Unique utility functions for extrapolators and fundamentalists

Following on from last week’s post, I wanted to begin the exploration of different utility functions for De Grauwe’s extrapolators and fundamentalists. The research from Social Psychology gives us some solid evidence that the discount rates for these two groups are likely to be different. The question now is, where do we start?

Going back to the basics of micro-economics is a reasonable place to begin. The following diagram provides a view of the consumption choices in a two period model.[1] It shows the range of possible consumptions options in periods 1 and 2, with a budget constraint that is influenced by the interest rate that the consumer faces. You can see that if a consumer decided to spend all of their income in period 1, and borrowed as much as possible, they could consume up to the point where the black line meets the x axis. Conversely, if they saved all of their period 1 income, due to the increase in available income from the interest earned, they could consume up the point where the black line meets the y axis.

consumption diagram

In reference to the current discussion, the two grey indifference curves are of interest. The grey curve in the upper left corner represents a saver, as they are consuming less than their income in period 1 and more than their income in period 2. This relates broadly to the concept of the fundamentalist. On the other hand, the grey indifference curve in the bottom right corner is roughly in line with an extrapolator. This individual is spending more than their income in period 1 and less than their income in period 2. Perhaps this is due to their optimism about the future.

So assuming that we are comfortable with these two groups being on different indifference curves, or having different utility functions, we now need to describe those utility functions.

Some equations

Again, we can use the article by Doppelhofer to describe the basics of the utility function more formally. The first equation shown below outlines the concept that we assume the preferences of the consumer can be added separately over the two periods. In other words, the goal for the consumer is to maximise the addition of these two period utilities.

utility 1

This first step is extended by combining it with the intertemporal budget constraint. This is the concept that any consumer has a limit to their spending in each period, which is determined by their income in each period and the interest rate. Again, it is shown as the thick black line in the diagram above. The formula for the budget constraint has been substituted into the right hand side in the place of c2.

utility 2

Following this, Doppelhofer then articulates the Euler equation, which is defined as the necessary first order condition for this problem. It says that a marginal increase in period 1 consumption must have a benefit equal to the cost of the decrease in period 2 of the same present value amount. In the diagram, this would relate to a consumer choosing more period 1 consumption, but the optimising aspect of this move implies that they choose the indifference curve that is still tangent to the budget constraint.

utility 3

Relating this to the heterogeneous households of extrapolators and fundamentalists, from last week’s post we saw that these two groups could conceivably have different discount factors, meaning different β’s in this equation.

utility 4

Where i = f, e, or fundamentalist, extrapolator. Moreover, because of the way these two groups view the future, βe> βf.

So now we have an intertemporal Euler equation for each household type. With the Woodford model that we are trying to emulate, the two utility functions also need to respond to different interest rates – unlike the diagram above that only has one interest rate. This is the next step.


[1] This diagram is sourced from Intertemporal Macroeconomics, May 2009, by Gernot Doppelhofer. Forthcoming in Cambridge Essays in Applied Economics.

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