7. The Time Inconsistency Problem and the Barro-Gordon Model

Earliest Central Banks

• The earliest central bank was founded when the Riksdag (Swedish Government) founded the Riksen Standers Bank, later renamed to Sveriges Riksbank in 1668

• The Bank of England emerged in 1694

Bank of England’s initial purpose

• Purchase government debt (lend money to fund wars in France)

• Finances were in such a bad state initially that the rate on bonds was 8%

• Eventually it developed into the bank for other banks

Recap of development

• Commodity money has problems, whilst paper money solves coincidence of want

• Various banks develop simultaneously, and the government needs to borrow money 

• A central bank is formed to lend to the government; because the Central Bank has links with the state and is trusted, people prefer CB notes, meaning other notes eventually decline 

• Central Banks control money and have a link to fiscal policy 

Key points about central banks

• Solved the problem of banks having different notes (frequently as receipts for gold or silver)

• Many CB’s are introduced to raise fiscal revenue (money for government spending)

• CB becomes a lender of last result, can be used for bailouts to prevent runs 

• The only institution allowed to print money

The Kydland-Prescott Model

• If a policymaker can act with discretion (without consequences) then there is an incentive for them to renege on promises

• Their actions would be time-inconsistent; their actions under discretion misalign with their promises

The Barro-Gordon Model

• Developed a model of policy-making under rational expectations, where the policymaker is emphasised as not the government 

• Allows for a comparison of outcomes under discretion and under commitment 

• Loss function: L_t = (U_t - U*)² + a (⊓_t - ⊓*)², Inverted Phillips Curve: U_{t =} U^N - b (⊓_t - ⊓_t^e)

• Policymaker choose ⊓_t, U* < U^N such that the government desires unemployment below the natural rate

• ⊓_t is effectively the policy makers tool

The Barro-Gordon Loss Function

• L_t = (U_t - U*)² + a (⊓_t - ⊓*)²   

• U* is unemployment target, ⊓* is the inflation weight, a is the weight on inflation deviations from target relative to unemployment deviations from target

• Because the function is quadratic, being above or below target are equally bad 

• Loss is minimised at (U*, ⊓*), where L_t = 0

The Barro-Gordon Phillips Curve

• U_{t =} U^N - b (⊓_t - ⊓_t^e), normal Phillips Curve is ⊓_t  = ⊓_t^e - 1/b (U^t - U^N)

• U_t is unemployment at time t, U^N is the natural rate of unemployment

• If actual inflation exceeds expectations, unemployment will be lower than the natural rate

• If real wages are influenced by inflation, then inflation above target means real wages fall because the nominal amount is too low

• -b is the sensitivity of unemployment to real wage changes

• -1/b is the sensitivity of inflation to the unemployment gap

• Downwards sloping linear function, with unemployment on x and inflation on y

Characteristics which can affect inflation expectations

• Naivety: Public wage-setters believe inflation will be what the government tells them

• Anchored: People might be insensitive to data

• Rational: Forward looking model of the world which is correct

Naive solution

• Public believe inflation will be what the government says, so ⊓^e = ⊓_t = ⊓*

• The aim is to minimise the loss function with respect to the constraint of the inverted Phillips curve given ⊓_t^e = ⊓*

• We substitute the Phillips Curve into the loss curve and minimise with respect to ⊓_t, then setting = 0 and rearranging to get an answer in terms of ⊓_t; this answer will be > ⊓*

• By substituting the expression for ⊓_t into the loss function, one can attain a value of U_t

• The government has an incentive to set an inflation rate greater than ⊓_t^e so that they can get an unemployment rate below the natural level 

Loss in the naive solution

• L > 0 and L = 0 is unachievable if U^N > U*

• There is incentive to renege on the promise once expectations have been set

• The government could achieve ⊓*, but only by setting U_t > U*, where U_t = U^N meaning that (U^N, ⊓*) is not the welfare-maximising choice

• They could achieve U*, but only by allowing ⊓_t = ⊓^b > ⊓*, thus (U*, ⊓^b) is not the welfare-maximising choice either

• The solution to the minimising expenditure problem represents the best outcome the government can achieve with a foolish and naive electorate (U_A, ⊓_A)

• This is time-inconsistent because the actual inflation rate is not the same as the one the government promised; this can only be maintained if the public can be convinced that ⊓_t^e = ⊓*

Rational Expectations Outline

• ⊓_t^e = E_t-1⊓_t

• E_t-1 just means that expectations are formed using available information at time t-1

• The public (wage-setters, private sector) assume their expectations before ⊓_t is realised and before the policy choice ⊓_t is made

• They are also assumed to have a correct model of the economy

Rational Expectations derivation

• Minimise the Loss function subject to the inverted Phillips curve given ⊓_t^e = ⊓_t

• Substitute the Phillips Curve into the Loss function and differentiate wrt ⊓_t, then set = 0

• Rearrange to get an expression in terms of ⊓_t  which is greater than ⊓*

Rational Expectations Solution

• Graphically, the solution is understood as a ‘time-consistent equilibrium’, where expectations = the inflation level

• The time-consistent solution has higher inflation and unemployment than both the optimum and naive equilibrium points

• This is because the rational wage-setters aren’t fooled by the policymaker; they understand the incentive to raise inflation surprisingly, and so they adjust their expectations accordingly

• An equilibrium is at (⊓_c, U^N)

Loss in the rational expectations equilibrium

• Because inflation is above optimum, loss will be strictly positive

• Despite inflation being above target, there is no compensated gain in lower unemployment because inflation expectations are equal to the actual inflation level

• If ⊓_t = ⊓^e, this naturally implies that unemployment is at the natural rate

• There is an inflation bias in the time-consistent equilibrium, which is the difference between actual inflation and optimum inflation (⊓_e - ⊓* = b/a (U^N - U*))

Anchored Expectations outline

• Can be treated in a similar way to naive expectations, but they do move eventually, if people start to realise they’re being fooled

• This consists in a movement to more rational expectations

• Beliefs are anchored when agents hold inflation expectations in line with the CB’s target rate

Imperfectly anchored expectations

• If beliefs are perfectly anchored, they are insensitive to any incoming data, like the naive case

• Partially anchored expectations are slightly responsive to changes in information which influences expectations, leading to slight shifts in the inverted Phillips Curve

What is stabilisation policy?

• Monetary/fiscal policy aimed to counteract shocks which affect the real economy 

The role of anchored expectations in reducing stabilisation policy costs

• If expectations aren’t anchored, a shock will shift the Phillips curve by the ‘amount’ of the shock

• If expectations are anchored during the shock, unemployment doesn’t need to rise to bring it back down; it will rise and fall itself

• If partially anchored, unemployment levels would rise, but less than if they were completely unanchored

• Disinflation (stabilisation) policy is less costly when expectations are anchored

Assumptions about policymakers

• In the Barro-Gordon model, they assume the policymaker is trying to reach a level of inflation below the natural rate; governments might want to do this to make it look like the economy is doing well before elections

• In stabilisation policy, generally policymakers are conceived of as trying to reach the natural rate of unemployment and optimal inflation (Barro-Gordon is (U_t > U^N, ⊓^a) compared to stabilisation policy’s target of (U^N, ⊓*)

What is the time-inconsistency problem?

• It concerns the sub-optimal outcomes which can occur if there is incentive at a later point in time to renege on promises made at an earlier point in time 

• The example is of a policymaker who sets an inflation target and then surprises people with higher inflation for their own ends