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Working Paper 9606
COMMITMENT AS INVESTMENT UNDER UNCERTAINTY
by Joseph A. Ritter and Joseph G. Haubrich
Joseph A. Ritter is a senior economist at the Federal
Reserve Bank of St. Louis. Joseph G. Haubrich is a
consultant and economist at the Federal Reserve Bank of
Cleveland. The authors thank Avinash Dixit and seminar
participants at North Carolina State University, the
University of Pittsburgh, and the Federal Reserve Bank of
Atlanta for helpful comments.
Working papers of the Federal Reserve Bank of Cleveland
are preliminary materials circulated to stimulate discussion
and critical comment. The views stated herein are those of
the authors and are not necessarily those of the Federal
Reserve Bank of Cleveland or of the Board of Governors of
the Federal Reserve System.
Working papers are now available electronically through
the Cleveland Feds home page on the World Wide Web:
http://www.clev.frb.org.
August 1996
Abstract
Irreversible investment and the techniques associated with pricing real options have led to
significant advances many areas. We broaden this range of applications, showing how
the techniques can apply to many policy problems in finance, macroeconomics, and trade
policy. With small changes, standard techniques can handle a wide range of strategic
problems related to policy. The decision to commit is like the decision to make an
irreversible investment. Explicitly considering and correctly valuing the option to wait
makes discretion relatively more attractive, implies that greater uncertainty increases the
gain to discretion and results in policy that displays hysteresis.
I. Introduction
Irreversible investment and the techniques associated with pricing real options
have led to significant advances in capital budgeting, environmental economics, and
industrial organization. We wish to broaden this range of applications further, showing
how the techniques can apply to many game theoretic problems in finance,
macroeconomics, and trade policy. We show how, with small changes, standard
techniques can handle a wide range of strategic problems related to policy.
More specifically, we consider problems of commitment. The decision to commit
is like the decision to make an irreversible investment. The previous literature on
commitment considers a once-and-for-all choice between rules and discretion, and does
not allow future agents to adopt rules. If the option to wait indeed has positive value--as
such options often do--it adds to the desirability of discretion. Furthermore, because no
policymaker can bind itself forever, we extend the analysis to consider entry and exit; not
from production, but from commitment to a policy rule.
Our paper proceeds as follows. Section II discusses a variety of models that fit the
general framework we propose. It looks at the static games that section III embeds in
continuous time. We choose games where commitment is sometimes useful, that is ,
where the standard NCE (noncooperative equilibrium, or Nash-Cournot equilibrium)
leads to a Pareto inferior outcome. Section III provides a very general way of thinking
about policy, allowing costly commitment with costly reversal. Continuous time
highlights the analogy with irreversible investment problems, as well as simplifying the
model. We illustrate how decisions to commit or renege depend on the cost of doing so
and on uncertainty in the environment.
1
In section IV, we conclude by emphasizing three general results. First, the option
to wait, which we have restored to the policymaker's decision problem, makes
commitment less attractive and also implies that increased uncertainty makes
commitment even less so. This is the "bad news principle" of irreversible investment
applied in a policy context. Second, by allowing the commitment decision to take place
in "real time," we note that the policy choice process displays hysteresis; the policy in
force at a given time depends on history, not just the prevailing state. Third, we show that
the ability to switch regimes means that small changes in the underlying state can induce
large changes in the relevant expectations; consequently, variables sensitive to
expectations (such as asset prices) can move quickly and asymmetrically, showing a
decided nonlinearity.
II. Preliminary Examples
In this section, we present several concrete examples in which commitment
matters and regret is possible. We begin with one from bank regulation. The banking
focus also shows how to use irreversibility for policy rather than investment decisions
(see Pindyck [1991] or McDonald and Siegel [1986]).
Bank Regulation
Consider the following game between a regulator and a bank (or the banking
system). The regulator may choose to be either tough (T) or weak (W). Tough regulators
do not bail out insolvent banks; weak regulators do. A bank chooses to be safe (S) or
risky (R). If banks are truly safe, the regulator prefers to relax his vigilance, take it easy,
2
and be weak. If banks are risky, the regulator prefers to be tough. If the regulator is
tough, banks have an incentive to stay safe, but if the regulator is weak, they would
rather choose risky. The strategic form of the game then looks like (G0).
(G0) Payoffs for Game between Regulator and Banks
Banks
Regulator
Safe
Risky
Tough
0, 0
-8, -4
Weak
4, -8
-8, -7
(See Mailath and Mester [1994] or Kane [1989] for more sophisticated approaches to
closure policy, which do not, however, address the dynamic commitment problem.) The
NCE is (Weak, Risky) but both parties would prefer (Tough, Safe). The regulator can
accomplish this by committing to play tough, binding itself to play T no matter what
happens.1 With a regulator dedicated to playing T, banks will choose S. Hence the value
of commitment.2
1
The notion behind this game is that tough regulators will not bail out an insolvent bank, leading the banks to
undertake safe investments that make bail-outs unnecessary. A weak regulator will bail out the banks, and so
banks choose the more profitable risky investment; some fail, and the regulator must bail them out.
2
Those familiar with game theory may notice that this is a game in which the Row player has staying
power. In the standard classification of the 78 distinct bimatrix games, it is Brams Number 68. A similar
game, Brams Number 63, would suit our purposes as well. See Brams (1983).
