math

Econ 507 Practice

ECON 507 - Mathematical Economics

Kuhn-Tucker Problems

1 Kuhn-Tucker Conditions and “Corner Solutions”

Suppose a consumer spends his/her income on beer (b) and chips (c) with prices pb and pc, respec- tively. The consumer’s income is W , hence their budget constraint is given by W = pbb + pcc and their preferences can be represented by the following utility function:

u(b,c) = bα(c + 2)β

1. Set up the consumer’s optimization problem and use the Kuhn-Tucker conditions to solve for the consumer’s optimal consumption of chips and beer.

2. How does your solution to the Kuhn-Tucker conditions depend on the parameters (i.e. W,pb,pc,β,α)

3. Now suppose that W = 100, α = 0.9, β = 0.1, pb = 10, and pc = 10. What is the solution?

2 Inequality Constrained Optimization

Find the maximum of 20 − x − (11 − x)2 subject to x ≥ 5 and x ≤ 10.

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