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Demand Function

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  1. Demand Function

  2. Properties of Demand Functions • Comparative statics analysis of ordinary demand functions -- the study of how ordinary demands x1*(p1,p2,y) and x2*(p1,p2,y) change as prices p1, p2 and income y change.

  3. Own-Price Changes • How does x1*(p1,p2,y) change as p1 changes, holding p2 and y constant? • Suppose only p1 increases, from p1’ to p1’’ and then to p1’’’.

  4. Own-Price Changes Fixed p2 and y. x2 p1x1 + p2x2 = y p1 = p1’ x1

  5. Own-Price Changes Fixed p2 and y. x2 p1x1 + p2x2 = y p1 = p1’ p1= p1’’ x1

  6. Own-Price Changes Fixed p2 and y. x2 p1x1 + p2x2 = y p1 = p1’ p1=p1’’’ p1= p1’’ x1

  7. Own-Price Changes Fixed p2 and y. p1 = p1’

  8. Own-Price Changes Fixed p2 and y. p1 = p1’ x1*(p1’)

  9. p1 Own-Price Changes Fixed p2 and y. p1 = p1’ p1’ x1* x1*(p1’) x1*(p1’)

  10. p1 Own-Price Changes Fixed p2 and y. p1 = p1’’ p1’ x1* x1*(p1’) x1*(p1’)

  11. p1 Own-Price Changes Fixed p2 and y. p1 = p1’’ p1’ x1* x1*(p1’) x1*(p1’) x1*(p1’’)

  12. p1 Own-Price Changes Fixed p2 and y. p1’’ p1’ x1* x1*(p1’) x1*(p1’’) x1*(p1’) x1*(p1’’)

  13. p1 Own-Price Changes Fixed p2 and y. p1 = p1’’’ p1’’ p1’ x1* x1*(p1’) x1*(p1’’) x1*(p1’) x1*(p1’’)

  14. p1 Own-Price Changes Fixed p2 and y. p1 = p1’’’ p1’’ p1’ x1* x1*(p1’) x1*(p1’’) x1*(p1’’’) x1*(p1’) x1*(p1’’)

  15. p1 Own-Price Changes Fixed p2 and y. p1’’’ p1’’ p1’ x1* x1*(p1’) x1*(p1’’’) x1*(p1’’) x1*(p1’’’) x1*(p1’) x1*(p1’’)

  16. p1 Own-Price Changes Ordinarydemand curvefor commodity 1 Fixed p2 and y. p1’’’ p1’’ p1’ x1* x1*(p1’) x1*(p1’’’) x1*(p1’’) x1*(p1’’’) x1*(p1’) x1*(p1’’)

  17. p1 Own-Price Changes Ordinarydemand curvefor commodity 1 Fixed p2 and y. p1’’’ p1’’ p1’ x1* x1*(p1’) x1*(p1’’’) x1*(p1’’) x1*(p1’’’) x1*(p1’) x1*(p1’’)

  18. p1 Own-Price Changes Ordinarydemand curvefor commodity 1 Fixed p2 and y. p1’’’ p1’’ p1 price offer curve p1’ x1* x1*(p1’) x1*(p1’’’) x1*(p1’’) x1*(p1’’’) x1*(p1’) x1*(p1’’)

  19. Own-Price Changes • The curve containing all the utility-maximizing bundles traced out as p1 changes, with p2 and y constant, is the p1- price offer curve. • The plot of the x1-coordinate of the p1- price offer curve against p1 is the ordinary demand curve for commodity 1.

  20. Own-Price Changes • What does a p1 price-offer curve look like for Cobb-Douglas preferences?

  21. Own-Price Changes • What does a p1 price-offer curve look like for Cobb-Douglas preferences? • TakeThen the ordinary demand functions for commodities 1 and 2 are

  22. Own-Price Changes and Notice that x2* does not vary with p1 so thep1 price offer curve is

  23. Own-Price Changes and Notice that x2* does not vary with p1 so thep1 price offer curve is flat

  24. Own-Price Changes and Notice that x2* does not vary with p1 so thep1 price offer curve is flat and the ordinarydemand curve for commodity 1 is a

  25. Own-Price Changes and Notice that x2* does not vary with p1 so thep1 price offer curve is flat and the ordinarydemand curve for commodity 1 is arectangular hyperbola.

