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73 for 73

07 Mar 2021

Here at Cambridge there is a campaign called the “73-73 Challenge” - it is an initiative aimed at raising awareness for the 73% of students who have struggled with mental health during these pandemic times. As part of that, it encourages you to spend 73 minutes doing something fun - seeing as it happens to be 7/3, it seemed only apt to spend that time on another 73. Specifically, the 1973 paper “Some International Evidence on Output-Inflation Tradeoffs” by Bob Lucas.

The year before, Lucas’s “Expectations and the Neutrality of Money” had paved the way by introducing what is arguably the first modern macroeconomic model, integrating the use of rational expectations as a modelling tool with a mathematically elegant general equilibrium model of the natural rate hypothesis, as well as a role for monetary policy. But although Lucas 1972 might have set the stage, it is his 1973 paper which spawned countless other pieces using the parsimonious supply function it developed. Furthermore, it was a sign of the structural macroeconometrics to come, tightly tying theory together with empirics by testing the model against cross-country data.

The model Lucas sets up has an aggregate demand curve and an aggregate supply curve. The AD curve is the standard downwards-sloping relationship in $(P,Y)$ space as implied by IS-LM. For convenience, it is assumed that the AD curve is unit elastic in the form $x_t=y_t + P_t$. This means the split of nominal output into real output and prices is explained entirely by the AS curve. (All of the variables here are log variables.)

The AS curve is designed by Lucas as follows. Consider rational suppliers each located on a separate competitive island market - the result of this decentralisation is that each supplier can only observe prices on their own island. However, they cannot observe the aggregate price level across all islands.

The supplier will be willing to supply an amount composed of a secular component which affects all suppliers and a cyclical component that is idiosyncratic to their particular market. For markets indexed by $z$, we have the following components.

\[y_t(z) = y_{nt} + y_{ct}(z)\]

The secular component is given by $y_{nt}=\alpha+\beta t$. We can think of this as representing economy-wide changes such as population growth etc. The cyclical component depends on expected relative prices i.e. the known local price minus the expected aggregate price level given the available information, as well as the lagged values of the previous cyclical components. Unsurprisingly, suppliers will be willing to supply more if the relative price is higher, since that represents more people demanding their goods.

\[y_{ct}(z)=\gamma[P\_t(z)-E(P\_t|I\_t(z))]+\lambda y_{c,t-1}(z)\]

The supplier has two sources of information $I_t(z)$. Firstly, they know the past values of demand shifts, the current value of the secular component and the past values of the cyclical component. This allows them to produce a distribution of the price level $P_t$ - we assume this is given by $N(\bar{P_t},\sigma^2)$. Secondly, they know that the percentage deviation of local prices from the price level to be $z$, which is independent of $P_t$ and distributed $N(0,\tau^2)$. As such, $P_t(z)=P_t+z$.

Suppliers can therefore produce a conditional expectation of $P_t$. By OLS, we can see that $\theta=\frac{\tau^2}{\sigma^2+\tau^2}$. We have the expected value of $ P_t $ as $E(P_t | I_t(z) )$ given below and the variance as $\theta \sigma^2$.

\[E(P\_t|I\_t(z)) = E(P\_t|P\_t(z),\bar{P_t}) = (1-\theta)P\_t(z) + \theta \bar{P_t}\]

This can be substituted back into the expression of quantity supplied for a particular market as $y_t(z)=y_{nt}+\theta \gamma [P_t(z)-\bar{P_t}]+\lambda y_{c,t-1}(z)$. When aggregated across markets, we get the AS curve.

\[y_t = y_{nt} + \theta \gamma [P_t-\bar{P_t}] + \lambda [y_{t-1}-y_{n,t-1}]\]

Recognise that we have an AS curve which is upwards sloping when there are relative price variations but is entirely vertical if all price changes reflect general price level changes. In this way, there is a short-run but not long-run trade-off between output and prices.

Lucas finds that the expectations of prices by suppliers are consistent with the model’s economy i.e. suppliers form rational and model-consistent expectations. He does so by considering $\Delta x_t \sim N(\delta,\sigma_x^2)$. We can write the price level as depending on past demand shifts, past real output and current secular output.

\[P_t = \pi_0 + \pi_1 x_t + pi_2 x_{t-1} + ... + \eta_1 y_{t-1} + \eta_2 y_{t-2} + ... + \xi_0 y_{nt}\]

As such, $\bar{P_t}$ will be the expectation of $P_t$ without knowing the current level of $x_t$ i.e. $\bar{P_t} = \bar{P_0} + \pi_1(x_{t-1}+\delta) + pi_2 x_{t-1} + … + \eta_1 y_{t-1} + \eta_2 y_{t-2} + … + \xi_0 y_{nt}$. Lucas solves this to produce the equilibrium values below, where $\pi=\frac{\theta \gamma}{1+\theta \gamma}$.

\[y_{ct} = -\pi \delta + \pi \Delta x_t + \lambda y_{c,t-1}\] \[\Delta P_t = -\beta + (1-\pi)\Delta x_t + \pi \Delta x_{t-1} - \lambda \Delta y_{c,t-1}\]

These tell us that a change in the rate of nominal output expansion has effects on real output as well as on prices, due to the lagged signal extraction of demand shocks. These also produce a natural rate of output, since the average rate of demand expansion $\delta$ should be cancelled out exactly by the current rate $\Delta x_t$ i.e. changes in the average rate of nominal output growth have no real effect. However, unexpected demand shifts do due to the problem of signal extraction, but only in a transitory manner.

We also know from the definition of $\theta$ that we can write $\pi$ as follows. This is helpful, because it is impossible to test whether or not the average deviation from trend is invariant to demand-side choices given the trend is fitted - however, the following expression means that we can test this natural rate hypothesis. What Lucas finds mostly corroborates this hypothesis, though the quality of the econometrics is as Lucas himself concedes later on, rather sloppy.

\[\pi = \frac{\tau^2 \gamma}{(1-\pi)^2 \sigma_x^2 +\tau^2(1+\gamma)}\]

Clearly this is imperfect as a model. In particular, the output fluctuations are modelled as serially uncorrelated, which is both unrealistic and fails to produce the necessary booms and busts. But insofar as modern business cycle macroeconomics is in many ways about dynamic stochastic general equilibrium models, the importance of expectations and the use of empirical data, all of this appears in this seminal paper. More than that, this is just very pretty - it gives us a way of explaining why the Phillips Curve was not exploitable by invoking Rational Expectations as a modelling tool!

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