The New Keynesian Model

Macroeconomics (M8674), May 2025

Vivaldo Mendes, ISCTE

vivaldo.mendes@iscte-iul.pt

1. Introduction

What is the New Keynesian Model (NKM)?

• It's a DSGEM
• It's a new version of the Old Keynesian model, which was:
  • Following John Maynard Keynes from the 1930s
  • Dominant until the late 1970's
  • Strictly based on Adaptive expectations
  • Based on price and wage rigidities
• The NKM allows:
  • Adapative and Rational Expectations
  • Allows price and wage rigidities

Is the NKM Relevant today?

"Those criticisms notwithstanding, the New Keynesian model arguably remains the dominant framework in the classroom, in academic research, and in policy modeling."

Jordi Galí (2018). The State of New Keynesian Economics: A Partial Assessment, Journal of Economic Perspectives, 32(3), 87–112.

2. The Structure of the NKM

The Baseline New Keynesian Model

• : nominal interest rate, natural real interest rate, output-gap, inflation rate, demand shock, supply shock, and a random disturbance.
• are parameters
• Forward-looking variables:
• Backward-looking variables:
• Static variables:

The IS curve

• represents the output-gap (% deviations of GDP from Potential GDP).
• is the conditional expectations operator.
• is the short term nominal interest rate set by the central bank.
• is the level of expected inflation for period .
• is the natural level of the real interest rate.
• is a demand shock.
• is a parameter (coefficient of relative risk aversion).

The AS curve

• is the inflation rate
• is the output-gap
• is a supply shock
• are parameters.
• gives the level of price rigidity in the economy:
  • : total flexibility
  • : total rigidity

The Monetary Policy Rule

• This specific form of rule is called the "Taylor Rule", following John Taylor (1993).
• is the nominal interest rate set by the central bank.
• is the inflation rate.
• is the natural level of the real interest rate.
• is the central bank's target level for the inflation rate.
• are parameters.
• When and , we have .

John Taylor (1993). Discretion versus policy rules in practice, Carnegie-Rochester conference series on public policy, 39, 195-214.

Policy Rules and How the Fed Uses Them

• In the website Policy Rules and How Policymakers Use Them, the Fed says they have 5 major rules they use in the decision making process:

  • Taylor rule
  • Balanced-approach rule
  • ELB-adjusted rule
  • Inertial rule
  • First-difference rule

• The last two perform much better than the others (include a persistence term)

• The Taylor rule is used in this course because it: (i) explains the logic of using rules , an (ii) simplifies the algebra of the NKM

Target Inflation Rate

All central banks in advanced countries have an optimal value for inflation they want to achieve. This is called the inflation target

Source: Central Bank News

Why 2% Inflation?

According to the Swedish Central Bank (Sveriges Central Bank):

3. Some (illustrative) results

IRFs from a positive demand shock

IRFs from a negative supply shock

IRFs from the two shocks

Time-series from the two shocks

Heatmap: cross-correlation function (two shocks)

Standard deviation (two shocks)

4. Some criticisms of the NKM

Dominant model ... but with alleged problems

Three main criticisms have been raised against the NKM:

• Counter-intuitive results in the AS curve.
• Counter-intuitive results in the IS curve.
• Lack of persistence

Some influential criticisms

The IS and AS curves (allegedly) lead to easily refutable results. See, e.g.:

• Ball, L. (1994). “Credible disinflation with staggered price-setting.”American Economic Review, 84(1), 282–289.
• Fuhrer, J. and Moore, G. (1995). “Inflation persistence.” Quarterly Journal of Economics, 110(1), 127–159, February.
• Fuhrer, J. (1995). The Persistence of Inflation and the Cost of Disinflation, New England Economic Review, January/February 1995, 3-16.
• Canzoneri, M. B., Cumby, R. E., and Diba, B. T. (2007). Euler equations and money market interest rates: A challenge for monetary policy models, Journal of Monetary Economics, 54(7), 1863-1881.

Counter-intuitive results in the AS curve

• Consider the AS curve and ignore any shocks :

"If the Federal Reserve engineers a disinflation, it does so by pursuing a contractionary monetary policy that lowers output below potential . But the flexible inflation model of the equation [above] says that when output falls short of potential, expected inflation in the next period must exceed current inflation. This does not sound like a disinflation". Fuhrer (1995), page 9

• For simplicity assume ,, . A central bank could reduce inflation and produce higher expected inflation at the same time (see next slide for details):

Furher's argument in more detail

• Suppose the economy is in a stable situation:

• Now, suppose inflation increases to . Normally, higher inflation forces the central bank to increase nominal interest rates, in order to reduce aggregate demand, which in turn will lead to a decrease in inflation and expected inflation.
• However, this is not what happens in the AS curve. If inflation goes up to , and the central bank creates a recession of , we have to get:

• It does not make sense: inflation goes up, the central bank creates a recession, and expected inflation goes up. In reality, should go down, not up!

Counter-intuitive results in the IS curve

Consider the IS curve. Put the term on its left hand side:

"An awkward implication of [this equation] is that, when the real interest rate rises above its steady-state level, the level of consumption must decrease, but its change must be expected to increase." Estrella and Furher (2002), page 1021

• For simplicity assume and : The IS seems to give a strange result:

• The central bank may increase to fight inflation, and the expected output-gap is supposed to increase. Does not make sense (more details in next slide).