3
Now lets complicate the example by bringing in the possibility of regret. So far,
the regulator is always happy about committing to be tough. Suppose, however, that in
some states of the world, the regulator regrets this. In good states, we prefer a tough
regulator who eliminates the costly wealth transfers from taxpayers to bank investors, but
in bad states, we prefer the weak regulator. Perhaps in the bad state (say a recession),
systemic risk means that being tough leads to a financial panic.
(G1) Payoff Functions for Game between Regulator and Banks
Banks
Regulator
Safe
Risky
Tough
-u2, -u2
-8-u2, -4-u2
Weak
+4- u2, -8-u2
-8-0.5u2, -7-0.5u2
For small values of u, this game has the same equilibrium as (G0), to which it
reduces when u is zero. This game has a Prisoners Dilemma flavor about it for small
values of u, in that both parties would very much prefer the Tough, Safe payoff. For large
shocks to the economy, however (that is, for large u), the Weak, Risky equilibrium
becomes preferable--perhaps reflecting that in a systemic crisis, we need to bail out the
4
banks, even if that means they make riskier investments.3 In this case, the regulator
would regret any commitment to a fixed rule of being tough.
In the next section, we derive the optimal policy when u follows a more general
process and when the regulator has a cost of committing and a cost of reneging on that
commitment, but some central insights arise if we consider a simple two-state example,
with u =0 or u=6.
Because the policy decision takes place in real time, we have two cases to
consider. Either the economy starts out in the good state, or it starts out in the bad state.
Suppose it starts out in the good state. If the regulator is weak, he gets a payoff of -8
today and chooses whether to be weak again or tough next period. If the regulator is
tough, he gets a payoff of 0 today and remains tough forever, as the only way to be tough
is to commit forever. This immediately shows where the option value enters: By being
weak today, the regulator retains the option to commit tomorrow, and this option has
value. The analogy with irreversible investment is direct.
The standard time-consistency literature, however, considers rules versus
discretion as a once-and-for-all choice. Unless the regulator commits to rules at the
beginning of time, the suboptimal or "weak" choice is made in each period. Making such
a decision forever seems simple-minded in this simple model, yet it is analogous to the
restriction implied by posing the rules-versus-discretion question in the standard way.
Drawing the analogy to investment under uncertainty highlights a flaw in the standard
approach.
3
Section III uses positive and negative shocks. In this example, it doesnt really make sense to consider
u < 0. Section III could easily accommodate one-sided shocks by using geometric Brownian motion.
5
A striking consequence of the option value--the bad news principle--also arises in
this example. We suspect this principle lies behind a tendency that is conspicuous in the
arguments over rules versus discretion. The rhetoric advocating discretion accentuates
the negative possibilities--the downside, the worst outcomes--of rules.
In this example, the bad news principle arises because the regulator sometimes
regrets the commitment to be tough. The regulator never regrets an initial decision to be
weak, since it can later commit to be tough. Increasing the payoff to toughness does not
affect the relative payoffs--and thus the choice--today. This illustrates the principle that
only news about bad outcomes affects the choice between rules and discretion.
The above formulation differs from the standard approach in a more subtle way,
necessary, but not sufficient, for irreversibility. The standard approach makes a timeless
comparison before the state of the economy is fully known. By contrast, in this paper the
government operates in "real time" and knows the current state of the economy, just as in
the irreversible investment literature the investor knows today's rate of return. Again, this
twist follows naturally from the investment analogy.
Continuing the example shows how the standard timeless comparison can lead to
the wrong conclusion by ignoring information the government can use. The standard
approach gives the regulator two strategies: Either commit to Tough or allow discretion,
which in our simple example amounts to playing Weak forever.
The regulator, though, has another possibility. Operating in real time, the
regulator can observe the economy and chose rules or discretion. If the good state turns
up, the regulator should be tough. If the bad state occurs, the regulator chooses Weak
today and chooses again next period.
6
Macroeconomic Policy
A game with a somewhat different flavor is presented in (G2).
(G2) Payoffs for Game between Fed and Treasury
Treasury
Fed
Tight
Easy
Tight
2-u2, 1-u2
-3u2, -3u2
Easy
-3u2, -3u2
0.5-0.5u2,1.5-0.5u2
This is a version of the game known as Chicken or Battle of the Sexes. Its
clearest macroeconomic interpretation was presented by Sargent (1986), who argued in
Reaganomics and Credibility (1986) that tight monetary policy is compatible with tight
fiscal policy but not with easy fiscal policy. Who gives in and accommodates the others
policy, the Fed or the Treasury? In (G2), such a conflict exists for small values of u, but
easy policy is better for large shocks, and indeed forms a Nash equilibrium. This captures
the intuitive idea that, for a massive real shock, easy policy is better. By committing to
Tight, the Fed can enforce its preferred equilibrium, but it regrets this choice in times of
large shocks.
Pindyck (1977) considered such a coordination problem in greater depth,
analyzing a dynamic game between the fiscal and monetary authority, each of which has
a different objective in controlling the economy. He did not consider the irreversibility
aspect of policy choice.
