Solving consumer optimization problems
(Available online: Consumer optimum calculators (Excel): with geometry and a smaller file without geometry)
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
| General steps | Example I = entertainment budget; X=video rentals; Y=CDs U = X2/3 Y1/3 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 = H0.6J0.4 I = $160; Ph = $2; Pj = $1
(1) Solve for the utility-maximizing consumption bundle: (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 |
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