哈佛大学：《中级微观经济学》（英文版）Lecture 2 Choice and Utility Functions

homogeneous of degree zero if x(ap, aw)=x(p, w) for any p, w and a>0 This property follows from the fact that choice is only a function of the budget set and Bow={x∈界:p·x≤w} is the same set as Ban. aw={x∈界:ap·x≤a} This fairly obvious claim is in many ways the underlying intellectual basis for the economic bias that the price level doesnt matter Differentiating x(ap, aw)=x(p, w)totally with respect to a gives us the following equation

homogeneous of degree zero if xp,w  xp,w for any p,w and   0. This property follows from the fact that choice is only a function of the budget set and Bp,w  x   L : p  x  w is the same set as Bp,w  x   L : p  x  w This fairly obvious claim is in many ways the underlying intellectual basis for the economic bias that the price level doesn’t matter. Differentiating xp,w  xp,w totally with respect to  gives us the following equation:

∑ Oxk(p, w D Ox(p, w) 0 L ∑ oxk(p,w) pi Oxk(p, w) api Xk =∑ ￡D,+￡ This tells you that for any commodity, the sum of own and cross price elasticities equals-1 times the income elasticity

 i1 L xkp,w pi pi  xkp,w w w  0  i1 L xkp,w pi pi xk  xkp,w w w xk   i1 L pi k  w k  0 This tells you that for any commodity, the sum of own and cross price elasticities equals -1 times the income elasticity

MWG Definition 2.E.2 Walras Law The Walrasian demand correspondence x(p, w)satisfies Walras law if for every>0 and w>0, we have p·x= w for all x∈x(p,w) This just says that the consumer spends all of his wealth Looking ahead, Walras' law will come about as long as consumers are not satiated in at least one of the goods

MWG Definition 2.E.2 Walras’ Law: The Walrasian Demand correspondence xp,w satisfies Walras’ law if for every p  0 and w  0, we have p  x  w for all x  xp,w. This just says that the consumer spends all of his wealth. Looking ahead, Walras’ law will come about as long as consumers are not “satiated” in at least one of the goods

Walras' Law and differentiability give us two convenient equalities Differentiating po x =w with respect to w yields ∑m2N2) or manipulating this slightly yields

Walras’ Law and differentiability give us two convenient equalities. Differentiating p  x  w with respect to w yields:  i1 L pi xip,w w  1 or manipulating this slightly yields:

L ∑ axi(p, w) wpix Xi w ∑:1 Where ni pix the budget share of good This means that income elasticities(when weighted by budget shares)sum to one All goods cant be luxuries, etc Sometimes this is known as engel aggregation(income effects are after all drawn with Engel curves)

 i1 L xip,w w w xi pixi w   i1 L w i i  0, where i  pixi w , the budget share of good i. This means that income elasticities (when weighted by budget shares) sum to one. All goods can’t be luxuries, etc. Sometimes this is known as Engel aggregation (income effects are after all drawn with Engel curves)