7
In Haubrich and Ritter (1996), we analyze commitment to monetary rules in the
traditional time inconsistency setup (Barro and Gordon, [1983]). A third monetary policy
application derives from observations about the fragility of fixed-exchange-rate regimes.
Obstfeld and Rogoff (1995) lay out the following case: (1) Maintaining a fixed exchange
rate is technically feasible for almost any country; (2) under normal circumstances,
countries gain (or think they gain) from fixing their exchange rate; but (3) the collateral
damage caused by an attempt to defend the peg, when threatened by a terms-of-trade
shift or some other shock, means the government's commitment to its rate may not be
credible. Thus, even the strongest legal commitments to fixed exchange rates--currency
boards, for example--will not always succeed (Zarazaga [1995]). Nevertheless, despite
compelling arguments that they will ultimately fail, countries continue to adopt fixedexchange-rate policies. We describe a framework that can provide a positive theory of
the switches between policy regimes.
Trade Policy
Some insight into the dynamics of trade agreements might be gained from (G3). The
players are countries, say Argentina and Brazil. Each chooses between high and low
tariffs. The noncooperative equilibrium of the game is high tariffs in both countries. Both
of them would ordinarily gain by coordinating on low tariffs, and this outcome can be
achieved by establishing a free trade area, that is, by committing. But when Brazil
experiences a recession, measured by its unemployment rate u, its Argentine imports fall,
tempting Argentina to leave the free trade area and raise tariffs. Brazil responds by
raising its own tariff rate. Because Brazil sees reduced imports as an advantage, its
8
payoffs for this game are increasing in u. We assume that Argentina's economy is stable,
and that Brazil always stands open to the trade pact, so that Argentina effectively decides
the extent of free trade. A fully satisfactory model here would clearly involve more
symmetry and give Brazil incentives for breaking the trade pact as well. We include it to
illustrate the range of problems our approach can address.
(G3) Payoffs in Free Trade Game
Brazil
Argentina
Low
High
Low
8-u2, 8+u2
-2, 9+3u2
High
9-u2, -2
0,0
As mentioned before, the payoff structure of examples (G1) , (G2), and (G3) has
more general applications. The tractability of the quadratic model makes it a natural
approximation for many commitment problems (and for many other economic problems.
Thus, we could illustrate our main point with such additional examples as adhering to the
Gold Standard (with regret in a war or depression), granting patents for the exclusive use
of new technology (with regret in cases like AZT), or allowing constitutions to bind future
legislatures.4
4
In fact, the tractability constraint does not bind us exclusively to quadratic payoffs. From our perspective,
the more binding constraint in the continuous time models was the need to posit an essentially static
underlying game. We conjecture that, with sufficient mathematical expertise, this need not be a binding
constraint either.
9
III. Entering and Exiting Commitment
Mechanisms to commit irrevocably are almost impossible to imagine. It is not
difficult, however, to cite examples of mechanisms that make it costly for a firm or a
government to alter its policy. A constitutional amendment, for example, is difficult to
pass and to remove. Ordinary legislation has lower costs at both ends. For a firm, the
corporate charter, financial agreements, and strategic plans play a similar role.
Institutions can effectively tie their hands loosely or tightly, being able to escape if they
are willing to bear the appropriate level of pain. For any particular decision, these costs
can usually be considered as given: passing a law, amending the constitution, issuing a
regulation. In future work, we hope to make the choice of commitment mechanism
endogenous.
We maintain the traditional semantics of commitment and discretion, but wish to
highlight a bias in tone that colors the discussion when commitment is not irrevocable.
We are forced to use words (like renege and weasel) having clear negative
connotations that we regard as unfortunate. We interpret the results of this section as a
model of optimal behavior and tolerate the terminology only to fit our paper into the
literature on rules and discretion.
A world in which policymakers can, at a cost, enter and exit commitment (or,
more generally, any policy regime) closely resembles Dixit's (1989a) model of the entry
and exit problem faced by competitive firms.
10
While a discrete time approach can sometimes handle particular versions of the
problems (as Lambson [1992] does for entry-exit decisions),5 the continuous time
approach set out in Dixit and Pindyck (1994) generally proves more convenient. A
generic quadratic payoff model captures the main points in a context that is simple yet
general.
A Quadratic Model
We implement these ideas in continuous time as follows: The policy authority
(the Fed, for example) will be following either rules or discretion. The payoff from
discretion, which depends on the state of the economy u, is
(1) P D (u) = d 0 + d1 u + d 2 u 2 .
The payoff from rules is
(2) P R (u ) = r0 + r1u + r2 u 2 .
Following examples (G1) and (G2), and by analogy with the previous sections, we assume
that rules tend to be preferred when the shock is small, so that for small u,
P R (u ) > P D (u ). We assume that u follows a simple Ito process
du = αdt + σdz ,
where a describes the drift of the process and s denotes its standard deviation, with dz
describing a white-noise Wiener process.
The optimal policy switches between the two quadratic payoff functions with cost
C of committing to rules, that is, of moving from discretion to rules, and cost W (for
5
For some specialized problems, the discrete time approach is more natural. One workhorse of the dynamic
inconsistency literature in macroeconomics, the unanticipated money model, does not easily generalize to
11
weaseling) of moving from rules to discretion.6 To solve this, we employ the general
methods of Dixit and Pindyck (1994). Our problem maps most naturally into an entry
and exit problem. Unlike the problem for firms, where uncertainty over prices is best
modeled by geometric Brownian motion, for many problems two-sided shocks are more
natural and therefore are best modeled with Brownian motion, which may turn negative.
