Expenditure Minimization problem and Expenditure function

The expenditure minimization function is the minimum money that is required to achieve a given level of utility and prices.

This is very similar to the utility maximization question that you would be familiar dealing with in an intermediate microeconomics class.

Normally, you are given a set of prices and some income and you are asked to find the maximum utility that a consumer could derive and how many units of X and Y would the consumer consume. It might look like follows:

U = XY subject to M = Px * X + Py * Y

and the question states Income (M) is $200, Price of X (Px) is $2, and Price of Y (Py) is $1 so that you get the following:

U = XY subject to 200 = 2X + 1Y

you can use the substitution method to solve this:

U = 200X - 2X^2

du/dx = 200 - 4X = 0

X =50, Y = 100, U = 5,000

(If you have trouble with this, you will probably need to practice some utility maximization problems first)

In terms of an expenditure minimization problem, you are given the prices and how much utility the consumer derives, and you are asked to figure out how much income the consumer would require to achieve this level of income.

Suppose we had the above example where:

U = XY subject to M = Px * X + Py * Y

Instead of being told how much income we have, we instead would have been told the consumer wants to achieve a level of 5,000 utility, the price of X (Px) is $2 and price of Y (Py) is $1. We then need to find the minimum income required to achieve that utility. We would write it out as follows:

M = 2X + 1Y subject to 5,000 = XY

The best approach to solve this problem is to use a Lagrangian

L = 2X + Y + h[5,000 - XY]

(1) dL/dX = 2 - hY = 0

(2) dL/dY = 1 - hx = 0

(3) dL/dh = 5000 - XY = 0

From (1) and (2)

1/X = h and 2/Y = h


1/x = 2/y which implies 2X = Y

Substitute the above into (3)

5000 - 2X^2 = 0

5000 = 2X^2

X^2 = 2500

X = sqrt(2500)

X = 50

since Y = 2X

Y = 100

If we substitute this back into the original equation we get:

M = 2*50 + 100

M = 200

As we can see, the minimum income required is $200 - which is the same from our utility maximization question! This is not the expenditure function.. yet!

What we did above was solve the expenditure minimization problem by finding the minimum income required to achieve a given level of utility. The expenditure function is more generalized. In essence, it's a function which says "given certain prices, how much income do we need to achieve a fixed level of utility". Essentially, what we need to do is solve the equation without substituting in values for Px, Py, or U. (instead of letting Px = $1, Py = $2, and U = 5000, we leave them as Px, Py and U)

What we do is take the function:

M = Px * X + Py * Y subject to U = XY

and we find the lagrangian.

L = Px * X + Py * Y + h[U - XY]

(1) dL/dX = Px - hY = 0

(2) dl/dy = Py - hX = 0

(3) dl/dh = U - XY = 0

From (1) and (2) we get:

Px / Y = h, Py/X = h and thus

Px/Y = Py/X

From this we get:

Px * X = Py * Y


X = (Py/Px) * Y

This can be substituted into (3)

U - (Py/Px) *Y^2 = 0

U = (Py/Px) * Y^2

we now need to solve for Y

Y^2 = U*(Px/Py) // take notice that Py/Px becomes Px/Py because we have moved it to the other side of the equation

Y = sqrt[U * (Px/Py)]

since X = (Py/Px)*Y we get the following value for X.

X = (Py/Px) * sqrt[U * (Px/Py)]

X = sqrt[U * Py/Px]

Substitute the values of X and Y into this into M = Px * X +  Py * Y

M = Px* sqrt[U * Py/Px] + Py * sqrt[U * Px/Py]

M = sqrt(U* Py * Px) + sqrt(U * Py * Px]

M = 2 * sqrt(U * Py * Px)

We could write M = E(U,Px,Py) which just says that the expenditure (income required) to gain the level of utility is dependent on the level of utility you are trying to attain and the price of goods.

Sorry, that was long and tedious. However, that is the expenditure function. 

We can test to ensure that the function is correct by substituting the prices and utility into the equation and seeing if we get an income of $200.

M = 2 * sqrt(5000 *2)

M = 2 * sqrt(10,000)

M = 2 * 100

M = 200

And there we have it. An expenditure minimization function for the utility function U = XY.  Now, given any level of utility and any set of prices, we can find the minimum income required to achieve that level of utility.

Why is this helpful?

The expenditure minimization function is used to work out things like compensating variation and equivalent variation which are used in public policy. For example, the concepts of compensating variation was used when determining the level of assistance to be provided to low income families when they introduced the carbon tax.

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