Integrability Problem in Economics

To a system of partial differential equations, there may or may not be a solution. In economics this can matter when looking at demand functions, or rather, systems of demand functions. For instance, suppose you have a demand system defined for a duopoly:

(1) Q1 = A1 - b1*P1 + c1*P2
(2) Q2 = A2 - b2*P2 + c2*P1

Equation (1) gives the demand for good 1, given the prices for goods 1 and 2, and equation (2) gives the demand for good 2. If the coefficients b1, b2, c1, c2 are all positive (as I intend them to be), then the two goods are substitutes, likely imperfect substitutes. Note that the quantity of good 1 purchased will go down when its price goes up. That's straightforward. The quantity will also go down when the price for good 2 goes down. That's because some people will switch from buying good 1 to buying good 2.

Now, we might want to say something about consumer surplus as it relates to this market. However, in order to do so, we need a measure of welfare. We need a utility function. Even better, we want a money metric utility function, because that will allow for better comparisons than just any old utility function.

The utility function we're looking for is a bit artificial (like they all aren't), since it applies to an amalgamation of all consumers, instead of one individual. It's called a representative consumer utility function, since there's hypothesized some consumer who represents the aggregate tastes of all consumers.

Give this representative consumer a utility function of the form:

U(Q1,Q2) = B(Q1,Q2) - P1*Q1 - P2*Q2

i.e. the utility is some Benefit function from consuming the goods, minus the amount spent on both of them.

Maximize utility by finding where the derivatives of the utility function are zero:

dU/dQ1 = dB/dQ1 - P1 = 0 -> dB/dQ1 = P1
dU/dQ2 = dB/dQ2 - P2 = 0 -> dB/dQ2 = P2

At this point, we want to be using an inverse demand system, instead of the demand system above. The equivalent inverse demand system is:

(1) P1 = B1 - e1*Q1 - f1*Q2
(1) P2 = B2 - e2*Q2 - f2*Q1

Redefining variables:

B1 = (a2*A1 + b1*A2)/(a1*a2 - b1*b2)
e1 = a2 / (a1*a2 - b1*b2)
f1 = b1 / (a1*a2 - b1*b2)

B2 = (a1*A2 + b2*A1)/(a1*a2 - b1*b2)
e2 = a1 / (a1*a2 - b1*b2)
f2 = b2 / (a1*a2 - b1*b2)

Since dB/dQ1 = P1, B = Integral(B1 - e1*Q1 - f1*Q2) with respect to Q1:

B = B1*Q1 - (e1/2)*Q1² - f1*Q1*Q2 + k1

Likewise, since dB/dQ2 = P2:

B = B2*Q2 - (e2/2)*Q2² - f2*Q1*Q2 + k2

So, how can these two (B)enefit functions be consistent? k1 is a constant with respect to Q1, so there's no problem letting k1 = B2*Q2 - (e2/2)*Q2², and likewise k2 = B1*Q1 - (e1/2)*Q1².

But, it's not so easy to deal with f1*Q1*Q2, f2*Q1*Q2. In order for there to be a defined benefit function (and thereby a defined utility function), this requires that f = f1 = f2. Then we can say:

B(Q1,Q2) = B1*Q1 - (e1/2)*Q1² - f*Q1*Q2 + B2*Q2 - (e2/2)*Q2²

U(Q1,Q2) = B1*Q1 - (e1/2)*Q1² - f*Q1*Q2 + B2*Q2 - (e2/2)*Q2² - P1*Q1 - P2*Q2

One good thing about these functions is they are in terms of money, since the money scale is included in the (P1*Q1 - P2*Q2) portion. That means this utility function is cardinal rather than simply ordinal. But, since we're now restricted to f = f1 = f2, we can see that in the original demand system, this implies b = b1 = b2. So, a representative consumer utility function only exists when cross-price effects are symmetric.

This does not mean that cross-price effects have to be symmetric to have a well-defined demand system. Instead, it is a problem of trying to represent aggregate consumer well-being with a utility function. This is one problem in social choice.