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The Barro-Gordon Model
M. McMahon
University of Warwick
July 29, 2014
This note outlines the Barro-Gordon model of time-consistent monetary policy, dis- cussing the meaning of the equations and how to solve the model. I also present a game-theoretic outline of what is going on in the model which may help some of you to understand the material more easily. This note is not a substitute for understanding the lecture material but will hopefully act as a complement.As always, I welcome comments and suggestions on ways of improving this note.
1 The Idea of Time Inconsistency
The key idea to take from this model is that when the government is given an opportunity to cheat workers, they will - but knowing that they will be cheated, rational agents will lead to a higher level of inflation but with no gain in terms of lower unemployment. In this model, the incentive to cheat derives from the fact that inflation expectations are set in advance of the governent setting current policy and so there is a chance for the government to exploit a trade-off between inflation and unemployment (the Phillips curve)
- the government will choose higher inflation (above expected inflation) to drive down the unemployment rate. All this, despite the fact that everyone knows that in advance, the government can do no better than achieve the inflation target and the natural rate of unemployment - in technical terms, there are differences between ex-ante and ex-post incentives to set good policy. This idea is known as the time inconsistency problem and is actually not due to Barro or Gordon! It is originally due to the work of Kydland and Prescott for which, along with real business cycle theory, they were awarded The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 2004.^1 The Kydland-Prescott work was not related only to monetary policy but to all types of government policy - they (^1) This is what we called the Economics Nobel prize but it is actually not a proper Nobel prize... but still worth winning!
showed that when the economic agents were rational, and where expectations influence the decision of government policy, discretionary policymaking made everyone worse off compared with a situation in which the government could commit to a particular policy path - i.e. there was a credibility problem, or a time inconsistency problem.^2 The 1983 works of Barro and Gordon^3 were focused on the issue of monetary policy and in particular highlighted the role for monetary rules as a potential means to overcome the time inconsistency problem in monetary policy. In the next few pages I will cover the solution to the basic model and hopefully drive home the intuition to the results obtained.
2 The Simple Model
2.1 Set-up
The model comprises of the following key elements:
Loss Function L = (U − U ∗)^2 + a(π − π∗)^2
This tells us that society (the economy) is better off in the sense that it has smaller losses when inflation (π) is close to target (π∗), and unemployment (U ) is also close to target (U ∗). In fact, the best that society can achieve according to this loss function is L=0 by setting π = π∗^ and U = U ∗. There are a number of important features of this loss function:
the parameter a determines the relative importance of inflation relative to unem- ployment - for example when a is very high, inflation deviations are very bad for social welfare. because of the square terms, the loss can never go negative;
also because of the square terms, the loss is quadratic and so increases very quickly as deviations from target increase - e.g. if π∗^ = 0 but π = 1, then the loss is 1, but when inflation doubles (π = 2) the loss quadrouples!
Phillips Curve U = U N^ − b(π − πe)
This is the economy that we have to deal with in this model - it is a very simply rela- tionship between the level of unemployment and inflation. It tells us that unemployment (^2) There is a great article written about the work of Kydland and Prescott which is available at: http://nobelprize.org/economics/laureates/2004/ecoadv.pdf 3 ”Rules, Discretion and Reputation in a Model of Monetary Policy”, JME 1983, and ”A Positive Theory of Monetary Policy in a Natural Rate Model”, JPE 1983.
- the most simple one being that unemployed workers might vote a government out of office and so governments will try to keep unemployment low to signal that they are doing a good job. Another very nice story surrounds unions who may put political pressure on governments to try to create higher employment in the economy and so drive U ∗^ down - this is a particularly neat story because as we mentioned above, unions can drive U N^ up and as we will see below the gap between U N^ and U ∗^ will be the key to determine the extent of the problem of the inflation bias. Notice that because of this assumption, the optimal solution of L = 0 is not attainable as setting π = π∗^ will result in U = U N^ > U ∗. Therefore, there will be some losses in this economy in steady-state because the government is trying to achieve something that is inherently unachievable.
Timing of the Model
The timing in the model is key to the models predictions - in particular, it is key that inflation expectations are formed before the goverment sets the current rate of inflation. Otherwise, there would be no opportunity for the government/central bank to cheat since worker expectations would be formed after current inflation is set and so they would never be fooled. The precise timing of the model is as follows:
Pre-Time Before the economy begins, the parameters a, b, c, as well as the targets U ∗^ and π∗ are determined.
Start of Period At the beginning of each period, the workers must form there expectations of infla- tion in the forthcoming period (πe)
Middle of Period The government/central bank selects the policy for the current period (π) taking πe^ as given.
End of Period Given the choice of π and πe. The Phillips curve determines U and social welfare (L) is therefore determined.
