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Rules vs Discretion: Barro Gordon one-shot Model in Monetary policy -

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Observations on ‘rules’ and ‘discretion’ in monetary policy (Below are some arguments for rules dictating monetary policy) o Tavlas notes that policy rules for which quantity of money plays a key role were a central feature of the Chicago monetary tradition from the mid 1930s to the late 1970s o 2 most prominent rules were formed by Henry Simons and Milton Friedman  Simon proposed a rule that targeted a constant price level in the SR  Friedman advocated a rule that targeted a constant rate of growth of money supply (the ‘k-percent’ rule) o More recently the rules versus discretion issue featured at the heart of a debated between former Fed chair Ben Bernanke (favouring ‘constrained discretion’) and John Taylor (favouring ‘rules-based’ monetary policy) Time consistency in other settings o Cecchetti and Schoenholtz suggest that the problem of time consistency is one of the most profound in social science  They feel this time consistency notion has applications in other areas as well as monetary policy e.g. in other areas of economic policy as well as counterterrorism. o Notion of time consistency may also be used in an educational-setting; can affect incentives of people o Time consistency can also be used in a political setting  Time inconsistency is stated most simply in a political example e.g. of terrorism;  Announced policy of many nations is that they will not negotiate over hostages  This announcement is intended to deter terrorists; if there is nothing to be gained from kidnapping hostages then rational terrorists won’t kidnap any  Argument versus above is how rational are people in practice  Above announcement has little effect unless policymakers commit to the policy; be time consistent and don’t pay terrorists ransom to release the hostages  If policymakers were truly unable to make concessions, the incentive for terrorists to take hostages would be largely eliminated Barro-Gordon Model: Rules vs discretion o Following game, based on Barro and Gordon (1983) focuses on whether policymakers should adhere to rules or conduct policy according to discretion o We introduce  A Philips curve (PC) specification which captures the relation between unemployment and inflation  U = U^n – a(pi – pi^e)  Saying rate of unemployment is equal to its natural rate – ‘a’ parameter multiplied by inflation – expected inflation  Public form expectations of what they expect bank to do with inflation and central bank control pi i.e. what inflation actually will be

Un may change for several reasons e.g. level of benefits, union legislation etc.  Or pi = pi^e – 1/a(U – U^n) A loss function which specifies the preferences of the monetary authorities over inflation and unemployment  Loss function is specified as  L = [b(U-kU^n)^2 + pi^2] where b is between 0 and infinity and k is between 0 and 1 Whenever policymaker sets pi they must be aware of expected inflation as 

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divergence from expectations will affect the level of unemployment as shown by the Philips curve specification ‘k’ is an exogenous parameter; taken as a given and not set by policymaker; means an assumption that the government wants to drive unemployment below natural rate Argument is that government wants to do this to achieve higher level of output; shown in Okuns law; government may also want to have this approach for political reasons to be elected into office again  Arguably central banks in practice do not have this desire to drive unemployment below natural level; argue that they have specific targets e.g. inflation targets and only focus on this Squared signs in equations are there as the function is a quadratic utility function;  It says that if inflation becomes larger it doesn’t matter if it is +1 or -1 as it will cause equal disutility and hence the square causes this mathematically Greater unemployment deviations from target rate of unemployment means greater disutility; hence want to set U = kU^n ideally to get that term = 0 as well as inflation = 0 to minimise the loss function

One shot game o Through making substitutions, taking expectations etc. we get the following solutions for inflation and expected inflation o The above solutions are shown on the next page

Level of inflation set is a function of expectations of the public as shown in equation 3 Scenario 1: Surprise inflation vs Monetary rules o Assume authorities announce they will set inflation to 0 and this is believed by private sector (pi^e = 0) o Authorities now have choice on whether to stick to promise (set pi=0) or renege on it (pi > 0) o If pi^e = 0 then inflation will be positive and pi = ab(1-k)*U^n/1 + a^2 * b  We can then substitute pi and pi^e = 0 into the Philips curve and hence we get rate of unemployment associated with inflation being positive and expected inflation being 0 

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Hence, if central bank cheats it can achieve unemployment below natural rate

If the government doesn’t set inflation as positive but actually keeps its promise from above to set inflation = 0 o It results in unemployment being equal to its natural rate as inflation = expected inflation o Value of loss function is smaller under cheating  It is in incentives of government to cheat rather than abide to binding rules if it can get away with it as it lowers the utility loss function  Private sector knows this and this affects inflation expectations Scenario 2: Private Sector is not stupid o Private sector has rational expectations and there is complete information; means people know loss function and Philips curve o Hence PS will know that if authorities announce 0 inflation and are not bound by a rule, the temptation for them to cheat will be too big o Hence under absence of rules the private sector expects positive inflation of o

