Macroeconomics

Econ 606

TEMPLE UNIVERSITY

Charles. E. Swanson

 

 

Time Inconsistency

 

Introduction

 

Time inconsistency is a situation that is fairly easy to describe, but fairly difficult to fully appreciate. The economics profession did not really have any idea what the nature of theproblem was prior to 1977 or until much after that how pervasive it is. Two of its earliest investigators, Finn Kydland and Edward Prescott, even confess that their original purpose in

their paper that first fully articulated the problem was to show that the matter was not very interesting.

The nature of the problem can be presented with a simple example. Consider a potential borrower who would like to borrow money to buy a house. A pair of obvious questions are: (1) would this borrower like to receive the loan, and (2) would the borrower like to not have to repay the loan. The answers are yes and yes. Unfortunately for borrowers, the second option is usually not open.

A pair of less obvious questions are: (1) at the time of borrowing, would the borrower like there to be an enforcement agency that would ensure that it repaid the load, and (2) at the time of repayment would the borrower like there to be an enforcement agency that would ensure the repayment of loans. The answer to the second question is likely to be no, for repayment is not desirable to the borrower. The answer to the first question, however, is likely to be yes. In order to get the funds to buy the house, the bank must have some reassurance that the loan will be repaid, and the enforcement agency will provide just that reassurance.

The borrower might have a rank ordering over three possible outcomes: A - no loan, no repayment, B - loan, repayment, C - loan, no repayment. C is the most preferred, while B is next. The worst outcome is A. In the borrowing period, the borrower prefers the that enforcement be present because case B is preferred to case A.

The time inconsistency is not that the borrower would prefer to not repay the loan. It is rather that in the first period (when the loan contract is written) the borrower prefers to have an enforcement agency in place while in the repayment period it prefers not to have one in place.

In general, time inconsistency refers to the situation in which some agent, planner or objective maximizers must make a choice about an action or decision in some future plan and in which what is optimal initially is no longer optimal at a later date. This change in what is optimal occurs despite the fact that nothing new is learned and no physical circumstances change, except that decisions of the past are locked in place.

 

 

 

Kydland and Prescott (Rules rather than discretion, 1977)

Kydland and Prescott illustrated time inconsistency in the context of a Phillips curve model (a setting which these two authors probably find less palatable than nearly everyone else in the profession does, but serviceable nonetheless).

A Phillips curve can be written in the form

 

u = u0 - d(p - E[p]), (1)

where p is the rate of inflation, E[p] is inflation expectations as of the beginning of the period, u is the rate of unemployment and u0 is the natural rate of unemployment. The equation says that unemployment can be reduced if inflation is raised above the level that is expected.

The social welfare function is the negative of

S = - u - c p2. (2)

Unemployment is bad and very high or very low inflation is bad. Ideal inflation in this setup is zero, although the problem could be re-worked so that ideal inflation is some positive or negative number.

Substituting (1) into (2) gives social welfare as

 

S = - u0 + d(p - E[p]) - c p2. (3)

 

The nature of the play is such that either the government commits publicly to a rate of inflation before expectations are formed, which implies that expected inflation is the rate that is committed to, or the government does not commit and expected inflation is the rate that people (correctly) believe the government will choose in the following period, given what is optimal at that time.

The commitment equilibrium is what happens if the government chooses a rate of inflation initially and is not allowed to deviated from it in the following periods. In this case, inflation is always equal to expected inflation: p - E[p] = 0. No matter what inflation rate is chosen, unemployment will be u0, so inflation should simply be set to the optimal rate which is zero.

In this case, social welfare is

S = - u0 - 0 = - u0. (4)

 

 

 

The discretionary equilibrium is what happens if the government does not commit to any particular inflation rate initially. People have to form some inflation expectation. It is natural to assume that people guess correctly, because there are no other obvious candidates. The choice of inflation under discretion is determined by differentiating (3) with respect to the choice variable p:

dS/dp = + d - 2 c p = 0 (5)

which gives

p = d/(2c). (6)

If equation (6) gives the optimal choice for the government when it comes time to making this choice, this is what people will expect the government to do. After all what else should the people expect? Thus, if 5 percent inflation is thought to be best, that is what people will expect. Therefore, expected inflation is

E[p] = d/(2c). (7)

Unemployment is u = - u0 - d(p - E[p]) = - u0, the same as before.

