Micro 04 Classical Demand 2 微观经济学

+ + kLecture 4The Utility Maximization ProblemGiven prices p » 0 and wealth w >0, a consumer’s utility maximization problem (UMP) is:Max x ≥0u (x ) ,s.t. p ⋅ x ≤ w .Budget setB p ,w = {x ∈ R L: p ⋅ x ≤w } is bounded and closed. If p » 0 and u (.) iscontinuous, then the UMP always has a solution.The solution, denoted by x(p,w), is the Walrasian demand correspondence . It is called Walrasian demand function when it is single-valued.For each (p, w ) » 0, the utility value of the UMP is denoted v ( p , w ) ∈ R , called indirect utilityfunction . It is equal to u (x * ) with x * ∈ x ( p , w ) .Proposition D2 Suppose that u (.) is a continuous utility function representing a locallynon-satiated preference relation > defined on consumption setdemand correspondence x(p,w) possesses the following properties:X = R L . Then the Walrasian(i) Homogeneity of degree zero in (p,w): x(ap, aw)=x(p,w) for any p, w and scalar a >0; (ii) Walras’ law : p.x=w for all x in x(p,w);(iii) Convexity/uniqueness : if > is convex, so that u (.) is quasi-concave, then x(p,w) is a convex set. Moreover, if > is strictly convex, so that u (.) is strictly quasi-concave, then x(p,w) consists of a single element.If utility function u (.) is continuously differentiable , the Kuhn-Tucker conditions imply ifx * ∈ x ( p , w ) , there exists a multiplier λ ≥ 0 such that for l = 1,..., L ,∂u (x * )≤ λ∂x lp l , with equality if * > 0 ,or equivalently∇u (x * ) ≤ λ p and x * ⋅[∇u (x * ) - λ p ] = 0 .∂u (x * ) / ∂xp Hence if x * > 0 and x *> 0 , we have l = l . l k ∂u (x *) / ∂x p x l kL (x , ) ⋅ * j i 1 2 ∑ 2[Kuhn-Tucker Conditions: The optimal point x * of problemMax xsatisfies the following conditions:f (x ) s.t.g i (x ) ≤ 0 , ( i = 1,..., m ),x ≥ 0 ，(1) ∂L (x *, λ* ) f - λ g i ≤ 0 , ∂ * λ* x = 0 , and x * ≥ 0 , for j = 1,..., n . ∂x * j i j i =1∂x * j j(2) g i (x * ) ≤ 0 , andλ* ≥ 0 ; g i (x * ) ⋅ λ*= 0 . ]Example 1 (Cobb-Douglas utility function): Suppose L =2 and u (x , x ) = x α x 1-α with α ∈ (0,1)and k >0. This utility function can be transformed to strictly concave function. The UMP is121 2u (x 1 , x 2 ) = α ln x 1 + (1 - α ) ln x 2 , which is aMax x 1 , x 2u (x 1 , x 2 ) = α ln x 1 + (1 - α ) ln x 2 , s.t.,p 1 x 1 + p 2 x 2 = w .The first order conditions are:= λ p 11and 1 - αx 2= λ p 2 .From the budget constraint, we have λ = 1. It is the “shadow price of wealth”, which meanswthat∂v ( p , w ) = 1. It is easy to obtain the Walrasian demand functions ∂w wx ( p , w ) = α w and x ( p , w ) = (1 - α )w. 1p 2The indirect utility function is: v ( p , w ) = ln w + ln αα (1- α )1-α + ln p -α p α -1.Example 2 (Quasi-linear utility function): Suppose a consumer has differentiable and monotoneutility function of u (x ) = x 1 + ϕ (x 2 ,..., x L ) . The UMP isMax x 1 ,..., x Lu (x ) = x 1 + ϕ (x 2 ,..., x L ) , s.t.,p 1 x 1 + ... + p L x L ≤ w .Suppose there are interior solutions. The first order conditions areλ p = 1and∂ϕ = λ p ,l = 2,..., L ,i.e., 1λ = 1/ p ∂x land l∂ϕ =p l ,l = 2,..., L . 1 ∂x pl 1m j = αx 1 p+ We see from the first order conditions that the demand for commodityl = 2,..., L only dependson the relative prices but not the wealth.Proposition D3 Suppose that u (.) is a continuous utility function representing a locallynon-satiated preference > defined onX = R L . The indirect utility function v ( p , w ) is(i)Homogeneous of degree zero in (p, w );(ii)Strictly increasing in w and non-increasing inp l for any l ;(iii)Quasiconvex, that is, the set {( p , w ) : v ( p , w ) ≤ v } is convex for any v ;(iv) Continuous in p and w .Proof: (iii) ∀v ∈ R , for any (p,w) and (p’,w’) such that v ( p , w ) ≤ v and v ( p ', w ') ≤ v , denote( p ", w ") = α ( p , w ) + (1 - α )( p ', w ') = (α p + (1 - α ) p ',α w + (1 - α )w ') .We need to show that v ( p ", w ") ≤ v (which means set {( p , w ) : v ( p , w ) ≤ v } is convex).Since p "⋅ x ( p ", w ") ≤ w " , i.e.,α p ⋅ x ( p ", w ") + (1- α ) p '⋅ x ( p ", w ") ≤ α w + (1- α )w ' ,we shall have eitherp ⋅ x ( p ", w ") ≤ w or p '⋅ x ( p ", w ") ≤ w ' , which means eitheru (x ( p ", w ")) ≤ v ( p , w ) or u (x ( p ", w ")) ≤ v ( p ', w ') . Hence we have v ( p ", w ") ≤ v .x 2x 1+ + The Expenditure Minimization ProblemAn expenditure minimization problem (EMP ) is for p » 0andu > u (0) :Min x ≥0p.x , s.t. u (x ) ≥ u .The EMP is the “dual” problem to the UMP. It captures the same aim of efficient use of the consumer’s purchasing power while reversing the roles of objective function and constraint. The solution of the problem is denoted as h ( p , u ) , and e ( p , u ) ≡ p ⋅ h ( p , u ) .Proposition E1 Suppose that u (.) is a continuous utility function representing a locallynon-satiated preference > defined on consumption setX = R L and that the price vector isp » 0 . We have(i) Give wealth w >0, ifx * = x ( p , w ) , then x * = h ( p , u (x * )) and e ( p , u (x * )) = w . Inwords, if x * is optimal in the UMP with wealth w >0, thenx * is optimal in the EMP withutility level u ( x * ), and the minimized expenditure level in this EMP is exactly w .