Weather, oil price shocks, trade flows, and interest rate shifts may all take positive or
negative values. Consequently, where Dixit and Pindycks problem has two boundaries,
one price at which the firm enters the market and another price at which the firm exits,
our problem has four boundaries: two above zero and two below zero.
In what follows, we derive the differential equations for the value functions, and
derive the smooth pasting and value matching conditions necessary for the optimum of
this stochastic control problem. The conditions give us the necessary equations to solve
numerically for the boundaries between the rules region and the discretion region. Full
details can be found in the appendix.
In the interior of the discretion region, the value function for the problem obeys
rV D = P D (u) +
1
E[ dV D ] .
dt
We apply Itos Lemma to find the differential equation for the value function
1
2
σ 2 Vuu D + αVuD − rV D = − P D .
A similar argument for the interior of the rules region yields the following, in which
subscripts denote partial derivatives:
continuous time. We examine it in a companion paper (Haubrich and Ritter [1995]).
6
Allowing weaseling adds a component similar to the escape clause models of Flood and Isard (1988) and
Lohman (1992), who consider a cost to renege. In one sense, we generalize those models by allowing a
12
1
2
σ 2Vuu D + αVuD − rV D = − P R .
Each of these is a second-order linear differential equation, and standard solution
techniques are available.
In the solution, there are three regions: a rules regions centered about zero for
small shocks, and discretion regions for large positive and large negative shocks. This
necessitates three solutions to the equations, depending on which region we are in. Each
solution takes the form of a general solution plus S R or S D , a quadratic particular
solution to the differential equation.7 For the rules region the solution is
(3) V R (u) = B1 e
β 1u
+ B2 e
β 2u
+ S R (u ) ,
with β 1 > 0 and β 2 < 0 .
For the high (positive) discretion region, we have the corresponding solution
V D (u) = A1h e
β1u
+ A2 h e
β 2u
+ S D (u ) .
The particular solution S D (u) turns out to be the value of discretion forever, so that the
two exponential terms are the value of the option to commit. (See also Dixit and Pindyck
[1994, chapter 6, section 2].) For very large shocks u approaching infinity, it becomes
exceedingly unlikely that the regulator will ever commit (recall that it prefers discretion
for large shocks), and so the value of that option approaches zero. This means the term
with the positive exponent, β 1 , must vanish for large u, implying that A1h must be zero.
This leads to the simplified expression for the value function in the high (positive)
discretion region:
positive cost of recommitment and allowing delay in recommitment. In another sense, those models are more
general, in that they allow more general state-contingent rules. We prefer to focus on the dynamics.
7
The particular solution is all that would change if we used a form of costs other than quadratic.
13
(4) VhD (u ) = A2 h e
β 2u
+ S D (u) .
After employing a similar argument for the lower (negative) discretion region, we have
(5) Vl D (u ) = A1l e 1 + S D (u) .
βu
Four boundaries define the regions. Two boundaries determine when the
regulator weasels out of rules and adopts discretion, one at the upper boundary u W and
one at the lower boundary u W . The other two boundaries determine when a discretionary
regulator commits to rules, entering the commitment region from above, u C , or from
below, u C .
With the general form of the value function in hand, we can find the boundary
values by imposing the value-matching and smooth-pasting conditions. For example, at
the upper commitment boundary, the value of continuing in discretion just equals the
value of adopting rules and paying the cost to commit:
(6) VhD (u c ) = V R (u c ) − C .
Likewise, the smooth pasting conditions impose equality on the derivatives of the value
functions:
(7) Vh′ D (uc ) = V ′ R (uc ) .
This is repeated for each boundary, producing eight equations (one value matching and
one smooth pasting condition for each boundary) in eight unknowns (four boundaries and
four undetermined coefficients). The appendix sets out these equations and proves the
existence and uniqueness of the solution.
Numerical Solution and Comparative Statics
14
As frequently happens in the stochastic control literature, closed-form solutions do
not seem to exist for this problem, and we resort to numerical methods. Gauss NLSYS
was able to solve the eight simultaneous equations, though convergence of the algorithm
was sensitive to starting values. The actual numerical solutions are less interesting than
the comparative static results. Starting from a base case of C = W = 0.01, α = 0 ,
σ 2 = 0.01 , and r =0.02, figures 1 through 4 depict the solutions under a variety of
parameter variations.
Figure 1 highlights the importance of history. It shows a solution and one sample
path for the shocks, the commitment and weasel boundaries, and uses shading to indicate
the time spent committed to rules. Because the weasel and commit boundaries differ, in
some states of the economy (levels of u) current policy depends on past policy. For
anything above the upper commit line and below the upper weasel line, a regulator
committed to rules sticks with rules and a regulator using discretion sticks with discretion.
Quite apparently, then, it is incorrect to judge policy simply on the current state of the
economy, and particularly inappropriate to naively contrast current policy with past
policies at a similar state of the economy or stage of the business cycle. In a word, our
model predicts policy hysteresis. This shifting reemphasizes a point stressed by Flood and
Garber (1984) in their work on the gold standard: To evaluate a policy rule, one must
analyze the entire dynamic policy sequence, including periods where discretion reigns.
