What is cost optimisation?

Economics help ! cost mininimisation?

• how do i find the cost-minimisation optimisation of this problem ? produces good x, with inputs a ,b and c and the price of this inputs is 1, 2 and 5 respectively . has to produce 800 units of good x and the production function is x= abc any help would be much appreciated :)

• Answer:

  Your task is to minimize the value of a+2b+5c given that abc=800. Since you have a function of a, b, and c, and a constraint on a, b, and c, you can solve this with Lagrange multipliers. Let F(a,b,c)=a+2b+5c and g(a,b,c)=abc. Do you know the Lagrange multiplier technique? It's derived from taking level curves of F at the point(s) where they're tangent to g so that the gradient vectors are parallel i.e. <dF/da,dF/db,dF/dc> = z*<dg/da,dg/db,dg/dc> where z is a scalar. Set dF/da = z*(dg/da) and so on for b and c. Typically we would use lambda rather than z, but there's not a convenient Greek alphabet here. Also the d in dF/da should be a lowercase delta to represent the partial derivative of F with respect to a. 1=z*bc (1) 2=z*ac (2) 5=z*ab (3) abc=800 Divide (2) by (1) to see that a/b = 2. Thus b = a/2. Divide (3) by (1) to see that a/c = 5. Thus c = a/5. So a(a/2)(a/5) = 800. Thus a^3/10 = 800 so a = 20. This means b = 10 and c = 4. And thus your minimal cost is 20+2(10)+5(4)=60.