Lecture 2
Lecture 2
x,denotes commodities, continuous numbers .=(vector of discrete commodities ·p=(p,p2,…p)vector of prices w=wealth available to be spent The budget constraint px=∑px,≤w i=l
• denotes commodities, continuous numbers • vector of discrete commodities • vector of prices • w=wealth available to be spent • The budget constraint ( ) i L x x , x , ....x = 1 2 ( ) L p p , p , .... p = 1 2 i x p x p xi w L i • = i =1
MWG Definition 2.D.1 The Walrasian Budget Set Bpm={x∈9R:p●x≤w is the set of all feasible consumption bundles for the consumer faces market prices p and has wealth w
MWG Definition 2.D.1 • The Walrasian Budget Set is the set of all feasible consumption bundles for the consumer faces market prices p and has wealth w. B x p x w L p,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)
• 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(B2.)which is defined for all (p,w),or at least for a full dimensional subset (p,w)∈gRH We generally assume that C(B.)has a single element (for convenience)but it doesn't need to. We write C(Bp)=x(p,w)=(x(p,w).x(p,w) We will also generally assume that demand is continuous and differentiable
• The Walrasian Demand Function is the set which is defined for all , or at least for a full dimensional subset • We generally assume that has a single element (for convenience) but it doesn’t need to. • We write • We will also generally assume that demand is continuous and differentiable. ( ) C Bp,w (p,w) ( ) 1 , + + L p w ( ) C Bp,w C(B ) x(p w) (x (p w) x (p w)) p w L , , ,...., , , = = 1
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 anda>o This property follows from the fact that choice is only a function of the budget set and B.=x∈R:p●x≤w}is the same set as Bp.a={x∈9R:Cp·x≤w}
MWG Definition 2.E.1: • The Walrasian Demand Function is homogeneous of degree zero if for any p, w and • This property follows from the fact that choice is only a function of the budget set and is the same set as x(p,w) = x(p,w) 0 B x p x w L p,w = + : • B x p x w L p,w = + : •
This fairly obvious claim is in many ways the underlying intellectual basis for the economic bias that the price level doesn't matter
• This fairly obvious claim is in many ways the underlying intellectual basis for the economic bias that the price level doesn’t matter
Differentiatingx(ap,aw)=x(p,w)totally with respect tox gives us the following equation: 之m”n+ w=0 ap Ow 四2+”-立结+-0 i=1 Xk Ow Xk This tells you that for any commodity,the sum of own and cross price elasticities equals -1 times the income elasticity
• Differentiating totally with respect to gives us the following equation: • This tells you that for any commodity, the sum of own and cross price elasticities equals -1 times the income elasticity. x(p,w) = x(p,w) ( ) ( ) 0 , , 1 = + = w w x p w p p x p w k L i i i k ( ) ( ) 0 , , 1 1 = + = + = = k w L i k p k k L i k i i k i x w w x p w x p p x p w
MWG Definition 2.E.2 Walras'Law:The Walrasian Demand correspondence x(p,w)satisfies Walras' law if for every p>>0 and w>,we have p●x=w for allx∈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 satisfies Walras’ law if for every and , we have for all . • 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. x(p,w) p 0 w 0 p • x = w xx(p,w)
Walras'Law and differentiability give us two convenient equalities. ·Differentiating p●x=wwith respect to w yields: n ax,(p,w）=1 i=l Ow or manipulating this slightly yields: ax,ww2,x=之n,=0 i=1 Ow xi W i三1 where=,the budget share of good w
• Walras’ Law and differentiability give us two convenient equalities. • Differentiating with respect to w yields: or manipulating this slightly yields: where , the budget share of good . p • x = w ( ) 1 , 1 = = w x p w p i L i i ( ) 0 , 1 1 = = = = i L i i w i i i L i i w p x x w w x p w w p xi i i = i