Implicit in the hysteresis is something so obvious that it might escape notice--that
the policymaker switches from rules to discretion, and from discretion to rules, over time.
Regimes shift. Discretion, commitment, and weaseling out of commitment will all occur.
15
Figure 2 plots the commitment and weasel boundaries as the commitment cost
changes, keeping the weasel cost fixed at 0.01. Notice that for any particular
commitment cost, the regulator adopts rules for small shocks on either side of zero. For
larger shocks, the Fed adopts discretion. This is a natural consequence of the quadratic
payoff function.
Another prominent feature is that the weasel boundary is farther out than the
commit boundary. Were there no cost of switching between regimes, the boundaries
would be the same, at VhD (u ) = V R (u ) , where the expected gain from continuing
discretion just matches the expected gain from using rules.8 Adding a commitment cost
drives a wedge between the two value functions and requires that the regulator gain even
more from rules. This means moving the boundary farther into the area where rules are
preferred, that is, closer to zero. Similarly, a cost to backing out of rules means shifting
the boundary even farther into the area where discretion is preferred, that is, away from
zero. Hence the weasel boundary is farther out than the commit boundary.
Figure 2 shows that the greater the cost of commitment, the less likely the
regulator is to commit. As the cost increases, the relative benefits of rules over discretion
must also increase, and so the commitment boundary shrinks towards zero. For a high
enough cost, commitment never occurs.
One other more practical advantage of the continuous-time formulation lies in its
ability to allow easy exploration of a broad range of questions, like changes in entry and
exit costs and variability of shocks.
8
In the zero-cost case, first-order conditions (value matching and smooth pasting) have multiple solutions,
including the solution to the original problem. For all positive costs, the solution is unique.
16
Figure 3 illustrates what happens when the weasel cost varies. As the cost of
switching out of rules rises, it takes an increasingly large benefit of discretion over rules to
make the switch worthwhile, and so the weasel boundary increases. Notice that in figures
2 and 3, a rise in commitment cost primarily moves the commitment boundary, and a rise
in weaseling cost primarily moves the weasel boundary. This reflects the relatively low
variance of u. At the commitment boundary, it is improbable that the process will soon
wander as far as the weasel boundary, and so this has little weight in the optimization
problem, making the weasel boundary almost perfectly flat. When the boundaries are
close, as for small values of C and W, both boundaries move more noticeably with an
increase in either cost. A higher variance for u makes the effect more pronounced.
Figure 4 illustrates a different exercise, in which the variance of the Brownian
motion governing the shocks is increased. As the variance rises, the commitment
boundaries decrease and the weasel boundaries increase. This is a consequence of the
options component of the decision. As the variance rises, so too does the option value of
not switching. For example, in the discretion region, a high variance means there is a
good chance of moving deeper into that region in the near future, but also a good chance
of moving into the rules region. The bad news principle enters here. Ending up deep in
the discretion region means regretting the commitment to rules. Ending up deep in the
rules region means committing to rules when you get there, so committing today doesnt
help. Thus, the high variance makes commitment less likely, and the commitment
boundary decreases correspondingly. With a high enough variance, the regulator never
commits.
Expected Time in Regime
17
Since we propose a model with discrete regime shifts, how long the current policy
regime is expected to last is critical in applications like asset pricing, where agents must
look into the future. For example, monetary policy conducted under discretion may result
in a higher inflation rate than policy conducted under rules. Most bond traders--and
academics studying the term structure of interest rates--concede the influence of
monetary policy. Most would also concede frustration in understanding that influence.
At times modest increases in the federal funds rate lead to sharp increases in long rates; at
other times modest changes provoke modest changes.9 Thinking about policymakers as
entering and exiting commitment, with its associated nonlinearities and hysteresis, can
shed some light--and one day may even yield some quantitative evidence--on the matter.
To obtain an idea of how the expected time in a regime behaves, we set up the
following simulation. We let the underlying shock follow Brownian motion with a
variance of 0.1. We sampled this process 120 times at monthly intervals, assuming
commitment boundaries of +2 and -2 and weasel boundaries of +3 and -3. This was
meant to capture the idea that the policymaker periodically, but not continually, reviews
policy based on the indicators of the underlying economy. For a given starting point, we
generated 1,000 runs of the Brownian motion path, keeping track (by month) of when the
path was in the rules and the discretion region (which is obviously path dependent).
Averaging over the 1,000 runs gives an estimate of the expected fraction of time spent in
each region over the next 10 years. Figure 5 reports the results. The X-axis shows the
starting value for the simulation, and the Y-axis shows the fraction of time spent in
discretion. For example, if the current value of the underlying shock is 1.5, the expected
9
See Goodfriend (1993) or Campbell (1995) for amplification of this point.
18
fraction of time in discretion is only 0.04 ( or 4.8 months out of 10 years). In other words,
the amount of time expected to be spent in discretion over the next 10 years is trivial,
given a starting point this far into the rules region. The figure reports two numbers for
starting values between 2 and 3, depending on whether the starting value is assumed to be
in the rules or the discretion region.
