Shadow Prices

[phk]

Let us say you are a person trying to choose between buying bananas and oranges. What you are trying to do is maximize a utility function u(x,y) where x represents bananas and y represents oranges.  You can not buy an infinite amount of each however.  This is subject to a budget constraint.  Thus, we have the following maximization problem.

max u(x,y) s.t. p1x + p2y ≤ I

Where p1 = price of bananas
Where p2 = price of oranges
Where x = amount of bananas
Where y = amount of oranges
Where I = Income

If we add functional form assumptions on the utility function we can form the following Lagrangian:

L= ln(x) + αln(y) - λ[p1x + p2y - I]
Our first order conditions are:

Lx:  1/x - λp1 =0
Ly:  α/y - λp2 =0
Lλ:  p1x + p2y - I =0
Our optimal level of bananas and oranges is:

x* = I/[(1+α)p1]
y* = Iα/[(1+α)p2]
λ* = (1+α)/I

We solved for x* (bananas), y* (oranges), and λ*, but what the heck is λ? The term λ is the shadow price. The shadow price represents the following: assume that instead of I dollars of income, we had I+ε dollars. If we had the extra ε of income, the shadow price λ tells us by how much the objective function (utility function) would increase since we could buy a few more apples and bananas.

[TeePee4Prez]

Let us say you are a person trying to choose between buying bananas and oranges. What you are trying to do is maximize a utility function u(x,y) where x represents bananas and y represents oranges.  You can not buy an infinite amount of each however.  This is subject to a budget constraint.  Thus, we have the following maximization problem.

max u(x,y) s.t. p1x + p2y ≤ I
If we add functional form assumptions on the utility function we can form the following Lagrangian:

L= ln(x) + αln(y) - λ[p1x + p2y - I]
Our first order conditions are:

Lx:  1/x - λp1 =0
Ly:  α/y - λp2 =0
Lλ:  p1x + p2y - I =0
Our optimal level of bananas and oranges is:

x* = I/[(1+α)p1]
y* = Iα/[(1+α)p2]
λ* = (1+α)/I

We solved for x* (bananas), y* (oranges), and λ*, but what the heck is λ? The term λ is the shadow price. The shadow price represents the following: assume that instead of I dollars of income, we had I+ε dollars. If we had the extra ε of income, the shadow price λ tells us by how much the objective function (utility function) would increase since we could buy a few more apples and bananas.

I don't understand a damn thing you just posted.  Is this econometrics?  Never took it, but I heard it was brutal. 

[opebo]

Flyers, pinkrocket is just showing off...

[CARLHAYDEN]

Let us say you are a person trying to choose between buying bananas and oranges. What you are trying to do is maximize a utility function u(x,y) where x represents bananas and y represents oranges.  You can not buy an infinite amount of each however.  This is subject to a budget constraint.  Thus, we have the following maximization problem.

max u(x,y) s.t. p1x + p2y ≤ I
If we add functional form assumptions on the utility function we can form the following Lagrangian:

L= ln(x) + αln(y) - λ[p1x + p2y - I]
Our first order conditions are:

Lx:  1/x - λp1 =0
Ly:  α/y - λp2 =0
Lλ:  p1x + p2y - I =0
Our optimal level of bananas and oranges is:

x* = I/[(1+α)p1]
y* = Iα/[(1+α)p2]
λ* = (1+α)/I

We solved for x* (bananas), y* (oranges), and λ*, but what the heck is λ? The term λ is the shadow price. The shadow price represents the following: assume that instead of I dollars of income, we had I+ε dollars. If we had the extra ε of income, the shadow price λ tells us by how much the objective function (utility function) would increase since we could buy a few more apples and bananas.

I don't understand a damn thing you just posted.  Is this econometrics?  Never took it, but I heard it was brutal. 

This is actually a microecon topic. Though the Behavioralists have slightly disproven Expected Utility Theory.

One of the problems with your calculations is that the value of the items is not adjusted.

For example, where one banana per day may be nutritionally good, three may be bad.