The difference is that under discretion, inflation is different from zero so social welfare is lower.

S = - u0 + d(p - E[p]) - c p2.

= - u0 + d(d/(2c) - d/(2c)) - c(d/(2c))2

.

= - u0 - d2/(4c) < - u0. (8)

 

Comments

1. The commitment equilibrium is better than the discretionary equilibrium. If the planner has two options: commit and have everyone observe that commitment and leave no possibility of not honoring that commitment, or do not commit and have everyone believe that next period's short term optimum will be chosen, then the planner should commit. This is often given as the reason that the Federal Reserve Board should be given some independence from Congress and the President.

2. For this particular example, if discretion prevails and commitment does not, it is best if the agency that chooses the inflation rate has a bias against inflation. A bias against inflation can be invoked by making the c in equation (2) especially high. The effect is to make the choice of inflation, given by (6), especially low. One way to reduce the effect time inconsistency is to impute into the Federal Reserve a bias toward low inflation.

 

 

3. The nature of the equilibrium should be made clear. If the planner has two choices, commitment (rules) or no commitment (discretion), the choice in this example is for commitment. However, in the second period of play, after expectations have been formed, it is best for all concerned that the commitment be broken.

The greatest social welfare can be achieved by committing to low (zero) inflation, convincing everyone that this low rate will prevail, and then reneging on this commitment and choosing inflation equal to d/(2c). This third possibility is not a rational expectations equilibrium, unlike the other two. It can be achieved if you can fool all of the people some of the time. It is difficult to see how a government can systematically promise and break promises for low inflation and still have these promises affect beliefs.

The situation is such that if the government is given the choice in the first period of whether it will be able to alter its own decision in the second period, it will choose to not have this option in period two. When period two arrives, the answer changes.

In period two, the government would like to have the option to change a decision that was made in the previous period.

4. Time inconsistency is present because the optimal choice (about whether the government should have discretion in period two) is one thing (no) in period one and something different (yes) in period two. It would be fairly easy (and therefore meaningless) to have what is optimal for the government change between periods one and two because the government's preferences changed, it learned something new, or some circumstances changed markedly. In this simple Phillips curve model of Kydland and Prescott, no such change occurs. The government has the same objective in both periods, learns nothing new and is only interested in aiding its constituents.

Calvo (1978)

Calvo presents a model with money demand. The key feature is that current money demand is a function future expected inflation. The horizon is infinite, and the fundamentals of the economy are constant throughout, although variables like the real value of money are allowed to change. The government can issue new money and use the proceeds to substitute for other tax revenue and thereby reduce aggregate tax distortions. It can also repurchase money if is desired. The questions is what happens with and without commitment.

With commitment, the government chooses one constant money growth rate for all time, and sticks with this rate. The economy operates in a fairly standard fashion, behaving much like the Sidrauski model.

 

Without commitment, there is no equilibrium. The proof takes two steps. First, Calvo shows that if there is an equilibrium, the value of money must be constant. This is fairly straightforward, since most aspects of the economy do not change through time; it is a simple matter to show that the optimal money growth rate should not change through time either. The second step is to show that

the government always has an inclination to alter the money growth rate over the immediate horizon.

If the current time is t and the government must decide on money growth between t and t+d (for some small number d), the decision will only affect money demand between time t and t+d. It will not affect money demand prior to time t because that money demand has already established itself in the annals of history. On the other hand, money demand between times 0 and t will depend on what money growth is anticipated for all future time, including the interval between t and t+d. The paradox is that a money demand decision between 0 and t depends on future money growth, but when the time comes to choose that money growth, the decision has already been made, so there is no need to take account of it when the money growth decision needs to be made.

For example, money demand in period 5 depends on money growth in period 8. In period 8, money demand decisions in period 5 have already been made. Therefore, the government in period 8 should ignore the effects of current money growth on period 5 money demand. In period 0, the government's plan for money growth in period 8 should reflect its effect on period 5 money demand, but in period 8 it should not. This is the source of time inconsistency.

