1 / 31

Utility Maximization

Utility Maximization. Continued July 5, 2005. Graphical Understanding. Normal Indifference Curves. Downward Slope with bend toward origin. Graphical. Non-normal Indifference Curves. Y & X Perfect Substitutes. Graphical. Non-normal. Only X Yields Utility. X & & are perfect

jubal
Download Presentation

Utility Maximization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Utility Maximization Continued July 5, 2005

  2. Graphical Understanding • Normal Indifference Curves Downward Slope with bend toward origin

  3. Graphical • Non-normal Indifference Curves Y & X Perfect Substitutes

  4. Graphical • Non-normal Only X Yields Utility

  5. X & & are perfect complementary goods Graphical • Non-normal

  6. Calculus caution • When dealing with non-normal utility functions the utility maximizing FOC that MRS = Px/Py will not hold • Then you would use other techniques, graphical or numerical, to check for corner solution.

  7. Cobb-Douglas • Saturday Session we know that if U(X,Y) = XaY(1-a) then X* = am/Px • m: income or budget (I) • Px: price of X • a: share of income devoted to X • Similarly for Y

  8. Cobb-Douglas • How is the demand for X related to the price of X? • How is the demand for X related to income? • How is the demand for X related to the price of Y?

  9. CES • Example U(x,y) = (x.5+y.5)2

  10. CES Demand Eg: Y = IPx/Py(1/(Px+Py)) Let’s derive this in class

  11. I=150 I=100 CES Demand | Px=5 • I=100 & I = 150

  12. Px=10 Px=5 CES | I = 100

  13. For CES Demand • If the price of X goes up and the demand for Y goes up, how are X and Y related? • On exam could you show how the demand for Y changes as the price of X changes? • dY/dPx

  14. When a price changes • Aside: when all prices change (including income) we should expect no real change. Homogeneous of degree zero. • When one prices changes there is an income effect and a substitution effect of the price change.

  15. Changes in income • When income increases demand usually increase, this defines a normal good. • ∂X/∂I > 0 • If income increases and demand decreases, this defines an inferior good.

  16. Normal goods As income increase (decreases) the demand for X increase (decreases)

  17. Inferior good As income increases the demand for X decreases – so X is called an inferior good

  18. A change in Px Here the price of X changes…the budget line rotates about the vertical intercept, m/Py.

  19. The change in Px • The change in the price of X yields two points on the Marshallian or ordinary demand function. • Almost always when Px increase the quantity demand of X decreases and vice versa. • So ∂X/∂Px < 0

  20. But here, ∂X/∂Px > 0 This time the Marshallian or ordinary demand function will have a positive instead of a negative slope. Note that this is similar to working with an inferior good.

  21. Decomposition • We want to be able to decompose the effect of a change in price • The income effect • The substitution effect • We also will explore Giffen’s paradox – for goods exhibiting positively sloping Marshallian demand functions.

  22. Decomposition • There are two demand functions • The Marshallian, or ordinary, demand function. • The Hicksian, or income compensated demand function.

  23. Compensated Demand • A compensated demand function is designed to isolate the substitution effect of a price change. • It isolates this effect by holding utility constant. • X* = hx(Px, Py, U) • X = dx(Px, Py, I)

  24. The indirect utility function • When we solve the consumer optimization problem, we arrive at optimal values of X and Y | I, Px, and Py. • When we substitute these values of X and Y into the utility function, we obtain the indirect utility function.

  25. The indirect utility function • This function is called a value function. It results from an optimization problem and tells us the highest level of utility than the consumer can reach. • For example if U = X1/2Y1/2 we know • V = (.5I/Px).5(.5I/Py).5 = .5I/Px.5Py.5

  26. Indirect Utility • V = 1/2I / (Px1/2Py1/2) • or • I = 2VPx1/2Py1/2 • This represents the amount of income required to achieve a level of utility, V, which is the highest level of utility that can be obtained.

  27. I = 2VPx1/2Py1/2 • Let’s derive the expenditure function, which is the “dual” of the utility max problem. • We will see the minimum level of expenditure required to reach a given level of utility.

  28. Minimize • We want to minimize • PxX + PyY • Subject to the utility constraint • U = X1/2Y1/2 • So we form • L = PxX + PyY + λ(U- X1/2Y1/2)

  29. Minimize Continued • Let’s do this in class… • We will find • E = 2UPx1/2Py1/2 • In other words the least amount of money that is required to reach U is the same as the highest level of U that can be reached given I.

  30. Hicksian Demand • The compensated demand function is obtained by taking the derivative of the expenditure function wrt Px • ∂E/∂Px = U(Py/Px)1/2 • Let’s look at some simple examples

  31. Ordinary & Compensated In this example our utility function is: U = X.5Y.5. We change the price of X from 5 to 10.

More Related