Crescimento inesperado, produção e nível de preços por dinheiro

3.2 Key reading

Lucas (1973) “Some International Evidence on Output-Inflation Trade-offs”, American Economic Review,

63, 326-334

3.3 Related reading

Lucas (1972) “Expectations and the neutrality of money”, Journal of Economic Theory, 4, 103-24

Lucas (1975) “An Equilibrium Model of the Business Cycle”, Journal of Political Economy, 83, 1113-

1144

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Barro (1978) “Unanticipated money growth, output and the price level”, Journal of Political Economy,

86, 549-80

3.4 The model

There exist N islands. On each island is a producer who charges price pt(z), where z denotes a particular

island. We shall denote the aggregate price as pt, which is simply the average of p(z) across all Z islands.

The main focus in the paper is on the supply side where the following “Lucas supply” function is

assumed

yt(z) = γ(pt(z) − pt) (3.1)

so that a producer increases his output when his own price is greater than the aggregate price in the

economy. Thus output is given by a standard upward sloping supply schedule. y(z) should be interpreted

as deviations in output from trend, so that when y(z) = 0 output is equal to its trend value. The innovation

in this model is to assume imperfect information so that a producer on an island knows at time t his own

price p(z) but does not know the economy-wide price level p. Instead, they have to form a guess of p based

on the information at their disposal. Let It(z) denote the information available at time t to the producer

in market z. Then E(pt |It(z)) denotes the expectation of the aggregate price pt given the information

available to the producer in island z. Because of imperfect competition, we have the incomplete information

supply curve

yt(z) = γ(pt(z) − E(pt |It(z))) (3.2)

However, the crucial question is how do agents form E(pt |It(z))? Without an expression for this guess

of aggregate prices, we can do very little with our model.

Lucas’ trick is to use Rational Expectations. There are two different but related interpretations of

Rational Expectations. The first is a statistical one and implies that, when agents have to make a forecast,

errors are unpredictable. In other words, agents use all the information at their disposal. The second interpretation

is more economic and is due to John Muth. According to this definition, Rational Expectations

is when agents use the economic model to form their price predictions.

Assuming rational expectations, it must be the case that pt = E(pt |It−1) + ²t where ² is a forecast

error which is on average zero and which cannot be predicted from It−1. We shall denote the variance of

² (E(²2)) as σ2.

Lucas also assumes that the price in each island, p(z), differs only randomly from the aggregate price

level, p. In other words, pt(z) = pt + zt where z is on average zero and has a variance equal to τ 2.

If the producer has perfect information about the aggregate price level then y(z) would respond only

to z, the relative price shock. However, due to imperfect information agents only observe the gap between

p(z) and their expectation of the aggregate price level, that is they observe the composite error z + ².

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The producers problem is to decide how much of this composite error is due to mistakes in forecasting the

aggregate price level (²) and how much is the relative price shock (z) and to only alter output in response

to the latter.

How does the producer decide how much of the composite shock is due to ² and how much is due to z?

Technically this problem is called “signal extraction”. The answer is to look at historical data. Our model

only assumes that agents do not know the current value of ², but they do observe historical data on z and

². Running a regression of z on (z +²) will provide a guess of what proportion of (z +²) is due to z. Using

standard OLS formulae we have

θ =

τ2

σ2 + τ 2, where zt = θ(zt + ²t) + ut (3.3)

Therefore an agents best guess of z is θ(z +²). By definition p(z) = p+z so that an agent’s best guess

of the aggregate price level given that they observe pt(z) is

E(pt |It−1(z), pt(z)) = pt(z) − E(zt |It−1(z), pt(z))

= pt(z) − θ(pt(z) − E(pt |It−1 )) (3.4)

= (1− θ)pt(z) + θE(pt |It−1 )

Equation (3.4) gives an expression for E(pt |It(z)) which we can insert into (3.2) to get a supply curve.

Aggregating across all islands gives an aggregate supply curve.

yt = γθ(pt − E(pt |It−1 )) (3.5)

so that output only responds to unexpected aggregate price shocks. Note that by adding and subtracting

pt−1 to the right-hand of (3.5) and re-arranging we have

(pt − pt−1) =

1