economies question. anyone how to solve

Mar 2018
4
0
malaysia
A firm has using two inputs to produce a single output, Q . The Firm’s production function is Q = x11/2 x21/2 If input price of x1 and x2 are MYR2 and MYR8 per unit, respectively, get the necessary conditions and use it to determine :

(a) Quantity of x1 and x​2 that minimizes the cost
(b) Minimum cost if the firm produces output of 20 units
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
I presume you mean that given quantity $x_1$ of the first input and quantity $x_2$ of the second, we can produce quantity $x_1^{1/2}x_2^{1/2}$ of Q. So we could, for example, produce the same amount of Q by using, say, more of $x_1$ and less of $x_2$ or vice versa. The cost of $x_1$ of the first input is $2x_1$ and the cost of $x_2$ of the second input is $8x_2$ so the cost of Q is $(2x_1)^{1/2}(8x_1)^{1/2}= (16x_1x_2)^{1/2}= 4(x_1x_2)^{1/2}$. Since the first input is so much less expensive than the second, we should want to use as much of the first and as little of the second as possible. Is there some restriction on how little of each we can use? For example are these required to be positive integers so that the minimum of $x_2$ is 1, in which case $x_1$ must be Q and the minimum cost is MYR 2Q. If there is no minimum amount then there is no "minimum cost". We can make the cost as low as we please by taking $x_2$ to be a small positive number- but there is no "smallest" positive number.