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Expression (12) above turns out to be larger than 2 previous loss functions from cheating and following a rule so isn’t minimising loss of policymaker and isn’t best solution; could mean ultimately it is best for policymaker to set positive inflation and not 0 inflation and just announce this as private sector isn’t stupid Scenario 3: Authorities still follow a rule given the PS rational expectation of pi^e = ab(1k)*U^n o This results in the economy being disinflated i.e. realised inflation is lower than what the private sector expects o Unemployment is pushed above its natural rate while inflation is positive o

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This implies that actual rate of inflation which policymaker will set will be equal to the rate which the private sector will expect The above results in 9 and 10 being put into the PC results in unemployment being equal to the natural rate as there is no deviation from expectations of public sector by the policy from monetary authorities Using the PC and inflation into loss function gives the rational expectations equilibrium as shown below in (12)

Loss function w/ disinflation is highest of any loss function in scenarios discussed so far o This is due to central bank sticking to its rule and not using discretion while public sector expect the authorities to go against what they initially announced o L rule can only be achieved with commitment technology as otherwise PS will expect the authorities to cheat o This model has big impact on central banking and how central bankers talk about their policy in speeches Graphical solution to Barro-Gordon Model o

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o Graph shows isoquants i.e. indifference curves associated with central bankers Criticisms of BG model o Role of time is typically ignored and model discussed above is a ‘one-shot’ game o In reality, central banks use SR IR as the main tool of monetary policy

ADDITIONAL NOTES Barro-Gordon Model: Rules vs Discretion -

Whole model based only on 2 economic relationships however generates lots of results which inform us on actions of policymakers They envision that economic policy, and in this case monetary policy conducted by the bank specifically, is like a game of the policymakers vs the private sector who are always forming expectations to try predict what policymakers are going to do

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Different to an IS-LM model as here the policymaker has a utility function and has some sort of feeling which they aim to maximise The PC relation states that deviations of unemployment from its natural rate only arise due to occasions of expectations of inflation deviating from the actual rate of inflation Hence when policymakers set policy they must be aware of expectations of inflation such as to not set policy which is very far from expectations which will cause divergence of unemployment from its natural rate

Loss function -

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The function relates directly to the utility that the monetary authorities feel and larger L values mean more disutility for the monetary authorities; they want this value as low as they can Above would imply trying to drive unemployment below natural level and many argue that in practice central banks and monetary authorities don’t want this however this is implied in the above model Loss function is an example of a quadratic utility function; only the level of difference from targets matter and not the direction of difference; same magnitude of difference from target in either direction will cause an equal amount of disutility

One shot game -

We take PC expression in (1) then we substitute into (2) then we differentiate the L function w/ respect to inflation and then solve for inflation which gives us (3) Finding in (3) is very important which shows that whatever inflation rate the central bank chooses to set will be influenced by and be a function of the public expectations of inflation (4) is achieved when we take expectations and rearrange for pi e from equation (3) Following expressions 3 and 4 being derived barrow and Gordon asked whether monetary policy should be done by discretion or by binding rules After looking at the loss function, authorities will set policy which minimises their loss function according to the model One shot game isn’t realistic as in reality it is more likely to be repeated interaction where policymaker will likely build up reputation, however the intuition from the model is useful

Scenario 1 -

Algebraic manipulation shows that in this scenario if the central bank doesn’t stick to its word and cheats on its promise then it will achieve a rate of inflation below its natural rate If it abides by its promise to set inflation = 0 then unemployment will be equal to its natural rate Loss function under cheating is less than under rules and hence it is in government interests based on above to cheat

Scenario 2 -

If we know what drives the central bank in making decisions and incentivising it as is the case in this scenario w/ perfect information and rational expectations

Summary of results so far -

Can see from the expressions which are listed on the following page for loss functions based on the several different scenarios considered that the best loss function for the authorities is

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from Ls which is where they announce a given level of inflation and then cheat on this promise However in reality private sector will know its game and know to expect other than what the central bank announces and with these rational expectations the best the bank can hope for really is LRat

However commitment technology and central bank legislation which forces monetary authorities to commit to promises they make could make the loss function of LRule attainable which would be an improvement on LRat; even though central bank may be incentivised to cheat it is prevented from doing so by the legislation Central bank independence is key for this in terms of making sure they are independent from government and not politically motivated hence simply focus on their own goals

Graphical solutions to the BG Model -

We are only interested in plotting values of the loss function of the central bank for positive values of inflation and output As we move further out in the graph of figure 1 the level of disutility is increasing Vertical line is the LR PC whereas the other diagonal solid black lines are SR PC’s Point A is the lowest level of disutility optimal choice and is where the authorities look to get a monetary surprise Graph on the right has inflation on y axis vs expected inflation on x axis and the 45 degree line captures all points where inflation = expected inflation i.e. pi = pi e Point B shows the optimal choice where pi = pi e at LRat

Conclusion

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Model assumes the central bank has perfect and direct control over Ms and hence has perfect control over level of inflation and other models may not make this assumption The private sector’s knowledge of the monetary authorities and how they think may also be imperfect and this is also not considered but instead the private sector knowledge of central bank/monetary authorities is assumed to be perfect...