Question

1. Suppose the equation p(x,y)=-2x^2+80x-3y^2+90+100 models profit when x represents the number of handmade chairs and y is the number of handmade rockers produced per week.

(1) How many chairs and how many rockers will give the maximum profit when there is not constraint?

(2) Due to an insufficient labor force they can only make a total of 20 chairs and rockers per week (x + y = 20). So how many chairs and how many rockers will give the realistic maximum profit? (You can round to the nearest integer if your solutions of x and y are not integer numbers.)

2. A Cobb-Douglas production function for new company is given by f(x,y)=70x^2/7y^5/7, where x represents the units of labor and y represents the units of capital. Suppose units of labor and capital cost $100 and $50 each respectively. If the budget constraint is $3,500, (1) find the optimal combination of labor and capital units to maximize the production level for this manufacture. (2) find the value of shadow price and interpret it.

3. The function U=f(x,y)=x^2y^2 represent the utility or customer satisfaction from the consumption of a certain amount of product x and a certain amount of product y. The budget constraint is x+4y=1,200. What are the optimal amount of consumption for x and y? What is the maximum utility? How to interpret the Lagrange Multiplier?

Answer 1