Appendix: Complements and Precisions

8 Appendix: Complements and Precisions

8.1 Details on the behavioral consumer

Here I state the assumptions regarding the behavioral consumer. I use the notions laid out in Gabaix (2016). The agent’s action is a t = (c t ,N t ), consumption and labor supply. The macroeconomy (ex- cluding the agent’s personal wealth) is parametrized by a state vector X t that contains productivity shocks, announcements etc. For instance (linearizing), (br t ,b ω t ) = bX t for some equilibrium vector b. The state vector is z t = (k t ,X t ), where k t is the agent’s wealth. The perceived laws of motions are:

where y t =ω t N t +Π t +T t . Here m= (m r ,m y ,¯ m) is the vector of attention parameter. When all the components are 1 (so m = ι := (1, ..., 1)), the agent is rational.

This agent pays full attention to his wealth, which ensures his dynamic budget constraint. 66 He also has a rational understanding and rational reaction to the steady state: he understand the state state value r of the interest rate, for instance.

But he’s a bit myopic to the deviation from it, e.g. he sees only a part m r br t of the deviation of the interest rate from trend. The true law of motion of the macro state vector is X

=F t+1 X (X t , ι) = AX t (linearizing), but the agent has a subjective perception of it that it different: it is X

X t+1 =F (¯ mX t , ι) = ¯ mAX t . This captures that the future is hard to forecast so at each round in the future, the agent “cognitively discounts” it – sees those deviations from the steady state less and less as they are more remove.

The agent chooses his action, at each period, according to:

a t = arg smax v (a, z t , m)

a;m

where the value function maximized is:

v (a, z z

t , m) := u (a) + βV (F (a, z t , m) , m)

and V r is the rational value function in the economy parametrized by vector m, i.e. the value

66 In his consumption function (Proposition 2.2, the term c d t = ¯ rk t

R +¯ y reflects that attention to his wealth.

function that the agent would attain if the world was indeed the world he perceives. As discussed in Gabaix (2016), other proxy value functions would only affect second order terms (in O kXk 2 )

in the agent’s decision. In this draft of the paper I take the attention policy m as fixed – a later draft will endogenize it. So basically the agent maximizes at each round under a slightly imperfect model of the world.

8.2 The “natural interest rate” in a behavioral economy

The natural interest rate is defined here as the interest rate that would prevail “if pricing frictions were removed”, but keeping cognitive frictions (and before any deficits). Let us examine this in

detail. Take the IS curve (23), coming back to the more basic notion of ˆ c t := ln c t − ln ¯ c:

ˆ c t =ME t [ˆ c t+1 ]+b d d t − σ (r t −¯ r) (IS curve)

where r t =i t −E t π t+1 is the real rate. Consider also the case with productivity shocks, so that C t =e ζ t N t , so that the optimum frictionless consumption (see the derivation of (69)) is

So, if we removed all pricing frictions, we’d have ˆ c t =ˆ c n t , and if we were in an environment with no deficit, we’d have:

t =ME t c ˆ t+1 − σ (r t −¯ r) ,

ME t [ ˆ c t+1 n ] −ˆ c t n

which gives us the value of the natural rate, r t =¯ r+

. Given we define the output gap as x := ˆ c −ˆ c t n t t , we have:

t =ME t [x t+1 ]+b d d t − σ (r t −r t )

which is the formulation in the paper. Note that we could have defined the “natural” rate as the rate that would prevail in an economy without pricing frictions, and given the actual deficits, i.e. defined it as the solution ˜ r n t of:

t =ME t ˆ c t+1 +b d d t − σ (˜ r t −¯ r) t =ME t ˆ c t+1 +b d d t − σ (˜ r t −¯ r)

t =¯ r+

. And then the IS curve would become:

t =ME t [x t+1 ] − σ (r t −r t )

This would be mathematically equivalent, but the language would become more complicated. Then

a policy change (via deficits) would change the natural rates. For instance, a temporary rise of the deficit would decrease the natural rate (as it makes people want to spend more). With that definition, the natural rate is not very “natural”.