Expenditure Minimization

Expenditure Minimization Set up optimization problem

Expenditure Minimization: SOC The FOC ensure that the optimal consumption bundle is at a tangency. The SOC ensure that the tangency is a minimum, and not a maximum by ensuring that away from the tangency, along the indifference curve, expenditure rises. Y E*<E’ E=E’ E=E* X

Expenditure Minimization: SOC The second order condition for constrained minimization will hold if the following bordered Hessian matrix is positive definite: Will hold if the Hessian of the Lagrangian is is Positive Definite Note, -(-Ux )2 =-Ux 2 < 0 and (so long as μ > 0), 2UxUxy Uy -Uy2Uxx-Ux2Uyy > 0, so these conditions are equivalent to checking that the utility function is strictly quasi-concave.

Expenditure Minimization Solve FOC to get:

Expenditure Minimization Back into the expenditure function determine minimum expenditure: Solve for Ū to get the indirect utility function:

Interpreting μ: Envelope Result Start with L*

Finding : Envelope Result Start with L*.

Expenditure Minimization Comparative Statics

Comparative Statics

Comparative Statics: Effect of a change in px Put in Matrix Notation Solve for

Expenditure Minimization: Example

Expenditure Minimization Combining with

Expenditure Minimization Expenditure Function And solving this for U would yield U* = V *(px,py,M)

Properties of Expenditure Functions Homogeneity a doubling of all prices will precisely double the value of required expenditures homogeneous of degree one Nondecreasing in prices E*/pi  0 for every good, i Concave in prices When the price of one good rises, consumers respond by consuming less of that good and more of other goods. Therefore, expenditure will not rise proportionally with the price of one good.

Concavity of Expenditure Function If the consumer continues to buy a fixed bundle as p1’ changes (e.g. goods are perfect compliments), the expenditure function would be Ef Ef E(px,py,U*) Since the consumption pattern will likely change, actual expenditures will be less than portrayed Ef such as E(px,py,U*). At the px where the quantity demanded of a good becomes 0, the expenditure function will flatten and have a slope of 0. E(p1,…) E(px’,py…U*) px px’

Max and Min Relationships Utility Max L = U(x) + λ(M-g(x)) x* = x(px, M) Expenditure Min L = g(x) + μ(U-U(x))) xc* = xc (px, U) Indirect Utility U* = U *(x*) V * = V *(px, M) Expenditure Function E* = E *(xc*) E * = E *(px, U) Expenditure Function Solve V * for M (M=E) E * = E *(px, U) Indirect Utility Solve E * for U (E=M) U * = V *(px, M)

Shephards Lemma and Roy’s Identity Two envelope theorem results allow: Derivation of ordinary demand curves from the expenditure function Derivation of compensated demand curves from the indirect utility function

Envelope Theorem Say we know that y = f(x; ω) We find y is maximized at x* = x(ω) So we know that y* = y(x*=x(ω),ω)). Now say we want to find out So when ω changes, the optimal x changes, which changes the y* function. Two methods to solve this…

Envelope Theorem Start with: y = f(x; ω) and calculate x* = x(ω) First option: y = f(x; ω), substitute in x* = x(ω) to get y* = y(x(ω); ω): Second option, turn it around: First, take then substitute x* = x(ω) into yω(x ; ω) to get And we get the identity

This is the basis for… Roy’s Identity Shephard’s Lemma Allows us to generate ordinary (Marshallian) demand curves from the indirect utility function. Shephard’s Lemma Allows us to generate compensated (Hicksian) demand curves from the expenditure function.

Roy’s Identity: Envelope Theorem 1 Option 1: Plug the optimal choice variable equations into the Lagrangian and THEN differentiate

Roy’s Identity: Envelope Theorem 2 Option 2: Differentiate the Lagrangian and THEN plug the optimal choice variable into the derivative equation

Envelope Theorem and Roy’s Identity

Shephard’s Lemma: Envelope Theorem 1 Option 1: Plug the optimal choice variable equations into the Lagrangian and THEN differentiate

Shephard’s Lemma: Envelope Theorem 2 Differentiate the Lagrangian and THEN plug the optimal choice variable into the derivative equation

Shephard’s Lemma Bring results of Option 1 and Option 2 together:

The Relationships When E* = M’ And U* = Ū Primal Dual Max U(x), s.t. M = px L=U(x)-λ(p•x-M) Marshallian Demand x* = x(p,M’) λ=UM Min E=p•x, s.t. Ū=U(x) L=px-μ(Ū=U(x)) Hicksian Demand x*=xc(p, Ū) μ=EU x(p,M’) = x* = xc (p,Ū) when E* = M’ and U* = Ū x* = x(p,M) x*=xc(p,E(P,U)) x* = xc (p,U) x*=x(p,V(p,M)) Indirect Utility Function U* = U(x*) U* = U(x*=x(p,M’)) U* = V(p, M’) Expenditure Function E* = p•x* where x*=xc(p, Ū) M’=E* = E(p, Ū) U* =V(p,M’) when solved for M’ is E*= E(p, Ū)

Indirect Utility Function The Relationships Primal Dual Indirect Utility Function U* = V*(p, M) Expenditure Function E* = E*(p, U) Roy’s Identity Shephard’s Lemma xi* = xi(p,M)= - xi* = xci (p,U) = ∂V*(p,M) ∂pi ∂M ∂E*(p,U) ∂pi

Ordinary (Marshallian) Demand Slope of budget line from px/py to steeper px’/py Income is fixed at M’, but utility falls y px/py Ū U2 px’/py x*=x(px,py,M’) px/py x xb xa xb xa x Qd falls from xa to xb Qd falls from xa to xb

Compensated (Hicksian) Demand Slope of budget line from px/py to steeper px’/py Utility is fixed at Ū, but expenditure rises y px/py x*=xc(px,py, Ū) px’/py x(px,py,M’)=xc(px,py,Ū) U1 px/py x xc xa xc xa x Qd falls from xa to xc Qd falls from xa to xc

Ordinary (Marshallian) Demand Slope of budget line from px/py to flatter px’’/py Income is fixed at M’, but utility rises y px/py U0 Ū px/py x*=x(px,py,M’) px’’/py xa x xb xa xb x Qd falls from xa to xb Qd falls from xa to xb

Compensated (Hicksian) Demand Slope of budget line from px/py to flatter px’’/py Utility is fixed at Ū, but expenditure falls y px/py x*=xc(px,py,Ū) Ū px/py x(px,py,Ī)=xc(px,py,Ū) px’’/py xa xc x xa xc x Qd rises from xa to xc Qd rises from xa to xc

Ordinary and Compensated If price changes and Qd changes along the ordinary demand curve, then utility changes and you jump to a new compensated demand curve. If price changes and Qd changes along the compensated demand curve, then expenditure needed changes and you jump to a new compensated demand curve. Which curve is more or less elastic depends on whether the good is normal or inferior.