# Solving Lagrangians in Sage

I have been running the following Sage code on cocalc.com to solve a standard utility maximization problem that is in an intermediate microeconomics course. However I'm having difficulty solving these problems in the context of Sage.

Below is the code that I have run to solve the problem and so far I believe it's correct.

x, y, l = var('x, y, l')
U = x^0.7 * y^0.3; U
m = 2*x+2*y; m
solve(m == 4000, y)
L = U - l * (m - 4000); L
dLdx = L.diff(x); dLdx
dLdy = L.diff(y); dLdy
dLdl = L.diff(l); dLdl
solve([dLdx == 0, dLdy == 0, dLdl == 0], x, y, l)


All my code seems to be running fine, however the last line of:

solve([dLdx == 0, dLdy == 0, dLdl == 0], x, y, l)


is not outputting the final demand equations. Is there any reason why this is happening?

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Sage symbolics handles fractional powers better than numerical ones, so if you replace

U=x^0.7*y^0.3;U


with

U=x^(7/10)*y^(3/10);U


it seems to work much better.

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