  26. Own-Price Changes Fixed p2 and y. x1*(p1’’’) x1*(p1’) x1*(p1’’)

  27. p1 Own-Price Changes Ordinarydemand curvefor commodity 1 is Fixed p2 and y. x1* x1*(p1’’’) x1*(p1’) x1*(p1’’)

  28. Own-Price Changes • What does a p1 price-offer curve look like for a perfect-complements utility function?

  29. Own-Price Changes • What does a p1 price-offer curve look like for a perfect-complements utility function? Then the ordinary demand functionsfor commodities 1 and 2 are

  30. Own-Price Changes

  31. Own-Price Changes With p2 and y fixed, higher p1 causessmaller x1* and x2*.

  32. Own-Price Changes With p2 and y fixed, higher p1 causessmaller x1* and x2*. As

  33. Own-Price Changes With p2 and y fixed, higher p1 causessmaller x1* and x2*. As As

  34. Own-Price Changes Fixed p2 and y. x2 x1

  35. p1 Own-Price Changes Fixed p2 and y. x2 p1 = p1’ y/p2 p1’ x1* ’ ’ x1 ’

  36. p1 Own-Price Changes Fixed p2 and y. x2 p1 = p1’’ p1’’ y/p2 p1’ x1* ’’ ’’ x1 ’’

  37. p1 Own-Price Changes Fixed p2 and y. p1’’’ x2 p1 = p1’’’ p1’’ y/p2 p1’ x1* ’’’ ’’’ x1 ’’’

  38. p1 Own-Price Changes Ordinarydemand curvefor commodity 1 is Fixed p2 and y. p1’’’ x2 p1’’ y/p2 p1’ x1* x1

  39. Own-Price Changes • What does a p1 price-offer curve look like for a perfect-substitutes utility function? Then the ordinary demand functionsfor commodities 1 and 2 are

  40. Own-Price Changes and

  41. Own-Price Changes Fixed p2 and y. Fixed p2 and y. x2 p1 = p1’ < p2 x1 ’

  42. p1 Own-Price Changes Fixed p2 and y. Fixed p2 and y. x2 p1 = p1’ < p2 p1’ x1* ’ x1 ’

  43. p1 Own-Price Changes Fixed p2 and y. Fixed p2 and y. x2 p1 = p1’’ = p2 p1’ x1* x1

  44. p1 Own-Price Changes Fixed p2 and y. Fixed p2 and y. x2 p1 = p1’’ = p2 p1’ x1* x1

  45. p1 Own-Price Changes Fixed p2 and y. Fixed p2 and y. x2 p1 = p1’’ = p2 p1’ x1* x1 ’’

  46. p1 Own-Price Changes Fixed p2 and y. Fixed p2 and y. x2 p1 = p1’’ = p2 p2 = p1’’ p1’ x1* x1

  47. p1 Own-Price Changes Fixed p2 and y. Fixed p2 and y. p1’’’ x2 p2 = p1’’ p1’ x1* x1

  48. p1 Own-Price Changes Ordinarydemand curvefor commodity 1 Fixed p2 and y. Fixed p2 and y. p1’’’ x2 p2 = p1’’ p1 price offer curve p1’ x1* x1

  49. Own-Price Changes • Usually we ask “Given the price for commodity 1 what is the quantity demanded of commodity 1?” • But we could also ask the inverse question “At what price for commodity 1 would a given quantity of commodity 1 be demanded?”

  50. Own-Price Changes p1 Given p1’, what quantity isdemanded of commodity 1? p1’ x1*