[ag]

Let us say you are a person trying to choose between buying bananas and oranges. What you are trying to do is maximize a utility function u(x,y) where x represents bananas and y represents oranges.  You can not buy an infinite amount of each however.  This is subject to a budget constraint.  Thus, we have the following maximization problem.

max u(x,y) s.t. p1x + p2y ≤ I

Where p1 = price of bananas
Where p2 = price of oranges
Where x = amount of bananas
Where y = amount of oranges
Where I = Income

If we add functional form assumptions on the utility function we can form the following Lagrangian:

L= ln(x) + αln(y) - λ[p1x + p2y - I]
Our first order conditions are:

Lx:  1/x - λp1 =0
Ly:  α/y - λp2 =0
Lλ:  p1x + p2y - I =0
Our optimal level of bananas and oranges is:

x* = I/[(1+α)p1]
y* = Iα/[(1+α)p2]
λ* = (1+α)/I

We solved for x* (bananas), y* (oranges), and λ*, but what the heck is λ? The term λ is the shadow price. The shadow price represents the following: assume that instead of I dollars of income, we had I+ε dollars. If we had the extra ε of income, the shadow price λ tells us by how much the objective function (utility function) would increase since we could buy a few more apples and bananas.

Could you explain the purpose of posting a page of a basic undergrad textbook?

[ag]

Let us say you are a person trying to choose between buying bananas and oranges. What you are trying to do is maximize a utility function u(x,y) where x represents bananas and y represents oranges.  You can not buy an infinite amount of each however.  This is subject to a budget constraint.  Thus, we have the following maximization problem.

max u(x,y) s.t. p1x + p2y ≤ I
If we add functional form assumptions on the utility function we can form the following Lagrangian:

L= ln(x) + αln(y) - λ[p1x + p2y - I]
Our first order conditions are:

Lx:  1/x - λp1 =0
Ly:  α/y - λp2 =0
Lλ:  p1x + p2y - I =0
Our optimal level of bananas and oranges is:

x* = I/[(1+α)p1]
y* = Iα/[(1+α)p2]
λ* = (1+α)/I

We solved for x* (bananas), y* (oranges), and λ*, but what the heck is λ? The term λ is the shadow price. The shadow price represents the following: assume that instead of I dollars of income, we had I+ε dollars. If we had the extra ε of income, the shadow price λ tells us by how much the objective function (utility function) would increase since we could buy a few more apples and bananas.

I don't understand a damn thing you just posted.  Is this econometrics?  Never took it, but I heard it was brutal. 

This is actually a microecon topic. Though the Behavioralists have slightly disproven Expected Utility Theory.

Considering there is no uncertainty here, what the hell does "expected utility theory" has to do w/ it?

[ag]

Flyers, pinkrocket is just showing off...

Showing off?

Ok, let me show off: "2x2=4"

[ag]

Let us say you are a person trying to choose between buying bananas and oranges. What you are trying to do is maximize a utility function u(x,y) where x represents bananas and y represents oranges.  You can not buy an infinite amount of each however.  This is subject to a budget constraint.  Thus, we have the following maximization problem.

max u(x,y) s.t. p1x + p2y ≤ I
If we add functional form assumptions on the utility function we can form the following Lagrangian:

L= ln(x) + αln(y) - λ[p1x + p2y - I]
Our first order conditions are:

Lx:  1/x - λp1 =0
Ly:  α/y - λp2 =0
Lλ:  p1x + p2y - I =0
Our optimal level of bananas and oranges is:

x* = I/[(1+α)p1]
y* = Iα/[(1+α)p2]
λ* = (1+α)/I

We solved for x* (bananas), y* (oranges), and λ*, but what the heck is λ? The term λ is the shadow price. The shadow price represents the following: assume that instead of I dollars of income, we had I+ε dollars. If we had the extra ε of income, the shadow price λ tells us by how much the objective function (utility function) would increase since we could buy a few more apples and bananas.

I don't understand a damn thing you just posted.  Is this econometrics?  Never took it, but I heard it was brutal. 

No, this is not 'metrics. This is micro. Very basic micro: first semester of the Intermediate Micro, to be precise. In my experience, this is the one thing anyone who's taken the class can parrot on the final. Why did somebody think worth posting this "profound observation" beats me.