Question

A factory makes a product out of three infinitely- divisible ingredients X, Y , and Z. The ingredients cost p, q, and r pounds per kg, respectively, with p, q, r > 0 all different, and the factory has a budget of B > 0 pounds. The value of the product is some function g(x, y, z)=exp(x+y+z). of the weight of the ingredients. Solve this problem when the factory wishes to maximize the value of their product, within their budget.

Answer 1

Solution:

Cost of ingredient X is p pounds per kg, of ingredient Y is q pounds per kg, and of Z is r pounds per kg. The budget of factory is B pounds.

Also, the production function is: g(x,y,z) = e^x+y+z

So, the problem of factory is to

Maximize e^x+y+z such that p*X + q*Y + r*Z <= B

With equality, budget line becomes p*X + q*Y + r*Z = B

Forming the Lagrangian, and solving using First order conditions (FOCs):

L = e^x+y+z + d*(B - p*X - q*Y - r*Z) where d is the Lagrangian multiplier.

Then finding the FOCs: x = 0, y = 0 and z = 0

x = e^x+y+z - d*p

So, we have e^x+y+z/p = d

y = e^x+y+z - d*q

So we have e^x+y+z/q = d

z = e^x+y+z - d*r

So we have e^x+y+z/r = d

And we also have pX + qY + rZ = B

Solving all 4 equations above, we can find 4 unknowns: X, Y, Z and d