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Right now im trying to solve one of the problems i've encountered using sage and it seems like im onto something. However I've encountered a kink at step3 in my code.

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);
step1=solve([dLdx == 0, dLdy == 0, dLdl == 0], p1, p2, R)
step2=solve(step1[0][0].rhs()/step1[0][1].rhs()==p1/p2,x1)
step3=solve(U.subs(step2)==R,x2)
step3

Out: x2^b == R/(a*p2*x2/(b*p1))^a

being that I want to isolate x2 im not sure why its power b is not just moved over.

why is this the case?

Right now im trying to solve one of the problems i've encountered using sage and it seems like im onto something. However I've encountered a kink at step3 in my code.

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);
step1=solve([dLdx == 0, dLdy == 0, dLdl == 0], p1, p2, R)
step2=solve(step1[0][0].rhs()/step1[0][1].rhs()==p1/p2,x1)
step3=solve(U.subs(step2)==R,x2)
step3

Out: x2^b == R/(a*p2*x2/(b*p1))^a

being that I want to isolate x2 im not sure why its power b is not just moved over.

why is this the case?

Right now im trying to solve one of the problems i've encountered using sage and it seems like im onto something. However I've encountered a kink at step3 in my code.

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);
step1=solve([dLdx == 0, dLdy == 0, dLdl == 0], p1, p2, R)
step2=solve(step1[0][0].rhs()/step1[0][1].rhs()==p1/p2,x1)
step3=solve(U.subs(step2)==R,x2)
step3

Out: x2^b == R/(a*p2*x2/(b*p1))^a

being that I want to isolate x2 im not sure why its power b is not just moved over.

why is this the case?

Right now im trying to solve one of the problems i've encountered using sage and it seems like im onto something. However I've encountered a kink at step3 in my code.

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);
step1=solve([dLdx == 0, dLdy == 0, dLdl == 0], p1, p2, R)
step2=solve(step1[0][0].rhs()/step1[0][1].rhs()==p1/p2,x1)
step3=solve(U.subs(step2)==R,x2)
step3

Out: x2^b == R/(a*p2*x2/(b*p1))^a

being that I want to isolate x2 im not sure why its power b is not just moved over.

why is this the case?

Right now im trying to solve one of the problems i've encountered using sage and it seems like im onto something. However I've encountered a kink at step3 in my code.

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);
step1=solve([dLdx == 0, dLdy == 0, dLdl == 0], p1, p2, R)
step2=solve(step1[0][0].rhs()/step1[0][1].rhs()==p1/p2,x1)
step3=solve(U.subs(step2)==R,x2)
step3

Out: x2^b == R/(a*p2*x2/(b*p1))^a

being that I want to isolate x2 im not sure why its power b is not just moved over.

why is this the case?

Right now im trying to solve one of the problems i've encountered using sage and it seems like im onto something. However I've encountered a kink at step3 in my code.

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);
step1=solve([dLdx == 0, dLdy == 0, dLdl == 0], p1, p2, R)
step2=solve(step1[0][0].rhs()/step1[0][1].rhs()==p1/p2,x1)
step3=solve(U.subs(step2)==R,x2)
step3

Out: x2^b == R/(a*p2*x2/(b*p1))^a

being that I want to isolate x2 im not sure why its power b is not just moved over.

why is this the case?

x1, x2, l, p1, p2, a, b,  Ubar= var('x1, x2, l, p1, p2, a, b,  Ubar')
assume(x1>0,x2>0,a>0,b>0)
U = x1*x2
m = p1*x1+p2*x2;
L = m+ l * (Ubar-U);
dLdx = L.diff(x1);
dLdy = L.diff(x2);
dLdl = L.diff(l);
step1=solve([dLdx == 0, dLdy == 0, dLdl == 0], p1, p2, Ubar)
step2=solve(step1[0][0].rhs()/step1[0][1].rhs()==p1/p2,x1)
step3good2=solve(U.subs(step2)==Ubar,x2)
step4good1=solve(step2[0].subs(step3Good2),x1)

step3good2

Out: [x2 == sqrt(R*p1/p2)]

step4good1

Out: [x1 == sqrt(R*p1/p2)*p2/p1]

Right now im trying to solve one of the problems i've encountered using sage and it seems like im onto something. However I've encountered a kink at step3 in my code.

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);
step1=solve([dLdx == 0, dLdy == 0, dLdl == 0], p1, p2, R)
step2=solve(step1[0][0].rhs()/step1[0][1].rhs()==p1/p2,x1)
step3=solve(U.subs(step2)==R,x2)
step3

Out: x2^b == R/(a*p2*x2/(b*p1))^a

being that I want to isolate x2 im not sure why its power b is not just moved over.

why is this the case?

Note: The following code works

x1, x2, l, p1, p2, a, b,  Ubar= var('x1, x2, l, p1, p2, a, b,  Ubar')
assume(x1>0,x2>0,a>0,b>0)
U = x1*x2
m = p1*x1+p2*x2;
L = m+ l * (Ubar-U);
dLdx = L.diff(x1);
dLdy = L.diff(x2);
dLdl = L.diff(l);
step1=solve([dLdx == 0, dLdy == 0, dLdl == 0], p1, p2, Ubar)
step2=solve(step1[0][0].rhs()/step1[0][1].rhs()==p1/p2,x1)
step3good2=solve(U.subs(step2)==Ubar,x2)
step4good1=solve(step2[0].subs(step3Good2),x1)

step3good2

Out: [x2 == sqrt(R*p1/p2)]

step4good1

Out: [x1 == sqrt(R*p1/p2)*p2/p1]

This works because I'm considering a case where our utility function has no exponents a and b. What am I missing from my code?