### Riccardian Equivalence (somewhat technical)

**Matthew Martin**7/20/2012 04:16:00 PM

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Here is a very bas illustration of the mathematics:

Let C(t) be consumption in period t (measured in units of goods), G(t) be government spending in period t (measured in units of goods), P(t) be the price of a unit of goods, and T(t) be taxes in period t. Also let Y(t) be the aggregate number of goods produced in period t. We will use the greek letter Σ in front of variables or expressions to denote the sum over all periods from the initial period to infinity.

For the purposes of RE, the government's spending plans are a fixed sequence {G(0), G(1),..., G(t)}. Note that if we change government spending at all in any of the periods, then RE is broken and the economy shifts to a different equilibrium (this seems to be a source of confusion for some people). Now, the government has to finance that sequence of expenditures with taxes and borrowing, which means that it's budget constraint looks like this:

That just says that total spending summed over all future periods must equal total revenue over all future periods (technically less than or equal to, but no government would tax more than it has to).

Households take the sequences of Y(t), T(t), P(t), and G(t) as given, and choose a sequence of C(t) that maximizes utility over all future periods. So the household solves

Here is where RE comes from: since both the sequence of T(t) and G(t) are known to the household in advance, and we know from the government budget constraint that ΣP(t)G(t)= ΣT(t), we can simply rewrite the household problem above as

Something amazing has happened here--the sequence of taxes is no longer part of the household's problem! Therefore, we can conclude that how the government finances itself--through deficits now or taxes now--doesn't matter. All the equilibrium prices and equilibrium allocations will be identical regardless. Intuitively what happens in this simplistic model is households respond to a deficit financed tax cut by putting all of that money into savings.

Now back to the real world:

Figure 1. An argument why Riccardian Equivalence never holds. |

RE never actually holds. That's just an empirical observation (shh! no one tell Robert Barro). There are lots of ways to structure a theory in which RE makes no sense. The most obvious one is a model where households are not infinitely lived--if households don't live to infinity, then it is quite easy to break RE by cutting taxes on one generation of households and financing that debt by taxing a later generation. Considering that in the real world households are mortal, that's a massive oversight for people who believe RE to make.

But that is besides the point. The bigger problem is the unwarranted assumption we make about information: how would people know how much the government is going to tax and spend over all future years? A more realistic alternative, in my view, is to treat government taxation as a random variable, and endow the representative household with expectations about the distribution of the random variable, but no knowledge of future realizations of it. This would produce the results we see in the data: if there is a temporary tax cut, this would not significantly affect expected future taxation, but would affect the current budget constraint, producing a small but positive impact on current consumption; if there is a permanent tax cut, there would be bigger effects on consumption since expected future taxes would decrease.