---
title: Reverse Engineering Nice Graph for Ganong et. al
---

## Analysis

We do a simple reverse engineering exercise.
Assets, income $a,y$ are the states and $a'$,
savings for tomorrow, is the choice variable.
The most straightforward way to visualize extrapolation
and its impact on MPCs out of stimulus payments
(modeled as wealth transfers for the moment)
is to model MPCs in the following functional form:
> [name=Brandon Kaplowitz]
> [time=Fri, Mar 15, 2024 12:03 PM]
> "modeled as wealth transfers for the moment"
> An obvious question is if we also want to do income transfers as T. Philippon suggested

$$MPC(a,y)=dc^{*}_t(a,y)/da=\alpha_t$$

Independent of the actual $a,y$ values,
we want MPCs to be constant in order to reflect 'full generalization'
and get a clear graph showing level shifts
in MPCs from a rational benchmark.
> [name=Brandon Kaplowitz]
> [time=Fri, Mar 15, 2024 12:03 PM]
> "constant"
>
> or do we rather want MPCs that change slope?
> I will say the level shifts might be visually clearer, though.
> You could do with quadratic or log-linear form for the consumption policy $c$ I think.
>
> The statement in Ganong et al. is more about the 'slope' being shallower than expected.
> A minimal statement of Ganong et al. would be if
> $MPC_{actual}(a) = A_{actual}a+b_{actual}$, then
$MPC_{rational}(a) = A_{rational}a+b_{rational}$
has $b_{actual}>b_{rational}$, and
$|A_{actual}|>|A_{rational}|$
for $A_{actual}<0, A_{rational}<0$.
This is 'shallower' slopes for the agent who learns.

This implies that $c$ should take the form:

$$c(a,y)=\alpha_t/R (Ra+ky)+c_0$$
that is, a constant share of total cash on hand.
In turn, this then implies that:

$$dQ/da'=\beta_1a'+\beta_2Ra+\beta_3y+\beta_0=0$$so that:

$$a' = -(\beta_2Ra+\beta_3y+\beta_0)/\beta_1$$or:

$$c=Ra+y-a' = (R+\beta_2/\beta_1)a+(1+\beta_3/\beta_1)y +\beta_0/\beta_1$$

Then clearly by matching coefficients we have
$$\begin{aligned}
c_0 &= \beta_0/\beta_1 \\
\alpha_t &= (R+\beta_2/\beta_1)\\
k\alpha_t/R&=(1+\beta_3/\beta_1), \text{ or }\\
k &=(1+\beta_3/\beta_1)R/\alpha_t
\end{aligned}$$

However, the actual values only matter a little
other than prescribing a relationship between the shape of c
and our MPCs under our parametric restriction.

Lastly, therefore, this implies, at a minimum,
the functional form for $Q$ is

$$\hat{Q}(a,y,a') = \beta_1 a'^2 + \beta_2 Ra a' + \beta_3 y a' + \beta_0 a' + Q_0$$
> [name=Brandon Kaplowitz] $Q_0$ is level shifts,
so it doesn't matter in policies
except to hit the rational Q graph at time 0.
$\beta_3$ could be dropped if you only care about MPCs
in unemployment changing after experiencing unemployment.

I already have a quadratic version coded up, so I may test this next.

Would you do with full optimization?
This is effectively fit the quadratic model over the entire space with least square.

Or would you do partial optimization--that is gradient descent like I have been doing?