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 = food budget; X=health food (H); Y=junk food (J) U = H0.6 J0.4 I = $160, Ph = $2, Pj = $1 |
| Step 1: take partial derivatives of U to
get the tangency condition (TC): MUx/MUy = Px/Py |
Bx Y Px --·- = -- => By X Py 0.6 J 2 |
| Step 2: rearrange the tangency condition to express Y in terms of X, Px, and Py | J
2 2 1.5·- = - => J = ---·H => J=4/3·H H 1 1.5 |
| Step 3: plug the expression for Y into the budget constraint to solve for X in terms of I, Px, and Py | I = Px·X + Py·Y 160 = 2·H + 1·J 160 = 2·H + 1·(4/3·H) 160 = 10/3·H H = (3/10)·160 = 48 |
| Step 4: plug the solution for X into the formula for Y derived in Step 2 to solve for Y in terms of Px, Py, and I. | J = 4/3·H = 4/3·(48) = 64 |
| Step 5: check your answers. Is the tangency condition met? Is all income spent? |
Check TC: 1.5·(64/48) = 1.5·4/3 = 2 Check BC: 2·H + 1·J = 2(48)+1(64) = 160 |