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asked 2020-07-15 22:17:52 +0200

EconJohn gravatar image

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')
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 isn't 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?

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 isn't 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?

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 isn't 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?

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, a, b, R') U = x1^ax2^b m = p1x1+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.

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: code:

x1, x2, l, p1, p2, a, b,  R= var('x1, x2, l, p1, p2, a, b,  R')
 U = x1^ax2^b
    x1^a*x2^b
m = p1x1+p2*x2;
    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)
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.

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, 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)

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.

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, 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)

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.

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, 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)

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)], 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) -1/4*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.

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, 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)

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.

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, a, b,  R')
U = x1^a*x2^b
x1^1/2*x2^1/2
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.

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 = x1^1/2*x2^1/2
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.

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