Comment

The effect of time-inconsistency in the Kydland Prescott Phillips curve model is that inflation in the discretionary equilibrium is too high. The effect of time-inconsistency in the Calvo model is

that the equilibrium simply does not exist: there is no money rowth path that satisfies the simple condition that at each date the government maximizes public welfare from t onward. With Kydland and Prescott, commitment allows a superior equilibrium to be achieved. With Calvo commitment allows something non-crazy to occur.

 

 

Lucas and Stokey (1983)

This is an example of time inconsistency with respect to tax rates. This is an example of a government, acting purely in the interest of its people, that will want to have one tax schedule in period 1 and a different one in period 2. (The tax schedule that changes pertains to period 2 taxes.) The problem is the government needs to raise tax revenue in period 1 to pay for something, such as a war, but a tax causes "distortions." A lower tax rate in period 1 requires that the country borrow and this requires that it raise taxes in period 2 to pay the bondholders. However, when the next period arrives it is best to reduce the tax rate to zero. The money is all owed to the country’s citizens and the tax revenue to pay will come from the same base of citizens; they owe themselves the bondholder money. Total default is best because it eliminates the need to impose a distortionary tax in period 2. The time inconsistency then is two different plans. Plan 1 (optimal from the point of view of periods 1 and 2, taken together) calls for a small tax rate in periods 1 and 2, some bond issuing in period 1 and some bond repayment in period 2. Plan 2 (optimal from the point of view of period 2 alone) is for the tax rate in period 2 to be zero, for the government to raise no revenue with which to repay bondholders, and for total default to occur.

 

 

This argument is made in an economy that is as simple as possible, but with the essential incentives and effects of distorting taxes.

 

Consider an economy with a very simple linear production function

each period

ct + gt = nt (1)

where gt is government spending, and ct is consumer spending. The production function here is F(k,n) = n, so there is no capital in this model.

The individual maximizes

 

U(c1,1 - n1) + U(c2,1 - n2) (2)

subject to

 

c1 + b1 wn1(1 - t1) (3)

 

c1 + wn2( 1 - t2) + b1(1 + r)(1 - tk). (4)

 

 

Government spending is positive in period 1, and zero in period 2.

Its budget constraint is such that taxes plus borrowing in period 1 cover spending, g1. Since there is no government spending in period 2, all period 2 taxes are used to repay indebtedness from period 1. The government's budget constraint can therefore be written as

g1 = b1 + n1 t1 (5)

 

b1(1 + r)(1 - tk) = n2 t2 (6)

 

 

Feasibility constraints follow from the production function

 

 

c1 + g1 = n1 (7)

 

c2 = n2 (8)

 

 

 

 

 

The objective of the (benevolent) government is to choose the tax rates t1, t2, tk so as to maximize (1), the utility of individuals, while taking as given

 

i) The fact that the individuals are maximizing (1) subject to their budget constraint,

 

ii) The government budget constraints (5) and (6), and

 

iii) The feasibility conditions (7) and (8).

 

 

To solve the government's problem (to find what tax rates are optimal), it is convenient to do a bunch of algebra and show that the problem of choosing the optimal tax rates given the other conditions is equivalent to choosing optimal consumption and labor input subject to a few conditions. For this situation, Lucas and Stokey show that the government's problem is equivalent to maximizing (1), as before, subject to

 

n1Ul(1) +n2Ul(2) - c1Uc(1) - c2Uc(2) = 0 (8a)

 

 

 

n1 - c1 - g1 = 0 (8b)

 

 

n2 - c2 = 0 (8c)

 

 

where Ul(1) is the derivative of U with respect to argument 1 (consumption), evaluated at (c1,1 - n1), and similarly for the other terms.

Constraints (8b) and (8c) are the feasibility constraints rewritten. Constraint (8a), the new condition, is obtained by substituting the first order conditions into the individual's lifetime budget constraint. The purpose of recasting the optimization problem this way is that the tax rates are removed from the statement of the problem. Of course, optimal policy means the optimal choice of taxes, so at some point in solving the problem taxes have to be brought back into the analysis.

