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,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 were 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410:16Homogeneity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 function 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 parallel 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 xQuantity 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 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 range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 and 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 derivative 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,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 xpxpx 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 ypxpyx1IIIIIIII ,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 implies 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