# Was the stimulus too slow

Paul Krugman seems to think so.  As I posted five months ago, if we use the analogy of persistent activity in the brain for the economy then the total amount of stimulus would only be one variable that is important in knocking the economy out of a recession and into a new equilibrium point.  Another variable is how fast the money is spent.  Thus far, only 30% of the stimulus has been disbursed and Krugman thinks that the impact of it has already been maximized.  The rest of the money will then be dissipated without a stimulatory effect.

The question of stimulus size and speed is similar to the debate over whether or not the threshold for firing of a neuron is a voltage or a current.  Consider the simple example of an integrate-and-fire neuron

$C \frac{dV}{dt} =I-gV$

where V is the membrane voltage, I is a time dependent input current, C is the capacitance, and g is the leak conductance. Traditionally, we consider a voltage threshold $v_T$ for the neuron to fire an action potential.  Thus for a neuron to fire we would require that the input current satisfy

$\frac{1}{C} \int_0^t I(s) e^{-g(t- s)/ C} ds > v_T$

So if the time constant (C/g) was very long then the integral over the input current (i.e. total charge)  would be the relevant quantity for firing.  Now if instead the neuron had a current threshold, then all that matters for it to fire is that the maximum of the current administered exceeds the threshold at some point in time. Suppose we have a fixed amount of charge $Q=\int I(t) dt$ that we have to administer.  Then for a voltage threshold, if we deliver that charge in a time less than the time constant then we would need at least $Q \simeq CV_T$.  However, even for a voltage threshold, there is still a trade off between total amount of charge and amount of time to administer it.  If we consider a current threshold $I_T$, then all we would have to do is to deliver it  such that $dQ/dt>I_T$ at some point.

In the analogy to the economy, the charge is the total amount of stimulus and the voltage is some measure of economic activity.  I don’t know enough about economics to know if we have a voltage or a current threshold.  If it is a voltage threshold then we need to hope that the time constant is longer than two years and the “capacitance” is low.  If it is a current threshold, then speed is more important.  In any case, we’ll know soon enough if they guessed correctly.

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## 3 thoughts on “Was the stimulus too slow”

1. Another possibility is that the economic system is complex and decentralized and cannot be simplistically analogized to idealized one-dimensional-continuum physical systems from which one can easily extract mathematical thresholds and infer that ‘the economy’ can somehow be tuned, thermostat-style, by an Englightened Central Planner, to produce any given desired result – whether or not that Central Planner listens to the advice of Paul Krugman.

If ‘stimulus fails’ it may not merely be because we chose the wrong model and ‘threshold’. It may be because the thinking and oversimplification behind the arguments for ‘stimulus’ is dangerous bunk to begin with.

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2. Oh, that is true, I guess, but only trivially so. “Waste a lot of taxpayer money and line the pockets of financiers” is one possible “something”, for example.

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