 

 

 

The Lagrangian for the government's problem is

 

L(c1,n1,c2,n2) = U(c1,1 - n1) + U(c2,1 - n2)

 

+ m[n1Ul(1) +n2Ul(2) - c1Uc(1) - c2Uc(2)]

 

+ l1[n1 - c1 - g1] + l2[n2 - c2], (9)

 

 

which can be dealt with the usual Kuhn-Tucker methods.

 

 

Optimal policy with commitment is found by using (9) to solve for the vector (c1,n1,c2,n2), and substituting these values back into the individual's first order conditions to find the tax rates. Technically, this is the reverse of what happens in the model. Behaviorally, the people take taxes as given and choose consumption and labor input, with the choice characterized by the first order conditions. Computationally, consumption and labor are taken as given and tax rates are derived from them using the first order conditions.

 

The solution to this problem, which depends on the utility function assumed, will typically involve positive labor taxes in periods 1 and 2. Intuitively, a half-size tax in both periods is better than a whole-size tax in one period. Individuals fund the government expenditure partly through their taxes and partly through the bonds they buy.

When period 2 arrives, the government does not have any decisions to make because all decisions have been made in period 1. However, if the government could alter decisions of the previous period, its problem would be characterized by the Lagrangian

 

 

 

L(c2,n2) = U(c2,1 - n2) + q[n2Ul(2) - c2Uc(2)] + a2[n2 - c2]. (10)

 

 

The optimal values of (c2,n2) derived from (10) are usually going to differ from those in (c1,n1,c2,n2) derived from (9). The time inconsistency problem arise again. In period two, the optimal thing to do is default on the debt, set the capital tax equal to 100%. The ability to commit therefore allows borrowing to occur.

 

 

Comments

 

1. One of the objectives of the Lucas and Stokey paper is to show that the time inconsistency problem can be resolved if the government can commit to the repayment of debt.

 

2. Care should be taken to state the nature of the problem. In period 2, the government, acting entirely in the interest of the public who is owed the debt, should effectively default completely on all debt obligations. Realizing this in period 1, individuals will not lend to the government unless the government forsakes its ability to optimize on their behalf in period 2 (and commits to repayment of debt). To prevent the worst of the three possible equilibria from occurring, the government must commit.

 

3. The three equilibria are:

 

(1) People expect the government to repay, they lend to the government and the government defaults in full. Obviously this is not a rational expectations equilibrium, or a Nash equilibrium.

 

(2) The government commits, people see this and lend to the government. The government would like to default in period 2 but it cannot because it has committed to repayment.

 

(3) The government does not commit. It states that it in period 2 it will act in the best interest of all individuals (who are all alike). The individual realize that this translates to defaulting on debt, so they do not lend to the government in period 1.

 

 

 

Addition Comments

 

1. The notion of time-consistency is used to justify an independent monetary authority.

 

2. Time-inconsistency states that not committing can lead to suboptimal equilibria. Committing can lead to other problems, such as committing to the wrong thing.

 

3. The time-consistency problem can be resolved in ways other than through commitment. Reputational forces of various kinds resolve the problem. These can take the form

 

a. If the government every defaults, no one will ever lend to it again.

b. Whatever the government does, people learn something about the government that they cannot directly observe. The government might repay more when its ability to repay is higher and this would signal that it is better suited to repay loans in the future.

 

 

 

 

 

 

 

A Definition

 

A policy plan is a sequence of state contingent functions {gt(ht)} t = 0, 1, 2,..., where the value of gt is a number or vector of numbers that contain tax rates, spending levels or anything that the government does. History of all events up to date t is denoted by ht. A time t continuation of a plan is the part of the plan from time t onward. A plan is time consistent with respect to some objective if

 

(1) that plan maximizes the objective, and

 

(2) for all periods t, the time t continuation maximizes the objective from time t onward.

 

 

Comments

 

  1. The definition says that if the government plans to do something, but is free at some later date to change its plans, it will not change its plans. If a proposed future policy function is optimal, it will remain optimal up to and including the date when the policy function will actually have to be implemented.
  2. A plan is time inconsistent if it is not time consistent, i.e., if the government wishes to change its plans as time passes. To repeat, a "change" in plans is not simply a matter of doing something different at a new date. It is changing what was intended for that date, conditional upon the circumstances that arise; the state-contingent action is changed.