Optimization with Inequality Constraints: Practice Problems and Solutions

Mathematical Economics Problems and Solutions - Section 4

See Mr. Stolyarov's complete list of Mathematical Economics Problems and Solutions.

Note: Here, I will present solve problems typical of those offered in a mathematical economics or advanced microeconomics course. The problems were authored by Dr. Charles N. Steele and are reprinted with his generous permission. The solutions to the problems are my own work and not necessarily the only way to solve the problems.

Problem 4. Use the Kuhn-Tucker conditions to find maxima for the following:
a. F(x, y) = xy, subject to (i) x² + y² ≤ 100 and (ii) x + 4y ≤ 24

b. F(x, y) = a*ln(x) + b*ln(y), subject to (iii) x + y ≤ 100 and (iv) x + 2y ≤ 140.

Assume that a + b = 1

Solution 4a. See the graph of the feasible space. The intersection of x² + y² = 100 and x + 4y = 24 where 4y = 24 -x, and y = 6 - x/4. Thus, x² + (6 - x/4)² = 100 and x = 9.300261, so y = 6 - 9.300261/4 = y = 3.67493475. At (x, y) = (9.300261, 3.67493475), xy = 34.17785233. We cannot, however, move to the right of (x, y) = (9.300261, 3.67493475) along the constraint x² + y² = 100 to find a higher value of xy, because the point

(x, y) = (9.300261, 3.67493475) is already well below the 45-degree line, at which the value of cos(θ)sin(θ) for any circle is maximized. Moving farther away from the 45-degree line by moving to the right will only reduce the value of 10cos(θ)sin(θ), which is equivalent to the value of xy along the quarter-circle denoted by x² + y² = 100.

Thus, we search for our maximum along the line x + 4y = 24 to the left of (x, y) = (9.300261, 3.67493475). This constraint can also be expressed as y = 6 - x/4.

But when we move to the left by Δ, we also increase y by Δ/4.

This means that xy becomes (x - Δ)(y + Δ/4) = xy - Δy + xΔ/4 - (Δ)²/4

At y > 3.67493475, Δy > 3.67493475Δ

At x < 9.300261, Δx/4 < 2.32506525Δ < 3.67493475Δ

Thus, the positive xΔ/4 term will necessarily be smaller than the negative Δy term.