Worker Expectations
In solving this model, the way in which workers form their expectations of inflaton (and therefore the wage increases that they seek) is key. In solving the model, we consider 3 different approaches - namely, naive expectations, rational expectations (RE) and also briefly adaptive expectations, though the RE solution is the most important one.
2.2 Naive Solution
Let us first do the naive solution to highlight the incentive to cheat. In this case, the workers believe that since the economy can never do better than setting π = π∗^ and letting U = U N^ .The problem for the government is now:
min {π} L = (U − U ∗)^2 + a(π − π∗)^2 (1) s.t.U = U N^ − b(π − πe) (2) and πe^ = π∗^ given (3)
So we substitute in the Phillips curve:
min {π} L = (U N^ − b(π − πe) − U ∗)^2 + a(π − π∗)^2
and take the derivative wrt π to minimise the function with πe^ = π∗^ taken as given (we should also really check the second order condition):
dL dπ = 0^ ⇔^ 2(U^
N (^) − b(π − πe) − U ∗)(−b) + 2a(π − π∗) = 0
⇒ (b^2 (π − πe)) + a(π − π∗) = b(U N^ − U ∗) (4) And then we can use the fact that πe^ = π∗^ :
⇒ (b^2 (π − π∗)) + a(π − π∗) = b(U N^ − U ∗) (5) ⇒ π = π∗^ + (^) a +b b 2 (U N^ − U ∗) > πe^ = π∗^ (6)
Equation 6 shows us how when the workers set πe^ = π∗, there is ex-post (after the event) incentive for the government to run higher inflation than expected and try to drive down unemployment below the natural rate (since π > πe, the Phillips curve tells us that unemployment falls as real wages have declined)..
2.3 Rational Expectations Solution
Obviously the naive result is somewhat foolish as it relies on workers being consistently fooled in thinking that the government will not cheat them despite the fact that they will have the incentive and the capability to do so. The rational expectations (RE) solution is the major contribution of Kydland and Prescott and formed part of the rational
2.5 Insights and policy implications
So to summarise the model, we can use the following table: Inflation (π) Unemployment (U ) Loss (L) Naive Solution π > πe^ = π∗^ U < U N^ lower than steady-state RE solution πe^ = π > π∗^ U = U N^ higher than optimal if could commit Therefore the key insight is that simply because the government cannot commit to a policy path, the economy ends up in a less optimal equilibrium. The key policy insight is therefore that we should look at ways of allowing the government/central bank to commit not cheating to lower inflation. Numerous suggestions have been put forward and used in the real world; for example, delegating monetary policy to an incependent central bank who only cares about an inflation target (inflation targetting ala the Bank of England). Of course, the other possible solutions to this model highlight what are the key as- sumptions for solving the problem:
Timing assumption - if the world was more flexible, and workers set the wages more regularly (or after the Central Bank has set policy), then there is no possiblility to cheat and the problem goes away.
Assumption that U < U N^ - as discussed above, the size of the inflation bias de- pends on the difference between U and U N^ , and therefore reducing this gap to zero eliminates the problem. See also footnote 5 above.
2.6 Alternative approaches
Of course, the simple model outlined here is not the only specification of the model that is possible. There are many similar specifications of the model that may work (and can be solved in the same way) or even applications of this idea to other situations such as fiscal policy and optimal taxation.
3 Game-theoretic outline of the model
Here is the basic ”game” played by the authorities and the wage setters (workers).^8 In the beginning of each period, it is optimal for the government or CB to set π = π∗^ - this is the ex-ante incentive. However, in the next stage of the game workers must set πe^ and then only after πe^ are set, will the government actually set π. Once πe^ is fixed, the government now has an incentive to lower L by choosing slightly higher inflation but getting U below U N^ and so closer toU ∗. This is the ex-post incentive and depends on the parameters of the model as we have seen above. Notice that although the most desirable outcome for social welfare is (1, 1)^9 , this outcome will not be achieved. That is because rational workers know that if they set πe^ = π∗, the government will then choose π > π∗^ which yields a higher outcome for the government (2) but lower for workers who lose out. Therefore, the workers will always choose to set πe^ > π∗, then the government is forced to choose π = πe^ > 0 which yields an outcome (0, 0). This is lower than the best outcome, but is time consistent! Of course, those of you who are good with game theory will realise that the problem of time inconsistency is reduced if we consider the policy game as a repeated game rather than a one-shot game. In a repeated game, we can build up a reputation which leads to the optimal outcome - in fact, building monetary policy credibility is onw one of the main aims of central banks around the world, a trend due largely to the work of Barro and Gordon and others.
(^8) This formulation is originally due to Taylor (1985) in the Federal Reserve Bank of Philadelphia Business Review, but is reproduced in ”A Modern Guide to Macroeconomics” by Snowdon, Vane and Wynarczyk (1994). 9 The payoffs are defined as: (payoff to government, payoff to worrkers)
