A New Keynesian toy model

I’ve been keeping a collection of “toy models” on my computer. I do this for two reasons. First, building them is a lot of fun and useful as a kind of intellectual work-out to develop the “model-building” regions of my brain. Second, I think they help clarify my own thinking about economic issues.

I’d like to share one of my favorite toy models with you. I learnt it from Cedric Tille when I was at the IfW Kiel. The purpose of this model is to show the basic intuition behind a strand of literature called “New Keynesian” macroeconomics. The NK approach can be thought of as a combination of the techniques of the “Real Business Cycle”  literature (rational expectations, continuous market clearing, dynamically optimizing agents) with “old” Keynesian economics (monetary policy has real effects, government spending has a multiplier effect, etc.). The model is simple enough to be taught to first-year econ students and at the same time rich enough to provide a basis for discussion of the effects of monetary policy, technology shocks, fiscal policy, the distinction between expected and unexpected shocks and more. It is also much closer to current macroeconomic research than the usual AS-AD model contained in most textbooks. The model has a natural extension to an open economy setting, which is contained in this paper by Corset & Pesenti.

Here goes.

Technology. An economy’s output (Y) is produced by labor (L) alone. The aggregate production function is

Y = A*L,                           (1)

where A is the technology parameter (labor productivity).

Households. Households consume output and supply labor. They trade off the marginal utility from consumption against the marginal disutility of working. Under usual assumptions about the shape of utility functions, consumption will be an increasing function of the real wage. Denoting the nominal wage by W and the price level by P, let household consumption (C) be given by

C = k*(W/P),                 (2)

where k is a positive parameter. The basic intuition behind this consumption function is that a higher real wage induces people to substitute consumption for leisure (substitution effect) and raises their real income (income effect). Both effects act to increase consumption, while the effect on labor supply is ambiguous.*

In order to purchase goods, households must hold money. Money demand (M) is a function of nominal consumption spending:

M = (1/v)*P*C,             (3)

where v is the (exogenous) velocity of money. Note that this is just a versions of the quantity theory of money. The money supply is set by the central bank and exogenous to the model. We will think of M as describing the stance of monetary policy.

Firms. Firms compete in a monopolistic way, i.e. each firm has a monopoly over the specific kind of consumption good it produces, but there is a large number of close substitutes. It can be shown that under this kind of competition, the aggregate price level will be set as a mark-up over marginal costs of production. Nominal marginal costs are equal to W/A — it takes 1/A hours to produce one unit of output and each hour costs W euros.

Crucially, firms must set prices before learning the labor productivity and the monetary policy stance. Hence, they must form expectations about nominal marginal costs. Let z be the mark-up, which indicates the market power of firms (which in turn depends on how “tough’’ competition is in the goods market). Then the price level is given by

P = z*E(W/A),               (4)

where E() denotes the expected value conditional on information available to firms when they set prices.

Closing the model. The model is closed by the goods market clearing condition:

Y = C.                              (5)

This is a model with five endogenous variables (Y, C, L, W, and P) and two exogenous variables (M and A). Let’s find the general equilibrium of this economy. First, combine (2) and (3) to get
W = (v/k)*M.                (6)
Taking expectations and inserting into (4) yields
P = (z*v/k)*E(M/A).               (7)
Next, combine (3), (5) and (6) to get
Y = (k/z)*M/E(M/A).              (8)
Re-inserting this into (1) yields
L = (k/z)*[M/A]/[E(M/A)].   (9)

Equilibrium. Suppose that, in the long run, expected values equal actual values, i.e. E(M/A)=M/A. This is just the rational expectations assumption which in this context means that firms don’t make persistent, systematic mistakes in forming expectations about productivity and monetary policy. With this assumption, (8) reduces to

Yn = (k/z)*A,

which we can call the natural rate of output or full-employment output. It increases in productivity and decreases in the degree of monopolistic distortions. The long-run (“natural”) level of employment is given via (9) by

Ln = k/z.

Using these results in (8) yields

Y/Yn = [M/A]/[E(M/A)].

This equation relates the ratio of actual to natural output (the output gap) to the monetary stance and the state of technology. What exactly does this mean?

  • An unexpected increase in money supply raises output over its natural level. The reason is that an increase in M while P is fixed makes households spend more which raises output and employment.
  • An unexpected increase in labor productivity reduces the output below its natural level. The reason is that a higher A increases potential output, but does nothing to stimulate household spending. Hence output stays the same while labor demand (and therefore employment) falls. So a positive technology shock produces underemployment in the short run.
  • Expected changes in monetary policy or technology have no effect on the output gap. In the long run, money is completely neutral with respect to Y and L.
  • If the central bank has a way of knowing A in advance (for instance, because they employ competent economists who can forecast A perfectly), they could set M in such a way as to completely stabilize the economy at the natural output level. They “simply” have to set M=b*A.

Fiscal policy. How do we get fiscal policy into the model? Easy. Just add government spending into the goods market clearing condition:

Y = C + G                                     (5*)

and assume for simplicity that the government makes spending proportional to total output G=g*Y. (You also must assume that the government finances its expenditure by lump-sum taxes on households only so that firms’ pricing decisions and households’ labor supply are not distorted.) In this case natural output becomes

Yn =(k/z)*A/(1-g),

which increases in g. Government spending doesn’t affect the output gap, though, because it moves actual and potential output by the same amount.


*) A utility function which gives rise to such a consumption function is U(C,L) = log(C) — (1/k)*L.


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