有预算约束线的存在， x价格变化对 y消费数量的交叉价格影响的大小是有限的 We can demonstrate this by differentiating the budget constraint with respect to px 可以通过预算约束对 px求导，来说明这一点5810:18Cournot Aggregation 古诺加总59xyxxx pypxpxpp 0Iyyppyppxxxppxp xxyxxxx III