If, as mentioned above, the discretion regime results in a higher inflation rate, the
data shown in figure 5 can easily be translated into a numerical inflation premium. Say
that rules produce zero inflation and discretion produces constant inflation of 10 percent.
Then the average expected inflation over the next 10 years is 0.4 percent when the
underlying state is 1.5, but it rises rapidly thereafter.
Figure 5 emphasizes and quantifies the importance of hysteresis for forwardlooking variables. For a starting value of 2.5 in the rules region, the policymaker expects
to be in discretion only about one-third of the time over the next 10 years. If that same
value of 2.5 is in the discretion region, the corresponding number is about two-thirds.
This implies that expectations are asymmetric during increases and decreases of the
shocks. Equally important, expectations can change quickly once the shocks approach a
boundary. The expected time in discretion changes from 0.001 to 0.009 in moving from 0
to 1, but changes from 0.16 to 0.86 in moving from 2 to 3. The relation between the
underlying shock and the result is decidedly nonlinear.
These results imply that inflationary expectations--and thus long-term interest
rates--can change dramatically without a shift in policy, as people anticipate that a new
policy regime is more likely. These shifts depend sensitively on the underlying state of
the economy and on the current policy regime.
19
Conclusion
Viewing commitment as irreversible investment has two major advantages: It
provides a new perspective on questions of commitment, rules, and discretion, clearing up
some troubling aspects of the literature. Equally important, that perspective represents a
useful new direction for the irreversible investment literature. It applies quite naturally to
strategic interactions--games--without the need for drastic revision. Though we dont
wish to downplay the difficulties arising in each specific case, such as dealing with
different stochastic processes or multiple boundaries, the basic concepts and techniques
of investment under uncertainty gain a wider applicability.
Thus, besides providing new answers to old questions, this approach also raises
new questions. By making the commitment versus discretion problem more amenable to
attack by the techniques of financial economics, a new set of tools (and problems)
naturally arises. For example, policy commitment should matter for asset prices; consider
a shift in monetary policy, a poison pill being activated, or a shift in bank closure policy.
Conversely, asset prices may allow us to estimate commitment probabilities and other
fundamentals of the model. What this means that we have a powerful set of techniques
ready to address significant questions in banking, finance, and economics.
20
Appendix : Theoretical Solution
1. Solving for the Value Functions
This part of the appendix solves the differential equations of section III to find the
value functions. For reference, those two equations are
1 2 D
σ Vuu + α VuD − rV D = − P D
2
for the interior of the discretion region and
1 2 R
σ Vuu + α VuR − rV R = − P R
2
for the interior of the rules region.
Both are equations of the form
ay²(x) + by¢ (x) + cy(x) = q0 + q1x + q2x2 º Q(x).
The solutions to the homogenous part are
y( x ) = A1e β1x + A2 e β 2 x ,
where bi are solutions to the characteristic equation
al2 + bl + c = 0.
Since c < 0 in our application, we have one positive and one negative root. Let
β 2 < 0 < β 1 . The particular solution can be a quadratic:
y(x) = s0 + s1x + s2x2
y¢ (x) = s1 + 2s2x
y²(x) = 2s2.
Substituting yields
q0 + q1x + q2x2 = a(2s2) + b(s1 + 2s2x) + c(s0 + s1x + s2x2)
= (2as2 + bs1 + cs0) + (2bs2 + cs1)x + (cs2)x2.
Matching coefficients yields
s2 =
s1 =
q2
c
q1 − 2bs 2
c
21
s0 =
q 2 − bs1 − 2as2
.
c
Since Q(x) is either -PD or -PR, we have one particular solution for discretion and one for
rules (call them S D and S R).
There are three regions: high-u discretion, low-u discretion, and rules. Take these
in order. For high-u discretion the solution is
V D( u) = A1h e β 1u + A2 h e β 2 u + S D(u ) .
Substituting s0, s1, and s2 into the quadratic particular solution makes it clear that S D(u)
turns out to be the value of discretion forever, so the other terms are the value of the
option to commit. (See Dixit and Pindyck [1984, chapter 6, section 2]). As u® ¥ this
option becomes worthless, so we need to have A1h = 0 (since b 1 > 0). So our solution is
equation (4) of the paper):
V hD( u) = A2 h e β 2 u + S D (u) .
For low-u discretion we need A21 = 0. Otherwise, the value option to commit explodes as
we get farther in the negative direction from the point at which we would want to commit.
So, in the low-u discretion region, we have equation (5) of the paper:
V lD( u) = A1l e β1u + S D (u) .
The rules region is bounded, so neither option term drops out, and the solution is equation
(3) of the paper:
V R(u ) = B1e β1u + B2 e β 2u + S R(u) .
The value function must also satisfy the following value-matching and smoothpasting conditions:
V
D
h
( u c ) = V R (u c ) − C
V
D
l
( u c ) = V R (u c ) − C
V R (u w ) = V
D
h
(u w ) − W
V R (u w ) = V
D
l
(u w ) − W
′
V
D
h
V
D
l
′
(u c ) = V
R
′
(u c ) = V
R
′
22
(u c )
(u c )
V
R
V
R
′
′
(u w ) = V
D
h
′
(u w )
′
(u w ) = V lD (u w ) /
We have eight equations and eight unknowns: A1l, A2h, B1, B2, u c , u c , u w , and u w .
