# Monthly Archives: June 2014

## The Index Number Problem and Inflation

Nick Rowe asks whether or not housing prices should be included in the inflation rate that the Bank of Canada targets. His discussion focuses on whether or not housing prices are sticky or whether they are flexible. His discussion is a standard story that follows from Woodford’s textbook on monetary theory and policy. The idea is that the price index that the central bank uses to target inflation should consist only of sticky prices. However, I find this viewpoint (while commonly accepted) to be counter to the conclusion of the index number problem discussed by Samuelson, Niehans and many others. In addition, I think that there is something to learn from the latter.

Consider a standard microeconomic story. An individual receives income, $I$, and gets utility from consuming goods $x$ and $y$. Let $p_x$ denote the price of $x$ and $p_y$ denote the price of $y$. Further, suppose the utility function is given as $u(x, y)$ and has the usual properties. Thus, the consumer maximization problem is $\max\limits_{x, y} u(x, y)$ $s.t. p_x x + p_y y \leq I$

The optimal allocation is therefore given as ${{u_x}\over{u_y}} = {{p_y}\over{p_x}}$

where $u_x$ is the marginal utility of $x$ and correspondingly for $y$.

Now you are probably wondering, what does this have to do with inflation? Well the answer is quite simple. In the problem above, there was no discussion of money. This was a real economy. Suppose instead that we are dealing with a monetary economy. In this case, income is money income (i.e. the number of dollars that you earn). In order to solve the allocation problem, we now need to deflate money income by some price index such that income is expressed in real terms. If a change in the money supply has an equiproportional impact on all prices, the choice of the price index is entirely arbitrary. The price of any individual good will suffice as a price index. In other words, we could re-write the budget constraint as $x + {{p_y}\over{p_x}} y = {{I}\over{p_x}}$

Solving the consumer’s maximization problem yields the same equilibrium condition as that above. In addition, since changes in the money supply have an equiproportionate effect on all prices, the relative price of good $y$ to good $x$ remains unchanged and doesn’t have any effect on the allocation of goods. Additionally, so long as $p_x$ is held constant, then money income will not have any effect on the allocation either.

However, suppose that changes in the money supply do not have equiproportionate effects on prices. To use Nick’s example, suppose that the price of $x$ is sticky and the price of $y$ is flexible. In this case, a change in the money supply will also affect relative prices. In this case, one cannot simply solve the allocation problem by deflating money income by the price of one of the goods. In this case, changes in the money supply will distort the allocation of goods. In addition, this means that it is not sufficient to simply target the sticky price.

The solution to this problem is to choose a price index to deflate money income such that when that index is held constant, there is not any distortion in the allocation of goods. In other words, the objective to choose $P$ such that the budget constraint can be re-written as ${{p_x}\over{P}} x + {{p_y}\over{P}} y = {{I}\over{P}}$

Given the correct choice of $P$, it is straightforward to show that (1) the allocation of goods is determined by the relative prices of the goods, and (2) when $P$ is constant, money income is constant as one moves along an indifference curve.

So how do we construct $P$? Well, Samuelson gave us a class of examples where there was a specific price index that could solve the problem. And it turns out that the correct price index to use is dependent on the preferences of the representative consumer in the model. In particular, consider the following utility function: $u = \sqrt{xy}$

It is straightforward to show that the correct price index to use in this case is $P = \sqrt{p_x p_y}$

Here is a brief sketch of why this is true. In a real economy, when there is an increase in income, the individual moves to a higher indifference curve (i.e. utility increases). Thus, when an individual moves along an indifference curve, it must be true that income is constant. A different way of stating the problem above is that the objective is to choose a price index such that when that index is held constant, money income is constant when an individual moves along an indifference curve. We can now show that this is true for the utility function and price index above.

Consider the budget constraint: $I = p_x x + p_y y = p\bigg({{p_x}\over{p}} x + {{p_y}\over{p}} y\bigg)$

Suppose the price index is given as $P = \sqrt{p_x p_y}$, then this can be re-written as $I = p\bigg({{\sqrt{p_x}}\over{\sqrt{p_y}}} x + {{\sqrt{p_y}}\over{\sqrt{p_x}}} y\bigg)$

Given the preferences assumed above, the equilibrium condition for the consumer is ${{p_x}\over{p_y}} = {{y}\over{x}}$

Substituting this into the budget constraint yields $I = p(2\sqrt{xy}) = p(2U)$

Thus, when $p$ is constant, a movement along an indifference curve is associated with a constant amount of money income.

So what does all of this mean?

What this means is that if changes in the money supply result in changes in the relative price of goods, then the optimal policy is one in which there is no inflation. However, the choice of how to measure inflation is not arbitrary in this case. Rather, there is a precise index number that must be used to calculate inflation.

Nick’s point, and the accepted wisdom of many in the discipline, is that when changes in the money supply distort relative prices due to price stickiness, the best thing to do is to target the sticky prices and let the flexible prices adjust. However, the example above rejects this idea. If, say, $p_x$ was a sticky price and $p_y$ was a flexible price, targeting $p_x$ would be insufficient. Doing so would prevent money income from affecting utility, but it would not prevent an adjustment in the relative prices of $x$ and $y$ and would therefore distort the allocation.

What the index number problem suggests is that the choice of the proper price index does not depend on which price is sticky or the source of the relative price variability. Instead, the index number problem suggests that the proper price index is derived from the preferences of the consumer. Thus, when asked if housing prices should be included in the price index used to calculate inflation, the relevant question is not whether housing prices are sticky, but rather whether housing enters a representative consumer’s utility function.