(ii) Give utility level u > u 0 , if x * = h ( p , u ) , then x * = x ( p , p ⋅ x * ) and v ( p , p ⋅ x * ) = u .In words, if x * is optimal in the EMP with utility levelu > u 0 , then x * is optimal in theUMP with wealth p ⋅ x * , and the maximized utility level in this UMP is exactly u .Proposition E1 says that e(p,u)=w if and only if v(p,w)=u.For any p » 0 , w > 0 , and u > u 0 , we havee ( p , v ( p , w )) = w and v ( p , e ( p , u )) = u .The set of optimal commodity vectors in the EMP,h ( p , u ) ⊂ R L , is known as the Hicksian(or compensated) demand correspondence . Using Proposition E1, we can relate the Hicksian and Walrasian demand correspondences as:h(p, u)=x(p, e(p, u))andx(p, w)=h(p, v(p, w)).+ x 21Proposition E2 Suppose that u (.) is a continuous utility function representing a locallynon-satiated preference relation > defined on the consumption setX = R L . The expenditurefunction e ( p , u ) is(i) Homogeneous of degree one in p ;(ii) Strictly increasing in u and non-decreasing in(iii) Concave in p ; (iv) Continuous in p and u .p l for any l ;Proof: (i) When e(p,u)=w , we have v(p,w)=u. Then e(ap,u)=e(ap,v(p,w))=e(ap,v(ap,aw))=aw .(ii) For any u”>u’, let w”=e(p,u”) and w’=e(p,u’). Since v(p,w”)>v(p,w’), we have w”>w’ , because v(p,w) is strictly increasing in w .Consider price vector p” and p’, with p " > p ' and p " = p ' for j ≠ l . Let w”=e(p”, u)lljjand w’=e(p’, u). Then u=v(p”, w”)=v(p’, w’). Since v(p, w) is non-increasing in p andstrictly increasing in w , we have w” ≥ w’. Hence e(p, w) is non-decreasing in p l for any l .(iii) For any utility level u and price p and p’, denote p”=ap+(1-a)p’. Let w”=e(p”, u),w=e(p, u) and w’=e(p’, u). We need to show that w " ≥ aw + (1 - a )w ' .Indeed, we have u=v(p”, w”)=v(p, w)=v(p’, w’). From the quasi-convexity of v (.,.), wehave v ( p ", aw + (1 - a )w ') ≤ u = v(p”, w”). Hence we must havew " ≥ aw + (1 - a )w 'from the monotonicity of v (.,.).+ * lpp 2Proposition E3 Suppose that u(.) is a continuous utility function representing a locallynon-satiated preference relation > defined on the consumption setX = R L . Then for anyp » 0 , the Hicksian demand correspondence h ( p ,u ) possesses the following properties:(i) Homogeneity of degree zero in p : h (α p , u ) = h ( p , u ) for any p, u and α > 0 .(ii) No excess utility: for any x ∈ h ( p , u ) , u(x)=u .(iii) Convexity/uniqueness: if > is convex, thenh ( p , u ) is a convex set; and if > is strictlyconvex, so that u (.) is strictly quasi-concave, then there is a unique element in h ( p , u ) .Proposition E4 Suppose that u (.) is a continuous utility function and that h(p,u) consists of asingle element for allp >> 0 . Then the Hicksian demand function h(p,u) satisfies thecompensated law of demand: For all p’ and p”, ( p "- p ')[h ( p ", u ) - h ( p ', u )] ≤ 0 .With an interior solution of the EMP, we shall have the first-order conditions:p = λ ∂u (x ) , for ∂x ll = 1,..., L ,where λ is the Lagrangean multiplier. Hence for any l , k ∈{1,..., L }, we havepu (x * ) l = l .Note that the first-order conditions are virtually the same as those of UMP.* p k u k (x )1 2 12αExample 3: (EMP with Cobb-Douglas utility function ) Consider following EMP:Min x 1 , x 2p 1 x 1 + p 1 x 2 , st. α ln x 1 + (1 - α ) ln x 2 ≥ ln uThe first order conditions are p = αλ x 1and p = (1 - α )λ. Solving for x 2p α p 1-αux 1 ,x 2and substitutingthem into the budget constraint, we have λ= 1 2 . Substituting it back to the first-orderαα (1 - α )1-αconditions, we obtain the Hicksian demand functions:h ( p , u ) = ϒ α p 2 /1-αu ,andh ( p , u ) = ϒ (1 - α ) p 1 / u . 1 ' (1- α ) p ∞ 2 ' α p ∞≤ The expenditure function is: 1 ƒ≤ 2 ƒe (p ,u )= u α -α (1 - α )α -1 p α p1-α .Homework : Page 97, 3.D.4, 3.E.5.。