Furher&Estrella's argument in more detail

• Suppose the economy is initially in a stable situation:

• Now, suppose the central bank increases nominal interest rates to . This should lead to the expected output gap to go down, because an increase in must constrain aggregate demand and economic activity.
• However, that is not what happens in the IS curve:

• An increase in leads to an increase in the expected output gap: it doses not make sense.

The shapes of the IRFs are wrong

• In macroeconomics, most variables respond to a shock gradually.

5. Refuting the criticisms

Does an increase in lead to a boom?

• What happens if the central bank decides (for whatever reason) a sharp increase in its nominal interest rate?
• It causes a recession and a slight reduction in inflation:
• It does not cause an increase in the expected output gap as the criticism of the IS curve above suggested.
• This is a vindication of the NKM and exposes the limitations of that criticism.
• See the following figure.

Does the fight of inflation lead to higher inflation?

• Suppose there is a negative supply shock that leads to higher inflation.
• The criticism above regarding the AS curve suggests that disinflationary measures should lead to higher inflation expectations.
• Is this correct?
• The central bank sharply raises nominal interest rates to fight inflation.
• The IFRs can be found below. Anything wrong with them?
• No! In fact,

The NKM with inflation and output persistence

• To overcome the problem of hump-shaped IRFs we need to do two things:
  • Introduce lagged inflation in the AS curve
  • Introduce lagged output in the IS curve
• So the problem is not with the model, but whether we omit (or not) fundamental parts of reality.

Hump-shaped IRFs in the NKM

Appendix A

Derivation of the AD Curve (not required in the evaluation process)

Maximization of Utility

• Households maximize utility which depends on consumption over time

• Subject to a constraint in every period

• is real consumption, is the price level, is the real wage, is the nominal interest rate, and is the level of nominal bonds

• : it does not make sense to have positive savings in some last period

The Lagrangian Function and FOCs

The maximization of utility is given by the Lagrangian function

and the First Order Conditions (FOCs) are:

The Euler equation

From the three FOCs

we can obtain:

simplifying

Now, , where is the inflation rate at , so:

Uncertainty and the Euler Equation

• The Euler equation gives the optimal trade-off between consumption today versus consumption in the future
• But the future is not known with certainty, so the variables expressed with will have to be taken under the expectations operator

Marginal utility of current consumption divided by its cost is equal to the discounted value of the expected marginal utility of future consumption, divided by its expected cost .

A CRRA Utility Function

A CRRA Utility Function and the Euler equation

• A typical utility function is a CRRA

• where is the coefficient of relative risk aversion:

• Inserting this marginal utility into the Euler equation gives:

Applying logs to the Euler equation

• Applying logs to the Euler equation

• If are small values, then

• So, the log version of the Euler equation will look like:

• The Euler eq. in the steady-state (notice that is the natural real interest rate):

Percentage Deviations from the Steady-state

• Subtract eq. (2) from (1), and get consumption as a % deviation from steady-state:

• Therefore:

• That is:

Aggregate demand

• To make things as easy as possible, consider an economy where aggregate demand (or GDP) is determined by

• is real GDP, is real consumption, stands for real government spending.

• This means that there are no Investment expenditures, nor Exports or Imports.

• Applying growth rates to the eq. (4), we get

• where and , and are the steady-state values.

The IS function (I)

• The simplest economy is one where GDP is equal to consumption.
• This is the case when in eq. (5):

• Therefore

• The output-gap depends upon the expected output gap , and the difference between the expected real interest rate and the natural real interest rate .

The IS function with Government Spending (I)

• Now, consider that , implying the existence of public spending.
• To derivate the IS function, insert eq. (3) into (5) which leads to

• Then, from eq. (5) we can obtain

• Now, insert eq. (8) into (7), and simplify to obtain the IS function:

The IS function with Government Spending (II)

• Another way of expressing the demand side of the economy is to have two equations (instead of just one, as in eq. 9). These two are:
• The linearized Euler equation:

• The linearized aggregate demand equation

• These two equations display the inter-relationships between consumption, output, and government spending, the expected real interest rate, and the natural real interest rate.

Current vs future consumption: graphical analysis

vs : prices	vs : preferences

The optimal consumption pattern: graphical analysis

Appendix B

Derivation of the AS Curve (not required in the evaluation process)

Price Rigity: three different ways ... similar results

John M. Roberts (1995). “New Keynesian economics and the Phillips curve,” Journal
of Money, Credit, and Banking, 27(4), 975–984.

Shows that the 3 papers below lead to similar results:

• John Taylor (1979). “Staggered Wage Setting in a Macro Model,” American Economic Review, P&P, Vol. 69, 108-113.
• Julio Rotemberg (1982), “Monopolistic price adjustment and aggregate output,” Review of Economic Studies, 49(4), 517–531.
• Guillermo Calvo (1983). “Staggered prices in a utility-maximizing framework”, Journal of Monetary Economics, 12(3), 383-398.

We will follow Calvo's pricing.

Main assumptions: Calvo Pricing

• A1. In each period, a proportion of firms do not reset their prices.
• A2. There is monopolistic competition in the goods market: firms set prices (the optimal price, with a markup over marginal costs . In logs:

• A3. Firms know that the price they set today remains constant during some time. So is set to minimize the profit loss for not resetting prices.
• A4. Real marginal cost in logs depends on the output gap

Monopolistic competition pricing

From: Hal Varian (2010). Intermediate Microeconomics: A Modern Approach, 8th Ed.

Minimizing the Loss function

• Given assumptions A1 and A3, the Loss function is given by

• is the (log) price level that minimizes the losses until
• is a time discount factor
• is the probability of having the price constant until
• To minimize