# Solve() not working for Expenditure Minimization Problem in Sage

Though I've figured out how to go through a standard utility maximization problem analytically in sage Im having difficulty solving its dual problem, expenditure minimization. The code I have is the following:

```
x1, x2, l, p1, p2, a, b, R= var('x1, x2, l, p1, p2, a, b, R')
U = x1^a*x2^b
m = p1*x1+p2*x2;
L = m+ l * (R-U);
dLdx = L.diff(x1);
dLdy = L.diff(x2);
dLdl = L.diff(l);
solve([dLdx == 0, dLdy == 0, dLdl == 0], x1, x2, l)
```

The output I get however is:

```
[-b*l*x1^a*x2^(b - 1) + p2 == 0, -a*l*x1^(a - 1)*x2^b + p1 == 0, -x1^a*x2^b + R == 0]
```

which are only the first order conditions and not solving for $x_1$ and $x_2$.

This code is exactly like my utility maximization code here however I only moved where `m`

and `U`

are.

why is this the case?

Note: if I were to consider the following code:

```
x1, x2, l, p1, p2, a, b, R= var('x1, x2, l, p1, p2, R')
U = x1^1/3*x2^2/3
m = p1*x1+p2*x2;
L = m+ l * (R-U);
dLdx = L.diff(x1);
dLdy = L.diff(x2);
dLdl = L.diff(l);
solve([dLdx == 0, dLdy == 0, dLdl == 0], x1, x2, l)
```

I obtain the following analytical solution:

```
[[x1 == 1/2*18^(1/3)*R/(R*p1/p2)^(2/3), x2 == 18^(1/3)*(R*p1/p2)^(1/3), l==1/2*18^(1/3)*p1/(R*p1/p2)^(2/3)],
[x1 == 1/4*18^(1/3)*R*(-I*sqrt(3) - 1)/(R*p1/p2)^(2/3), x2 == -1/2*18^(1/3)*
(R*p1/p2)^(1/3)*(I*sqrt(3) + 1), l == 1/4*18^(1/3)*p1*(-I*sqrt(3) - 1)/(R*p1/p2)^(2/3)],
[x1 == -1/4*18^(1/3)*R*(-I*sqrt(3) + 1)/(R*p1/p2)^(2/3), x2 == 1/2*18^(1/3)*(R*p1/p2)^(1/3)*(I*sqrt(3) - 1), l == -1/4*18^(1/3)*p1*(- I*sqrt(3) + 1)/(R*p1/p2)^(2/3)]]
```

My guess is that there is a bug with regards to the rules of multiplying undefined exponents.