Cost function from CES production function

If you are interested in the case where $rho geq 1$ then look at the post CES $ rho geq 1$. For the standard case where $0 < rho < 1$ you should get a result like this

$$C(w_1,w_2,y) = left(w_1^{frac{rho}{rho -1}} +w_2^{frac{rho}{rho -1}}right)^{frac{rho - 1}{rho}} y.$$

To see this you should start by setting up the cost minimization problem

$$min_{x_1,x_2} w_1x_1 + w_2x_2 [8pt] s.t. (x_1^rho + x_2^rho)^{1/rho} geq y$$

for this problem the Lagrangian function is

$$mathcal L(x_1,x_2,lambda) = w_1x_1 + w_2x_2 - lambda((x_1^rho + x_2^rho)^{1/rho} -y).$$

From the first order conditions of the Lagrangian you can show the constraint is binding in optimum $(x_1^rho + x_2^rho)^{1/rho} = y$ and get MRS equal to relative prices

$$(1) frac{w_1}{w_2} = frac{x_1^{rho - 1}}{x_2^{rho - 1}},$$ given this information you should be able to solve for $x_1$ and $x_2$ as a function of the parameters of the problem which in this case is $rho,y,w_1,w_2$.

Try to get $(x_1^rho + x_2^rho)^{1/rho}$ to appear in the MRS equal to relative prices. So manipulate (1) to get

$$w_1^{frac{rho}{rho -1}}x_2^rho = w_2^{frac{rho}{rho -1}}x_1^rho,$$ then add $w_2^{frac{rho}{rho-1}}x_2^rho$ to both sides of equation

$$w_1^{frac{rho}{rho -1}}x_2^rho +w_2^{frac{rho}{rho -1}}x_2^rho = w_2^{frac{rho}{rho -1}}x_1^rho + w_2^{frac{rho}{rho -1}}x_2^rho,$$

isolate factors on both sides and exponentiate with exponent $1/rho$ to get

$$left(w_1^{frac{rho}{rho -1}} +w_2^{frac{rho}{rho -1}}right)^{1/rho}x_2 = w_2^{frac{1}{rho -1}}(x_2^rho + x_1^rho)^{1/rho} = w_2^{frac{1}{rho -1}} y ,$$

from here you can solve for conditional demand $x_2^star(w_1,w_2,y)$. However, it is easier to oberserve that the factor $left(w_1^{frac{rho}{rho -1}} +w_2^{frac{rho}{rho -1}}right)^{1/rho}$ do not change when interchanging indexes - it is symmetric. So define $a := left(w_1^{frac{rho}{rho -1}} +w_2^{frac{rho}{rho -1}}right)^{1/rho}$ and conclude that

$$ax_1 = w_1^{frac{1}{rho -1}} y [8pt] ax_2 = w_2^{frac{1}{rho -1}} y,$$ multiply first equation with $w_1$ and second with $w_2$ and add them to get

$$a(w_1x_1 + w_2x_2) = (w_1^{frac{rho}{rho -1}}+w_2^{frac{rho}{rho -1}})y = a^rho y$$

solve for $(w_1x_1 + w_2x_2)$ which are the costs to get the result that

$$C(w_1,w_2,y) = a^{rho -1} y = left(w_1^{frac{rho}{rho -1}} +w_2^{frac{rho}{rho -1}}right)^{frac{rho - 1}{rho}} y$$