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

Economics 2010a Fa2003 Edward L. Glaeser ecture 2

Economics 2010a Fall 2003 Edward L. Glaeser Lecture 2

2. Choice and Utility Functions Choice in Consumer Demand Theory and Walrasian Demand b. Properties of demand from continuity and properties from WARP C. Representing Preferences with a Utility Function d. Demand as Derived from Utility Maximization e. Application: Fertility

2. Choice and Utility Functions a. Choice in Consumer Demand Theory and Walrasian Demand b. Properties of demand from continuity and properties from WARP c. Representing Preferences with a Utility Function d. Demand as Derived from Utility Maximization e. Application: Fertility

x i denotes commodities continuous numbers x=(x1, x2,..xL) vector of discrete commodities p=(p1, p2,.pr) vector of prices w= wealth available to be spent The budget constraint pox ∑ Pix≤w

xi denotes commodities, continuous numbers x  x1, x2,.... xL  vector of discrete commodities p  p1, p2,....pL  vector of prices w  wealth available to be spent The budget constraint p  x   i1 L pixi  w

MWG Definition 2.D. 1 The Walrasian Budget set Bpm={x∈界:p·x≤w} is the set of all feasible consumption bundles for the consumer faces market prices p and has wealth w Note: We will be treating all prices and consumption levels as being weakly posItive. Prices are treated as exogenous-as they will be in the production case. While neither consumer nor producer chooses prices (generally) prices are the extra parameter in each sides problem that ensures that demand and supply are equal Non-linear prices are certainly possible

MWG Definition 2.D.1 The Walrasian Budget Set Bp,w  x   L : p  x  w is the set of all feasible consumption bundles for the consumer faces market prices p and has wealth w. Note: We will be treating all prices and consumption levels as being weakly positive. Prices are treated as exogenous– as they will be in the production case. While neither consumer nor producer chooses prices (generally) prices are the extra parameter in each side’s problem that ensures that demand and supply are equal. Non-linear prices are certainly possible

(example 2. D 4 The Walrasian Demand function is the set C(Bp, w)which is defined for all (p, w), or at least for a full dimensional subset L+1 (P,)∈9 We generally assume that C(Bp, w) has a single element(for convenience) but it doesnt need to We write (Bp.w)=x(p, w)=(xi(p, w),.xL(p, w) We will also generally assume that demand is continuous and differentiable MWG Definition 2.E1: The Walrasian Demand function is

(example 2.D.4). The Walrasian Demand Function is the set CBp,w which is defined for all p,w, or at least for a full dimensional subset p,w   L1 We generally assume that CBp,w has a single element (for convenience) but it doesn’t need to. We write CBp,w  xp,w  x1p,w,... xLp,w We will also generally assume that demand is continuous and differentiable. MWG Definition 2.E.1: The Walrasian Demand Function is

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)