# What can we know for sure among ambiguous impulse responses in RBC?

I'm trying to understand the section on responses to productivity disturbances in Understanding Real Business Cycles by Plossor.

It seems to me ambiguity of representative agent's response in consumption, saving and leisure rooted in two reasons:

1. Wealth effect is ambiguous. As far as I know, wealth effect means a temporary shock or a permanent change make the choice set determined by production techonology and budget bigger or smaller. The curvature of indifference curve is to some degree arbitrary, so the reallocation of consumption, saving, working, leisure could be arbitrary.
2. The relative strength of wealth effect against substitution effect is also arbitrary.

Here's my attempt: Suppose that, a higher value of (unexpected) temporary technological shock hits the economy at time $t$. $$\sigma_{high}F(K_t, N_t)=C_t+(K_{t+1}-(1-\delta)K_t)$$ Substition effect indicates more labour at $t$, but wealth effect makes this indetermined. $\{K_{t+1+i}\}_{i\in \mathbb N}$ is higher in each period for sure. But it's unclear for $\{C_{t+i}\}_{i\in \mathbb N}$ and $\{N_{t+i}\}_{i\in \mathbb N}$. It could be the case both wind up with higher value for all $i$, or only one of them wind up with higher value for all $i$.

It seems to me the qualitive result for a permanent increase of productivity is the same.

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Yes and no.

Temporary Shock

It is correct, that a temporary increase in TFP has unclear effects on leisure:

• The **income effect* will increase consumption and leisure
• The **substitution effect* will decrease leisure.

On total, effects onto leisure are unclear. But, unless you have very specific preferences, consumption should increase. What are the cases in which it doesn't? Let $l$ denote leisure, $1-N$:

1. Violation of local nonsatiation: Households do not prefer more consumption at their current $(C,l)$ locus, so an income effect will not make them consume more (shut down income effect)
2. Strong and negative elasticity of substitution between $(C, N)$: People that work more, prefer more consumption. Hence, if substitution effect increases $N$, it also decreases $C$ (putting in $C$ into the substitution effect)

1. is generally not accepted, both empirically and theoretically (remember, we're here not talking about individual preferences, but under aggregation). 2. is more disputed, but elasticities of substitution found typically range between $0$ and small positive amounts, not strongly negative as required here.

So, to conclude, temporarily, $C$ responds positively to a positive TFP shock.

Permanent Shock

Yes, a temporary shock could end up however, given how we calibrate it. But here's the gist:

• We calibrate models to data.

One standard calibration goal are the Kaldor facts, among which we will find:

• Despite an strong rise in productivity during the 20th century, working hours per capita have remained constant.

This means that in calibrating an RBC model, we will chose specifications that give us exactly this result. A simple way of doing so is having an additive separate utility function of King-Rebelo-Plosser type (Note that Prescott in his original paper claimed that only log-log preferences would reproduce this independence, which is incorrect).

Temporary Shock revisited

To conclude, we calibrate the model such that an income and substitution effect on leisure cancel out exactly. The income effect of a permanent shock is bigger than the income effect of a temporary shock. Hence, if it holds that permanent increases in TFP do not affect working hours, it must be the case that temporary increases in TFP do positively affect working hours, since the substitution effect is the same, but the income effect is smaller.

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Awesome answer. Thank you! –  Epicurus Feb 14 at 21:51