Carefully explain the concepts of time inconsistency and inflation bias. Develop one option to address inflation bias and evaluate its empirical validity
Kydland and Prescott made a great impact in the economic world when they decided to publish information on the impact of time inconsistency to the economy (WALSH, 2010). They were able to identify various areas in government that had time inconsistencies such as the output inflation trade-off in monetary policy and capital taxation that existed in fiscal policy (ARNON, 2010). On introducing the concept of time inconsistency, they insisted that it was important for the Central Bank to exhibit credibility and adopt the shared strategy in advance. This article shall however look at the concept of time inconsistency and inflation bias. In order to do so it is vital to understand the meaning of time consistency in the economic world. Time consistency means that a scheduled response to a given policy remains optimal even once new information has reached. In retrospect a policy is time inconsistent if at time t+ i it would not be ideal to respond as planned initially (VIEGI, 2012).
Understanding the importance of time inconsistency allows economists to come up with positive theories of the inflation rate (BANQUE CENTRALE DU LUXEMBOURG, 2006). It allows one to understand why certain countries tend to realize high inflation rates over a span of time. The knowledge of time inconsistency also leads to the generation of normative theories (BANQUE CENTRALE DU LUXEMBOURG, 2006). The latter clearly demonstrates the best design for the central bank, and gives information on the generation of incentives.
Inflationary bias on the other hand is usually as a result of the enactment of a discrete government policy that increases inflation rates in the labor market. Furthermore, during this phase there is no transitory income (VIEGI, 2012). Inflationary bias may also be as a result of nations full of debt instilling policies that increase inflation within a short or medium term (FROYEN & GUENDER, 2007).
Walsh (2010) clearly depicts how inflationary bias is created through the use of an optimal incentive structure. The basic structure of the model consists of a supply relationship that controls money growth and inflation. There is also an objective function that is dependent on both output fluctuations and inflation inconsistency (ROMER, 2012). This basically means that the private sectors anticipation is usually determined before the Central Banks choice of the growth rate. This shall be discussed further on in the paper.
At this point it is vital to consider the disadvantage of time inconsistency in monetary policy. One of the disadvantages is the fact that normal inflation usually increases past the rate desired socially. In such a scenario, inflation bias shall arise due to the aspiration to expand economically (ROMER, 2012) and (HEIJDRA, 2009). This usually strains the economy since its output level is usually low at this point. Furthermore, at this point the Central Bank usually lacks the capacity to sustain low inflation.
Time inconsistency would also mean that the central bank is able to inflate the economic rate and make it high if at all the public were looking forward to low inflation (ROMER, 2012). Time inconsistency would also mean that members of the public would be more aware of the situation if the Central Bank were to increase inflation (ROC, 2005). This would mean that they would be expecting the Central bank to increase the inflation rate.
Below is a simple model of the inflation bias;
Phillips Curve relationship model

Policy Maker Loss Function

The Barro-Gordon Model of Inflation Bias

The solution to the policy problem would be to substitute the first figure (1) into the second (2).This would then lead to optimization.

The next step would be to take the derivative of (3) in concurrence with Pie ( ).This would provide us with the first condition for an optimum.

Therefore, the first condition to be derived owing to the expectations of the private sector shall be as follows;

In the above scenario, if then the equilibrium point of inflation shall be obtained through the substitution of output (5) in (1) as shown below;

By including equilibrium values in the loss function (2) we are able to get the social welfare level from the same.

The latter equation would mean that which undoubtedly will be disadvantageous to the public in terms of inflation rates.
However, the model below will show what would happen if the public was aware of the Central bank incentives;

If we were to substitute the inflation in equilibrium values in (5) of the equation above, then we would have;

Substitution of would give us the following equation;

Below is a calculation of welfare losses;

This would mean that;

In retrospect, it shows that had the bank been compelled to a zero inflation policy with support from the private sector then the society at large would enjoy a better economy. It is also important to note that results of the Phillips curve also change when uncertainty is introduced (VIEGI, 2012). In such a scenario time inconsistency also arises when the central Bank decides to stabilize the economy through the use of the monetary policy after the public has been made aware of the situation. Inconsistency also arises in a scenario where inflation anticipations are created in the private sector together with fixed wages (VIEGI, 2012).
In the case of uncertainty the Phillips curve is represented below;

In the above equation, ,represents a random variable whose mean is equal to zero and the variance is equal to The loss of the Central Bank is therefore highlighted as follows;

Therefore, the first order condition for optimization according to the model shall be as follows;

This shall mean that the private sector will have high expectations of the central bank as shown below;

The equilibrium level of inflation shall also be as below;

The equilibrium level of output will therefore be,

When the two values are substituted into the objective function we get the social losses of the policy as shown;

From the above scenario, since inflation is equal to zero then the expected losses would be as follows;

Evaluation of the empirical validity of inflation bias
One option that can be used to address inflation bias is through the implementation of targeting rules. This would mean that policy flexibility is restricted and hence there would be fixed exchange rates, inflation targeting, and minimal GDP and money growth rules (WALSH, 2010). If we were to include strict inflation targeting where the inflation target is and at all times, then there would be no room for inflation bias (WALSH, 2010). The loss function of the targeter is satisfied in a condition that involves deviations from (WALSH, 2010).The condition is therefore as follows;

Inclusion of the supply function and expectations gives the following condition;

Here, under pure discretion the value of the loss function becomes;

The difference in expectation is thus satisfied in the equation below (WALSH, 2010);

From the above condition it is clear that the gain from strict inflation targeting can be either positive or negative based on the premise that if there is a larger , then strict targeting will provide a positive outcome (WALSH, 2010). Furthermore, if the supply chain is volatile, then it is highly likely that a loss in flexibility will supersede the achievements in getting rid of inflation bias.

In conclusion it is vital to note that in order to control inflation, it is important for governments to pledge to work towards low inflation as it brings better outcomes for the society. Furthermore, it is also vital that policy makers in government be come up with a flexible set up that could cushion an economy through tough times (VIEGI, 2012). Lastly, we have also noted from the models and equations that money policies that are discrete in nature tend to have unsuccessful outcomes in the long run.

References
ARNON, ARIE. (2010).Monetary Theory and Policy from Hume and Smith to Wicksell: Money, Credit and the Economy. Cambridge, Cambridge University press.pp 9-20
BANQUE CENTRALE DU LUXEMBOURG,( 2006). Monetary Policy and the Inconsistency in an uncertain environment. [Online]
Available at: http://www.bcl.lu/fr/media/discours/2006/Nobelux/
[Accessed 03 April 2014].
FROYEN, R. T., & GUENDER, A. V. (2007). Optimal monetary policy under uncertainty. Cheltenham, UK, Edward Elgarpp.127
HEIJDRA, B J, (2009), The Foundations of Modern Macroeconomics, Oxford University Press.pp.200
ROC A, M. B., (2005). www.newyorkfed.org. [Online]
Available at: http://www.newyorkfed.org/research/staff_reports/sr230.pdf
[Accessed 03 April 2014].
ROMER, D. (2012), Advanced Macroeconomics, 4th edition, New York: McGraw-Hill
WALSH, C. E. (2010). Monetary theory and policy. Cambridge, Mass, MIT Press.PP.302-304
VIEGI, M., 2012. Monetary economis;The Theory of Central bank Independence. [Online]
Available at: http://www.nviegi.net/teaching/lec07.pdf
[Accessed 03 April 2014].