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Quick Review Utility Maximization

Quick Review Utility Maximization. Econ 40565 Fall 2007. Quick review – Consumer maximization. Max U(x,y) subject to I = P x X + P y Y Slope of the budget constraint is –P x /P y What does the slope of the budget constraint represent?

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Quick Review Utility Maximization

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  1. Quick ReviewUtility Maximization Econ 40565 Fall 2007

  2. Quick review – Consumer maximization • Max U(x,y) subject to I = PxX + PyY • Slope of the budget constraint is –Px/Py • What does the slope of the budget constraint represent? • How much Y do you need to give up to get one more unit of X • Suppose Px=$6 and Py=$2, slope =-3 • You need to sacrifice 3 units of y to get one more x

  3. Y I/Py Slope = -Px/Py I/Px X

  4. Indifference curve, collection of points that represent equal utility • Slope of the Indifference curve • Line just tangent to the curve • Slope equals Marginal rate of substitution • MRS = -MUx/MUy

  5. Represents the amount of y you need to give up to consume one more unit of X and keep utility the same • MUx =0.5 and MUy= 2, MRS = - 1/4 • To get one more 1 x, you need to give up 1 quarter y – holding utility constant

  6. Y U1 X

  7. Y I/Py b a U1 I/Px X

  8. Notice at point B • MUy/MUx > Px/Py • MUy/Px > MUy/Py • Extra utility from spending $1 on X is greater than the utility from taking $1 from Y • Therefore, should increase spending on X • Notice at point A • MUy/MUx < Px/Py • MUy/Px < MUy/Py • Extra utility from spending $1 on Y is greater than the utility from taking $1 from X • Therefore, should increase spending on Y

  9. Y I/Py C Y1 X1 I/Px X

  10. Notice at point C • MUy/MUx = Px/Py • MUy/Px = MUy/Py • Extra utility from spending $1 on X is equal to the extra utility from taking $1 from Y • Therefore, you cannot re-arrange spending and make yourself better off

  11. Y U2 U1 I*/Py I/Py Y2 Y1 X2 X X1 I/Px I/Px*

  12. Suppose price of X falls to Px* • Maximum amount of Y you can purchase is still I/Py • Maximum amount of X you can purchase is not I/Px* • Budget constraint rotates about point C • Slope of budget constraint is now –Px*/Py • Amount of y you need to give up to get X has now increased

  13. Y I/Py I/Px* X I/Px

  14. Y U2 U1 I/Py Y1 Y2 X1 X2 I/Px* X I/Px

  15. Price of X has fallen relative to Y • Substitution effect, encourage more consumption of X, less of Y • Every dollar now goes farther, real income has increase, X is normal good, so income effect says you should increase X and Y • Net effect • Unambiguous increase in X • Uncertain impact on Y

  16. Y U2 U1 I/Py b Y1 Y2 Y3 S I X1 X2 I/Px* X3 X I/Px b

  17. To isolate income effect, draw line parallel to new budget constrain until just tangent to old indifference curve (thin line bb) • Movement along old indifference curve is only attributable to change in prices • Substitution effect • Utility is the same, therefore, movement is due only to price changes

  18. Compare demands between the two parallel budget constraints • Only difference is income (price ratios the same)

  19. Along X axis • X1 to X3 same utility, different prices, sub effect • Notice that sub effect is (+) • X3 to X2 same prices, different income, income eff • Notice income effect is (+)

  20. Along Y axis • Y1 to Y3 same utility, different prices, sub effect • Notice that sub effect is (-) • Y3 to Y2 same prices, different income, income eff • Notice income effect is (+)

  21. Example 2 • Price of Y increases to Py* • Substitution effect • Increase x, decrease y • Income effect • Relative income has dropped • If Y and X are normal goods, demand for both should drop

  22. Substitution effect • Y1 to Y3 (-) • X1 to X3 (+) • Income Effect • Y3 to Y2 (-) • X3 to X2 (-)

  23. Y I/Py Y1 U1 X1 I/Px X

  24. Y I/Py Y1 I/Py* Y2 U1 U2 X2 X1 I/Px X

  25. Y X1 to X3 (Sub Effect) I/Py X3 to X2 (Income Effect) Y1 to Y3 (Sub Effect) Y3 to Y2 (Income Effect) Y1 I/Py* Y3 Y2 U1 U2 X3 X2 X1 I/Px X

  26. Y I/Py Y1 I/Py* Sub Y3 Inc Y2 U1 U2 Sub Inc X3 X2 X1 I/Px X

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