2. Existence and Uniqueness of Solutions
To establish the existence and uniqueness of the solution, we use a variation on
the approach used by Dixit (1989b, unpublished appendix).
Preliminaries
First, we define two functions that measure the difference between the value
functions (analogues of Dixits G(P) function) for the upper and lower boundary pairs:
H(u ) = V R (u) − VhD (u )
= B1e β1u + B2 e β 2u + S R ( u) − A2 h e β 2 u − S D (u)
= B1e β1u + D2 e β 2 u + Q(u) and
L(u ) = V R (u ) − Vl D (u )
= B1e β1u + B2 eβ 2 u + S R (u ) − A1l eβ 1u − S D (u)
= D1e β 1u + B2 e β 2 u + Q(u) ,
where D1 = B1 - A1l, D2 = B2 - A2h, and Q(u) = S R (u ) − S D (u ) . S R and S D are the
particular solutions for the differential equations that lead to the value functions.
Next, we need to establish that Q(u) is convex. Convexity follows from our
assumption that the rules loss function is more convex than the discretion loss function
and from the formulae for S R and S D .
The introduction of Di separates the problem of finding the upper boundaries from
that of finding the lower boundaries. Without loss of generality, we consider only the
upper boundaries, concentrating on the function H(u;B1,D2). Where there is no chance of
confusion, we suppress the dependence of H on its parameters and write H(u).
Existence
23
Consider the upper boundaries. We define a sequence of functions Hi (and the
corresponding B1i and D2i ), which converge to a function H that which satisfies the
smooth-pasting and value-matching conditions. Let
H0(u) = Q(u).
Keeping D2 = 0, set B11 so that
H1 (u) = B11 e β 1u + H0 ( u)
is tangent to the horizontal line at +W. This can be accomplished by some B11 < 0,
because H1′ (u) increases without bound as we increase B11 and decreases without bound
as we decrease B11 . This produces a local maximum, since B11 < 0. (Note that we cannot
start with D2, because Q may not intersect -C.)
Now let
H2 (u ) = D22 e β 2 u + H1 (u ) .
H2 is increasing in D22 , and H2′ is decreasing (since b2 < 0) without bound in D22 .
Increase D22 to make H2 tangent to -C. This will be a local minimum. Notice that this
puts H2 above H1 at the point where H1 is tangent to +W.
Now let
H3 (u ) = B13e β1u + H2 (u) .
Decrease B13 to restore tangency with +W. Continue this process, thus generating the
sequence.
Note that Hi goes off to +¥ to the left of the tangencies and off to -¥ to the right,
as illustrated in figure A1. At each stage of this construction, there is an increasing
segment of Hi to the right of the local minimum and to the left of the local maximum. Let
{B1i, D2i} be the accumulation of the B1i and D2i in Hi. We have shown that this
sequence is always moving northwest in B1 - D2 space. This sequence cannot, by
construction, go into a region where Hi′(u ) < 0 for all u.
To show convergence, we need to bound the {B1i, D2i} sequence. Notice that
both exponential terms are downward-sloping, so we can find bounds on B1i and D2i
separately. The only interval on which H could possibly be increasing on the B1 steps
24
(that is, Hi for odd i) is between the minimum of Q, denoted by uQ, and the largest
solution to H1′( u) = β 1 B11e β1u + Q′ (u ) = 0 , denoted by u*. (There are generally two
solutions, because Q¢(u) is linear while −β 1 B11e β1u is convex. See figure A2.) We know
that u Q < u * , because β 1 B11e β 1u > 0 and B11 was chosen so that Hi(u*)=0. Hence,
β 1 B11 e β1u intersects Q(u). See figure A1.
A simple bounding argument will eliminate the possibility of an increasing H, even
on this interval. For uÎ [uQ, u*], we have that Hi′(u) < 0 for u > u* and i > 1:
Hi′(u ) = β 1 B1i e β1u + β 2 D2i e β 2u + Q′(u )
[
]
< β 1 B11e β1u + Q′(u ) + β 2 D 2 i e β 2u
< β 1 B11e β1u + Q′(u )
< β 1 B11e β1u* + Q′ (u*)
= H1′ (u*) = 0 .
The first inequality comes from the fact that {B1i } is a decreasing sequence of negative
numbers. The second follows from the fact that D2 i > 0 with b2 < 0. The third comes
from the fact that −β 1 B11e β 1u cuts Q¢(u) from below at u*, so that both increase on [uQ,
u*]. Again, see figure A2. On [uQ, u*], e β 1u and e β 2 u are minimized and maximized,
respectively, at uQ, since both are monotonic. Similarly, the slope of Q is maximized at
u*.
Hence, there can be no increasing portion of Hi(u) if
β 1 B1 e β 1u + Q ′ (u*) < 0
for all u Î [uQ, u*]. This condition holds if
B1 <
−Q′( u*) − Q′(u*)
= negative constant.
Q
βu <
β 1e 1
β 1e β 1u
Similarly, for the D2 steps (Hi for even i) there can be no increasing portion of
Hi(u) on [uQ, u*] if
β 2 D2 e β 2 u + Q′ (u*) < 0
or
25
D2 <
−Q′(u*) − Q′(u*)
= positive constant.
