Chapter3

# 20 40 60 80 100 120 1 2 3 4 5 6 7 8 9 10 upsxy

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20 40 60 80 100 120 1 2 3 4 5 6 7 8 9 10 U(PS)=X+Y U=X^1/2Y^1/2

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Solve the Rational Consumer’s Problem. Method 1 (for Cobb Douglas Utility). Use the equilibrium condition: With ( , ) X U X Y MU X ( , ) Y U X Y MU Y
Solve the Rational Consumer’s Problem. Method 2 (for Cobb Douglas Utility). From the budget constraint, isolate one of the good as the function of income, prices and the other good. Substitute the expression of Y into the utility function U(X,Y). Maximize the utility function which is now a function of only one of the good (good X in this case). The F.O.C yields the solution for good X. Substitute the value of good X onto the budget constraint to get the solution for good Y. X Y Y P M Y X P P

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Solve the Rational Consumer’s Problem. Advanced Mathematical Method. Lagrangean Multipliers (for Cobb Douglas Utility).. Set and And let (1)/(2) Similar equilibrium condition as the graphical method. Solve for the quantity of two goods given prices and income level.
Solve the Rational Consumer’s Problem.

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Solve the Rational Consumer’s Problem. Example: Dan spends all his 100 weekly income on two goods, X and Y. His utility function is given by: If Px = \$4/unit and Py = \$10/unit, how much of each good should he buys. 1 1 2 2 ( , ) U X Y X Y 1 1 2 2 ( , ) U X Y X Y
Solve the Rational Consumer’s Problem. For other types of utility functions: Perfect Substitute Linear Utility Function The slope of the Utility Function, MRS, is a constant. It is either equal, greater, or less than the slope of the budget line (good Y is on the vertical axis).

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