1The Lucas Supply Curve
Miller GR5215
The Lucas supply curve
2
Lucas formalized Friedman’s insight that the
Phillips curve would not be stable
This model is not very popular currently, but
every macroeconomist should have basic
familiarity with it
This version is modified from the Lucas
original to make it match with the
differentiated product model (see Romer
6.9)
Model setup
3
There is a continuum of differentiated products,
indexed by i on [0,1], but this time we’ll make
each produced by one household
Yi = Li
Each has a linear production function:
Households maximize: Ui =Ci − 1γ Liγ , γ >1
And: Ui = PiP Yi − 1γ Yiγ , γ >1
Latter follows from households having income
only from selling their good
FOC
4
Ui = PiP Yi − 1γ Yiγ , γ >1
FOC: PiP =Yiγ −1
So: Yi = PiP⎛⎝⎜ ⎞⎠⎟
1/(γ −1)
In logs: yi = 1(γ −1) pi − p( )
Demand
5
Add aggregate demand via the quantity theory:
y =m− p
But there are shocks to individual demand as well
– call these zi, so:

Y = M
P
yi =m− p+ zi +η(pi − p)
Which uses the constant elasticity demand
equation we derived previously
In logs:
Simplify
6
With heterogeneous demand our indices would be
messy, so we’ll approximate:
y = yi
Where the overbar indicates an average (we’re in
logs, so this is a geometric average)
p= pi
Key assumption: household does not observe m
and z separately, but simply sees overall demand
for its good, in the form of the price. So: pi = p+(pi − p)
Call the second term ri
The signal-extraction problem
7
The household would like to base its production
decision solely on ri, but it doesn’t see it
To simplify, assume that the household calculates
the mathematical expectation of r and then acts as
if it knew that with certainty
yi = 1(γ −1) pi − p( ) yi = 1(γ −1)E[ri |pi ]
Assume m and the zi ’s are normally distributed,
m has mean of E[m] and variance Vm. zi has mean
zero and variance of Vz; m and zi are independent
The signal-extraction problem
(II)
8
A result from probability theory (the signal
extraction problem) tells us that:
E[ri |pi ]= VrVr +Vp (pi −E[pi ])
Calling the constant factor b and averaging over
households:
Sub into production decision equation:
yi = 1(γ −1) VrVr +Vp (pi −E[pi ]) y = b(p−E[p])
the “Lucas supply curve”
Equilibrium
9
AS:
p= 11+bm+ b1+bE[p]
y = 11+bm− b1+bE[p]
Take expectations of both sides of the price
equation: E[p]= 11+bE[m]+ b1+bE[p] ⇒ E[p]=E[m]
y = b(p−E[p]) AD: y =m− p
Use AD to sub out y and then p in AS, to get:
Equilibrium (II)
10
Use E[p]=E[m] to rewrite the price equation:
p= E[m]+ 11+b(m−E[m])
And for output:
y = b1+b(m−E[m])
Last step is to verify this works with the
fundamental variances for m and z. See Romer.
This result implies that a central bank can only
stabilize if it knows more than households: only
monetary “surprises” matter
Empirical work
11
Lucas paper said output should be less
affected by aggregate demand shocks in
countries where monetary variability is high
He used a small sample of countries to show
this was empirically true
This follows directly from the signal
extraction problem
Output-inflation tradeoff and
variability of aggregate demand
12
Output inflation tradeoff and
average inflation
13
Empirical work
14
Ball, Mankiw, Romer (1988) showed that
average inflation was a better indicator that
the variability of inflation
They interpret this as an evidence for the
role of price-stickiness: at higher levels of
inflation, prices become less sticky

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