Solving consumer optimization problems

General steps	Example I = entertainment budget; X=video rentals; Y=CDs U = X^2/3 Y^1/3 I = $120 Px = $2, Py = $10
Step 1: Set up TC: take partial derivatives of U to get the tangency condition (TC): MUx/MUy = Px/Py	(1) Recall from previous worksheet that for a Cobb-Douglas utility function: MRS = MUx/MUy = (Bx/By) ^. Y/X. So here, MRS = ((2/3)/(1/3)) ^. Y/X = 2^.Y/X Px/Py = $2/$10 So TC is: 2^.Y/X = 2/10
Step 2: isolate Y: rearrange the tangency condition to express Y as a dependent variable.	(2) 2·Y/X = 2/10 => Y=X/10
Step 3: solve for X: plug the expression for Y into the budget constraint and solve for X.	(3) I = Px X + Py Y 120 = 2X + 10Y 120 = 2x + 10(X/10) 120 = 3X X = 40
Step 4: solve for Y: plug the solution for X into the formula for Y derived in Step 2 and solve for Y.	(4) Y = X/10 = 40/10 = 4
Step 5: check your answers: Is the tangency condition met? Is all income spent?	(5) check results: TC: MRS: 2^.Y/X = 2^.(4/40) = 8/40 = 1/5 Px/Py = 2/10 = 1/5 BC: 120 = 2x + 10y = 2(40) + 10(4) = 120

    To do: Try the following example:
        I = food budget; Good X = health food (H); Good Y = junk food (J)
        U = H^0.6J^0.4
        I = $160; Ph = $2; Pj = $1

(1) Solve for the utility-maximizing consumption bundle:
H = ______ J = ______

(2) Depict the optimum in the diagram to the right. Use actual numerical values to label (a) your budget line endpoints, and (b) the values of X and Y at your optimum.

(3) What would the total expenditure (TE) and MRS be at a consumption bundle of
H=40 and J=80? TE = ______ MRS = ______

(4) Draw an indifference curve that passes through the point H=40 and J=80.

Situation:		Goal: maximize U = U(X,Y)
		Constraint: I = PxX + PyY

Key relationships:		(1) Tangency Condition (TC): MUx/MUy = (dU/dX)/(dU/dY) = Px/Py
		(2) Budget Constraint (BC): I = Px X + Py Y