Help with production optimization problem?
"The production amount, Q, of an item manufactured by a company is modeled by the CobbDouglas function
Q = 200(K^0.6)(L^0.4)(T^0.3)
where K is the quantity of capital investment, L is the quantity of labor used, and T is the investment in training. Capital costs are $17 per unit, labor costs are $24 per unit, and training costs are $8 per unit. The company wants to keep the total cost at the current $20,000 budget while producing as much of the item as possible. Suppose you are asked to consult for the company, and you learn that 800 units of capital, 200 units of labor, and 200 units of training are being used.
Should the plant use more or less labor? More or less capital?
More or less training? By what percent can the company increase production over the current amount without changing the budget? At the optimal allocation, by approximately how much will the production quantity change with a dollar increase in the total budget?"
From what I can gather, the constraint here is 17K+24L+8T=2000, so the resulting equation would be Q = 200(K^0.6)(L^0.4)(T^0.3) - (17K+24L+8T). Our professor encourages us to use a program called Maple 10, and by using that, I've gotten the partial derivatives with respect to x, y, z.
Here are the program's results:
fx = ((120.0 * L^0.4 * T^0.3)/(K^0.4)) - 17
fy = ((80.0 K^0.6 T^0.3)/(L^0.6))-24
fz = ((60.0 K^0.6 L^0.4)/(T^0.7))-8
My next step was to solve these equations by setting them to 0. But after that, I'm not sure what to do, nor am I certain that what's been done so far is correct. Any input would be appreciated!