Q
β u <
β 2e 2
β 2 eβ 2 u
Therefore, the sequence (which is moving northwest) is bounded in the region
0 > B1 > negative constant, 0 < D2 < positive constant.
Uniqueness
Recall that the definition of D2 above reduced the problem to separate sets of
four equations and four unknowns, two boundaries (two values of u) and two
undetermined constants. The uniqueness proof first shows that, for any given value of the
constants, the boundaries are unique, and then shows that the constants are unique.
Define u c ( B1 , D2 ) and u w ( B1 , D2 ) as the respective values of u where the local
minimum and maximum of H(u;B1,D2) occur. First, we show that there can be only one
minimum u c ( B1 , D2 ) , and one maximum u w ( B1 , D2 ) for H, given B1 and D2.
Lemma: For given values of B1 < 0 and D2 > 0, H ′ (u; B1 , D2 ) = 0 has at most
three solutions.
Proof: Write H¢ = 0 as
− Q ′( u) = β 1 B1 e β 1u + β 2 D2 e β 2 u .
Since Q is convex, the LHS is a decreasing line. The RHS is downward-sloping, convex
to the left, and concave to the right -- like a cotangent function. Obviously, there will be
no more than three solutions. Û
Given the shape of H--that is, lim u→−∞ H (u) = +∞, lim u →+∞ H(u ) = −∞ (again see
figure A1)--solutions to H¢(u)=0 come in pairs. Thus, have more than one minimum and
one maximum, we would need at least four solutions to H¢ = 0. But the lemma shows that
we can have at most three, and since we have already proven existence, we know that
exactly two solutions exist, a unique maximum and a unique minimum. This implies that
u c ( B1 , D2 ) and u w ( B1 , D2 ) are well defined, single-valued functions.
To complete the proof, we show that B1 and D2 are unique. The proof proceeds
by contradiction:
Define
26
Γc ( B1 , D2 ) ≡ B1e β1 uc ( B1, D2 ) + D2e β 2 uc ( B1D2 ) + Q(u c ( B1 , D2 ))
Γw ( B1 , D2 ) ≡ B1e β1 u w ( B1 D2 ) + D2 eβ 2 u w ( B1 D2 ) + Q( u w ( B1 , D2 ))
In this notation, the value-matching conditions are
Γc ( B1 , D2 ) = − C
Γw ( B1 , D2 ) = W .
Also,
∂ Γc
∂ uc
= Hu
+ H B1
∂ B1
∂ B1
= H B1
= e β 1 u c ( B1 ,D2 ) .
Hu = 0 because u c ( B1 , D2 ) and u w ( B1 , D2 ) are chosen so that the smooth-pasting
conditions hold when H is evaluated at u c ( B1 , D2 ) or u w ( B1 , D2 ) . Similarly,
∂ Γc
= e β 2 u c ( B1 ,D2 )
∂ D2
∂ Γw
= e β1 u w ( B1 , D2 )
∂ B1
∂ Γw
= e β 2 u w ( B1 , D2 ) .
∂ D2
Now we show that a second solution cannot exist. Note that if ( B1′, D2′ ) is a
second solution to the value-matching and smooth-pasting conditions with B1′ > B1 , we
must have D2′ < D2 to maintain the value-matching conditions: Γc ( B1′, D2′ ) = − C and
Γw ( B1′, D2′ ) = W .
Let b = B1′ − B1 > 0 and d = D2′ − D2 < 0 . The line segment joining the solutions
is ( B1 + tb, D2 + td ) . We have
∂Γ
∂ Γc
d
Γc ( B1 + tb, D2 + td ) = b c + d
∂ B1
∂ D2
dt
= be β 1 uc ( B1 + tb, D2 + td ) + de β 2 uc ( B1 + tb, D2 + td ) .
Given our hypothesis that Γc ( B1 , D2 ) = Γc ( B1′, D2′ ) = C ,
27
0 = Γc ( B1′, D2′ ) − Γc ( B1 , D2 ) .
1
(
)
= ∫ be β1 uc ( B1 + tb, D2 + td ) + deβ 2 u c ( B1 + tb, D2 + td ) dt .
0
Similarly,
0 = Γw ( B1′, D2′ ) − Γw ( B1 , D2 )
1
= ∫ [be β 1 u w ( B1 + tb ,D2 + td ) + de β 2 u w ( B1 +tb , D2 +td ) ]dt .
0
Again, because of the shape of H,
u c ( B1 , D2 ) < u w ( B1 , D2 ) ⇒ e β 1 uc ( B1 , D2 ) < e β 1 u w ( B1 , D2 ) and e β 2 u c ( B1 , D2 ) > e β 2 u w ( B1 , D2 ) .
Recall that b > 0 and d < 0. Subtracting the two integrals, we get
1
[(
)]
) (
0 = ∫ b e β1 uc ( B1 , D2 ) − eβ 1 uw ( B1 , D2 ) + d e β 2 uc ( B1, D2 ) − e β 2 u w ( B1 , D2 ) dt .
0
The integrand is always negative, so the integral cannot be 0. That is, both solutions
satisfy the value-matching conditions only if they are identical.Û
Thus, B1 and D2 are unique, and so uniquely define u c ( B1 , D2 ) and u w ( B1 , D2 ) ,
making the entire solution unique.
28
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