## Friday, September 11, 2015

### Stuff measured by GDP and the emergent representative agent

I am turning one of my comment responses into a post because I think it has broader interest and relates to this post on the emergent representative agent. Commenter Jamie discusses what does and doesn't contribute to GDP -- a discussion that was briefly the topic of the blogs earlier this summer.

Let's count Jamie's examples that contribute to GDP as "consumption" C and those that don't as "other". If we have two time periods consumption goods 1 and 2 the consumption by people is going to be constrained by

C₁ + C₂ ≤ GDP

The actually realized consumption in an economy is likely to be less -- if C₁ = σ₁ and C₂ = σ₂, then we have:

σ₁ + σ₂ < GDP

or

σ₁ + σ₂ + ɛ = GDP

where ɛ is that "other stuff" that is not a part of GDP (ɛ is the distance from the budget constraint line). I drew a picture of this scenario here:

In the two period good case, the ensemble average would give you a point in the middle of the triangle with ɛ > 0 and σ₁ + σ₂ < GDP.

However! (And this is pretty neat -- it depends on the mathematical properties of higher dimensional spaces.)

If you have a lot of consumption periods goods C₁, C₂, C₃, ... Cn subject to the budget constraint Σ Ci ≤ GDP, then the ensemble average (expected value) is

ɛ' + Σ σi = GDP

with ɛ' ≈ 0 simply because most of the points in the higher dimensional space are near the budget constraint (now a hyperplane). This scenario is illustrated here:

The ensemble average (the "representative agent" at the blue point) spends all of their consumption on things that count towards GDP, but individual agents (e.g. σi) do not necessarily do so.

Update 9/13/2015:

I changed the problem to be a single time period and instead had i index different consumption goods. The intertemporal version doesn't really illustrate the point I was trying to make.

1. Jason,

Why just summing up sigmas without any discount factor? Are agents to indifferent as to whether they consume something today or in the future?

Also, if I think of GDP as a measure of the value of all transactions in a economy, then individual budgets, if not allocated to personal to personal consumption/investment, then are either allocated into personal savings (which finance other people's consumption/investment) or left to "rot" in money value. If the ensemble average is closer to the average budget line through financial transfers, that I can understand. But not if money is left unused, and yet that would be the case if agents were in interior points of their budget set. So what is going on? Perhaps I'm confusing "average" with "total", and there could be unused resources in the whole economy while any random agent, conceived as an average agent, would seem to be on a point on his budget line. If so, then how this goes for policy? Perhaps taxation, at "moderate" levels, would counter-intuitively raise GDP by transferring this money to agents guaranteed to exhaust their budget, e. g. government?

1. This was a simplified illustration -- you can construct an intertemporal version with investment ('interest rate') and a discount factor. Those serve to (explicitly) break the symmetry of the underlying system. This is the limit of R → 0 and β → 1.

I also think I should have written this as a single time problem with i indexing the various goods and services and that it was a mistake to talk about time periods. I'm going to edit that bit.

The main point was supposed to be that even though agents aren't necessarily required to maximize their consumption on stuff that counts toward GDP, if there are enough dimensions of consumption (different goods, different time periods) the average total consumption does meet the macroeconomic constraint (here GDP).