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1

$begingroup$

Preamble, set-up:
In economics, a well known optimization problem writes as follows:
$$
max_x_j=1^n,yU(x_1,y)+sum_i=2^nlambda_icdot U(x_i,y)+mucdot(y-sum_j=1^n(w_j-x_j))
$$

where $lambda_i$ and $mu$ are (binding) lagrangian multipliers
such that $mu>0$, and $lambda_i>0$ for all $i$, and $U$ is a differentiable and concave function.

An easy way to solve this problem taking first order conditions:
$$
x_j:lambda_jcdotfracpartial U(x_j,y)partial x_j+mu=0, quad textfor $j=1,dots,n$, and $lambda_1=1$
$$
$$
y:sum_j=1^nlambda_jcdotfracpartial U(x_j,y)partial y+mu=0, quad textfor $j=1,dots,n$, and $lambda_1=1$
$$

Notice that provided $mu>0$ we can divide the last equation by $mu$ and the use the $n$ firs order conditions for $x_j$ in order to get a compact necessary condition:

$$
sum_j=1^nfracpartial U(x_j,y)/partial ypartial U(x_j,y)/partial x_j=1
$$

My interest:
I would like to expand this result to a setting were there is an infinite number of consumers. Lets say that consumers are indexed by $jin[0,1]$ according to an atomless CDF $F(j)$. Essentialy, I would like to retrieve an equivalent necesary optimality condition like:

$$
int_0^1fracpartial U(x_j,y)/partial ypartial U(x_j,y)/partial x_jdF(j)=1
$$

The optimization problem would look like:

$$
max_x_jin[0,1],yint_0^1lambda_jcdot U(x_j,y)dF(j)+mucdotleft(y-int_0^1(w_j-x_j)dF(j)right)
$$

However if I want to retrieve the desired expression it seems like I need that:

$$
fracint_0^1(w_i-x_i)dF(i)dx_i=-1
$$

and I do not know under which conditions I can get this. Any suggestions? Thanks!

measure-theory optimization economics continuum-theory

share|cite|improve this question

asked Mar 15 at 17:02

Weierstraß RamirezWeierstraß Ramirez

146420

$endgroup$

add a comment | 

1

$begingroup$

Preamble, set-up:
In economics, a well known optimization problem writes as follows:
$$
max_x_j=1^n,yU(x_1,y)+sum_i=2^nlambda_icdot U(x_i,y)+mucdot(y-sum_j=1^n(w_j-x_j))
$$

where $lambda_i$ and $mu$ are (binding) lagrangian multipliers
such that $mu>0$, and $lambda_i>0$ for all $i$, and $U$ is a differentiable and concave function.

An easy way to solve this problem taking first order conditions:
$$
x_j:lambda_jcdotfracpartial U(x_j,y)partial x_j+mu=0, quad textfor $j=1,dots,n$, and $lambda_1=1$
$$
$$
y:sum_j=1^nlambda_jcdotfracpartial U(x_j,y)partial y+mu=0, quad textfor $j=1,dots,n$, and $lambda_1=1$
$$

Notice that provided $mu>0$ we can divide the last equation by $mu$ and the use the $n$ firs order conditions for $x_j$ in order to get a compact necessary condition:

$$
sum_j=1^nfracpartial U(x_j,y)/partial ypartial U(x_j,y)/partial x_j=1
$$

My interest:
I would like to expand this result to a setting were there is an infinite number of consumers. Lets say that consumers are indexed by $jin[0,1]$ according to an atomless CDF $F(j)$. Essentialy, I would like to retrieve an equivalent necesary optimality condition like:

$$
int_0^1fracpartial U(x_j,y)/partial ypartial U(x_j,y)/partial x_jdF(j)=1
$$

The optimization problem would look like:

$$
max_x_jin[0,1],yint_0^1lambda_jcdot U(x_j,y)dF(j)+mucdotleft(y-int_0^1(w_j-x_j)dF(j)right)
$$

However if I want to retrieve the desired expression it seems like I need that:

$$
fracint_0^1(w_i-x_i)dF(i)dx_i=-1
$$

and I do not know under which conditions I can get this. Any suggestions? Thanks!

measure-theory optimization economics continuum-theory

share|cite|improve this question

asked Mar 15 at 17:02

Weierstraß RamirezWeierstraß Ramirez

146420

$endgroup$

add a comment | 

$begingroup$

Preamble, set-up:
In economics, a well known optimization problem writes as follows:
$$
max_x_j=1^n,yU(x_1,y)+sum_i=2^nlambda_icdot U(x_i,y)+mucdot(y-sum_j=1^n(w_j-x_j))
$$

where $lambda_i$ and $mu$ are (binding) lagrangian multipliers
such that $mu>0$, and $lambda_i>0$ for all $i$, and $U$ is a differentiable and concave function.

An easy way to solve this problem taking first order conditions:
$$
x_j:lambda_jcdotfracpartial U(x_j,y)partial x_j+mu=0, quad textfor $j=1,dots,n$, and $lambda_1=1$
$$
$$
y:sum_j=1^nlambda_jcdotfracpartial U(x_j,y)partial y+mu=0, quad textfor $j=1,dots,n$, and $lambda_1=1$
$$

Notice that provided $mu>0$ we can divide the last equation by $mu$ and the use the $n$ firs order conditions for $x_j$ in order to get a compact necessary condition:

$$
sum_j=1^nfracpartial U(x_j,y)/partial ypartial U(x_j,y)/partial x_j=1
$$

My interest:
I would like to expand this result to a setting were there is an infinite number of consumers. Lets say that consumers are indexed by $jin[0,1]$ according to an atomless CDF $F(j)$. Essentialy, I would like to retrieve an equivalent necesary optimality condition like:

$$
int_0^1fracpartial U(x_j,y)/partial ypartial U(x_j,y)/partial x_jdF(j)=1
$$

The optimization problem would look like:

$$
max_x_jin[0,1],yint_0^1lambda_jcdot U(x_j,y)dF(j)+mucdotleft(y-int_0^1(w_j-x_j)dF(j)right)
$$

However if I want to retrieve the desired expression it seems like I need that:

$$
fracint_0^1(w_i-x_i)dF(i)dx_i=-1
$$

and I do not know under which conditions I can get this. Any suggestions? Thanks!

measure-theory optimization economics continuum-theory

share|cite|improve this question

asked Mar 15 at 17:02

Weierstraß RamirezWeierstraß Ramirez

146420

$endgroup$

share|cite|improve this question

asked Mar 15 at 17:02

Weierstraß RamirezWeierstraß Ramirez

146420

share|cite|improve this question

asked Mar 15 at 17:02

Weierstraß RamirezWeierstraß Ramirez

146420

asked Mar 15 at 17:02

Weierstraß RamirezWeierstraß Ramirez

146420

asked Mar 15 at 17:02

Weierstraß RamirezWeierstraß Ramirez

146420