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Expenditure Minimization

Expenditure Minimization. Expenditure Minimization. Set up optimization problem. Expenditure Minimization: SOC. The FOC ensure that the optimal consumption bundle is at a tangency.

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Expenditure Minimization

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  1. Expenditure Minimization

  2. Expenditure Minimization • Set up optimization problem

  3. 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

  4. 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 =-Ux2 < 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.

  5. Expenditure Minimization • Solve FOC to get:

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

  7. Interpreting μ: Envelope Result • Start with L*

  8. Finding : Envelope Result • Start with L*.

  9. Expenditure Minimization • Comparative Statics

  10. Comparative Statics

  11. Comparative Statics: Effect of a change in pxPut in Matrix Notation • Solve for

  12. Expenditure Minimization: Example

  13. Expenditure Minimization • Combining with

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

  15. 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.

  16. 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 Efsuch 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’

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

  18. 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

  19. 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…

  20. 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

  21. This is the basis for… • Roy’s Identity • 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.

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

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

  24. Envelope Theorem and Roy’s Identity

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

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

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

  28. 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)) Expenditure Function E* = p•x* where x*=xc(p, Ū) M’=E* = E(p, Ū) Indirect Utility Function U* = U(x*) U* = U(x*=x(p,M’)) U* = V(p, M’) U* =V(p,M’) when solved for M’ is E*= E(p, Ū)

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

  30. 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 x xb xa xb xa Qd falls from xa to xb Qd falls from xa to xb

  31. 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 x xc xa xc xa Qd falls from xa to xc Qd falls from xa to xc

  32. 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 x xa xb xa xb x Qd falls from xa to xb Qd falls from xa to xb

  33. 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 xa xc x x Qd rises from xa to xc Qd rises from xa to xc

  34. 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.

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