Suppose that you own an electronics manufacturing company and produce two types of circuit boards: board A and board B. To produce either type you need some number of worker-hours and some amount of money, and these trade off (e.g., if you spend more money on automation, you need fewer worker hours; and in general, producing more lowers the cost per item). We generally want to minimize the cost of production. The constraints are (1) that we have contracts to produce some number of each type of the circuit boards; and (2) that we have a fixed number of worker hours available. The first of these is modeled by the Cobb-Douglas production model: if L is the number of labor hours spent to produce the boards and K the dollars of capital invested, (number of boards) = kL^K^ , for some constants k, , and . Let's say that we have a contract to produce 15,000 circuit board As, and 25,000 circuit board Bs, and that the Cobb-Douglas models for these are 15000 = 5L^0.3 A K^0.7 A and 25000 = 2L^0.4 B K^0.6 B . Suppose you have 2000 total labor hours available, so that 2000 LA + LB . Finally, the cost of production is the capital invested in the production of each type of board, and the cost of the labor. Suppose that each hour of labor on circuit A costs $20, while each hour on circuit B costs $25. (In the following you may assume that the gradients of the constraints are linearly independent, as they are.)

Exercise 5: Write equation for the total cost of production, and the constraints. What are the variables in the optimization problem?

Exercise 6: Consider the case 2000 > La + LB . Find possible locations of the minimum cost by using the technique from exercise 2.

Exercise 7: Now consider the boundary 2000 = La + LB . Find possible locations of the minimum cost, and by using the results from 6 find the minimum production cost.