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Lecture 4

Lecture 4. Demand Curve July, 2008. Key concepts. Demand functions vs. correspondences Market demand Income and price elasticities Demand curve estimation Consumer surplus. Continuity of the demand function.

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Lecture 4

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  1. Lecture 4 Demand Curve July, 2008

  2. Key concepts • Demand functions vs. correspondences • Market demand • Income and price elasticities • Demand curve estimation • Consumer surplus

  3. Continuity of the demand function • Continuity of Demand Function. Suppose preference set is continuous and weakly convex, and (p;m) > 0. Then, the demand curve x(p;m) is a upper hemi-continuous convex-valued correspondence. Furthermore, if the weak convexity is replaced by the strict convexity, x(p;m) is a continuous single-valued function.

  4. Weakly convex preferences • Recall: • CONVEXITY. Given x; x0in X such that x0> x, then it follows that tx +(1- t)x0> x for all 0 < t < 1. • STRICT CONVEXITY. Given x; x0in X such that x0 > or~x, then it follows that tx +(1- t)x0> x for all 0 < t < 1. • WEAK CONVEXITY. Given x; x0in X such that x0 > or~x, then it follows that tx +(1- t)x0> or~ x for all 0 < t < 1.

  5. Illustrations px y • Discontinuous demand due to non convex preferences b a pxa a c pxb b d c pxd d x xb xc xd xa xb xc xd xa x

  6. Inverse demand function • x=f(p;m) is the demand function • p=p(x) is the inverse demand function

  7. Illustration x p Demand function Inverse demand x p

  8. Integrability problem • Given a system of demand functions x(p;m). Is there necessarily a utility function from which these demand functions can be derived? --- integrability problem. • Yes. if the system of demand functions satisfy these properties: • 1. Nonnegativity: x(p;m) >= 0. • 2. Homogeneity: x(tp; tm) = x(p;m). • 3. Budget Balancedness: px(p;m) = m. • 4. Symmetry: The Slutsky matrix is symmetric. • 5. Negative Semi-definite: The Slutsky matrix S is negative semi-definite.

  9. Revealed Preference • Criticism of preference theory: “too strong on the grounds that individuals are unlikely to make choices through conscious use of a preference relation. • Samuelson developed revealed preference theory as alternative based on weaker set of hypotheses.

  10. Principle behind revealed preference • preference statements are constructed only from observable decisions, that is, from actual choice made by a consumer. • An individual preference relation, even if it exists, can never be directly observed in the market. • Consumers make actual choices given prices and income. • Revealed preference is recovering the preference behavior from actual choices.

  11. Assumptions made in revealed preference theory • In general there are no restrictions: anything is possible. • Need to rule out this trivial case. • The underlying utility function to be locally non-satiated

  12. Definitions • Direct Revealed Preference: If ptxt=> ptx, then u(xt) => u(x). We will say that xtis directly revealed preferred to x, i.e. xtRDx. Assuming that the data were generated by utility maximization, we can conclude that if xtRDx implies u(xt) => u(x). • Strictly Direct Revealed Preference: If ptxt> ptx, then u(xt) > u(x) or xtis strictly direct revealed preferred to x, i.e. xtPDx. • Revealed Preference: xtis said to be revealed preferred to x if there exists a finite number of bundles xt ,x1; x2,…, xnsuch that xtRDx1; x1RDx2,…, xn-1RDxn; Or xtRx. • Transitive closure of the relation RD. xtRx implies u(xt) => u(x).

  13. GARP: Generalized Axiom of Revealed Preference • If xt R xs then xs cannot be strictly directly revealed preferred to xt. • xt R xsimplies not xs PD xt. In other words, xt R xs implies psxs<= psxt. • Two standard conditions of GARP: WARP and SARP.

  14. WARP: Weak Axiom of Revealed Preference • If xtRD xs and xt is not equal to xs, then it is not the case that xs RD xt, i.e., pt xt=> pt xsimplies ps xt=> ps xs. • Requires: the demand function is single valued.

  15. SARP: Strong Axiom of Revealed Preference • If xtRxs and xt is not equal to xs, then it is not the case that xs R xt, i.e., pt xt > pt xsimplies ps xt> ps xs. • Requires: the demand function is single valued.

  16. Recoverability of preference relation • Boundaries. RP revealed preferred, RW revealed worse • Relative to xo x’ xs

  17. Recoverability of preference relation • Boundaries. RP revealed preferred, RW revealed worse • With many observed points the NRP and NRW becomes tighter.

  18. Income – Leisure Model • Slutsky’s equation

  19. Illustration: price x increase y A’ A v v B UA’ UA UB x xB xA’ xA

  20. Income – Leisure model • x is consumption basket • w is wage • L is Leisure • Tbar is endowment of time • H or hours of work is (Tbar-L)

  21. Income effect is positive; assuming Leisure is normal good Possible that income dominates the substitution effect And thus the demand for Leisure will have portion where it is like that of Giffen good Supply of hours of work is the negative of the demand of Leisure Upward sloping supply of hours of work Income-Leisure Model

  22. Demand for Leisure w L

  23. Individual Hours of work supply w • H=Tbar-L • Backward bending supply function of Hours of Work H=Tbar-L

  24. End of Lecture 4 Demand Curve

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