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Real Effects of Price Stability with
Endogenous Nominal Indexation∗
C´esaire A. Meh
Bank of Canada
Vincenzo Quadrini
University of Southern California
Yaz Terajima
Bank of Canada
October 29, 2008
VERY PRELIMINARY AND INCOMPLETE
Abstract
We study a model with repeated moral hazard where financial
contracts are not fully indexed to inflation because nominal prices are
observed with delay as in Jovanovic & Ueda (1997). More constrained
firms sign contracts that are less indexed to the nominal price and, as
a result, their investment is more sensitive to nominal price shocks.
We also find that the overall degree of nominal indexation increases
with the uncertainty of the price level. An implication of this is that
economies with higher price-level uncertainty are less vulnerable to a
price shock of a given magnitude, that is, aggregate investment and
output respond to a lesser degree.
1 Introduction
This paper studies how nominal price-level uncertainty affects the real sector
of the economy in a model in which optimal financial contracts are not fully
indexed to inflation. The important feature of the model is that limited
∗The views expressed in this paper are those of the authors. No responsibility should
be attributed to the Bank of Canada.
1
indexation is not imposed by assumption but is determined endogenously
as part of the optimal contract. This allows us to study how the degree of
indexation depends on the properties of the monetary policy regime and how
different regimes affect the response of the economy to nominal price shocks.
The model features entrepreneurs who finance investment by entering
into contractual relations with financial intermediaries. Because of agency
problems, the contracts are constrained optimal. We follow Jovanovic & Ueda
(1997) and assume that the aggregate nominal price level is observed with
delay, after resolving the agency problem. Bullard (1994) provides evidence
that there is a substantial time lag before the aggregate price level becomes
public information.1This timing lag creates a time-inconsistency problem in
the optimal long-term contract which leads to renegotiation.
We first characterize the optimal long-term contract in which the parties
commit not to renegotiate in future periods. The contract is fully indexed,
and therefore, inflation is neutral. After showing that the long-term contract
is not immune from renegotiation, we characterize the renegotiation-proof
contract. A key property of the renegotiation-proof contract is the limited
indexation to inflation, that is, real payments depend on nominal quantities.
A consequence of this is that, unexpected movements in the nominal price
level have real consequences for an individual firm as well as for the aggregate
economy.
The central mechanic of transmission is the debt-deflation channel. An
unexpected increase in prices reduces the real value of nominal liabilities im-
proving the net worth of entrepreneurs. The higher net worth then facilitates
investments and leads to a macroeconomic expansion.
This result can also be obtained with a model in which we impose ex-
ogenously that the only source of funds for entrepreneurs are nominal debt
contracts. However, with this simpler framework, we would not be able to
study how different monetary policy regimes or policies affect the degree
of indexation, and therefore, how the economy would respond to inflation
shocks given the prevailing monetary policy regime.
Although the basic theoretical foundation for limited indexation is simi-
lar to Jovanovic and Ueda, the structure of our economy and the questions
addressed in the paper are different. First, in our environment all agents are
risk neutral but they operate a concave investment technology. Therefore,
1According to Bullard it takes about a year before the GDP deflator is reliably mea-
sured.
2
the role that the concavity of preferences plays in Jovanovic and Ueda it is
now played by the concavity of the investment technology. Second, we con-
sider agents that are infinitely lived, and therefore, we solve for a repeated
moral hazard problem. This allows us to study how inflation shocks impact
investment and aggregate output dynamically over time. It also allows us
to distinguish the short-term versus long-term effects of different monetary
regimes. Third, in our model entrepreneurs/firms are ex-ante identical but
ex-post heterogeneous. At each point in time, some firms face tighter con-
straints and invest less while other face weaker constraints and invest more.
This allows us to study how inflation shocks impact firms at different stages
of growth.
There are several findings we are able to show within this framework. The
first finding is that the optimal contract allows for lower nominal indexation
in firms that are more financially constrained (and tend to be smaller in
size). As a result, these firms are much more vulnerable to inflation shocks.
This finding is also relevant for cross-country comparisons. More specifically,
a country with less developed financial markets is likely to have a larger
share of firms with tighter financial constraints. Thus, controlling for the
monetary regime, the economies of these countries are more vulnerable to
inflation shocks.
The second finding is that the degree of nominal price indexation increases
with the degree of nominal price uncertainty. This implies that the impact
of a given inflation shock is bigger in economies with lower price volatility
(since contracts are less indexed in these economies). On average, however,
economies with greater price uncertainly also face larger shocks on average.
Therefore, the overall aggregate volatility induced by inflation shocks is not
necessarily smaller in these economies. In fact, we show in the numerical ex-
ercise that the relation between inflation uncertainty and aggregate volatility
is not monotone: it first increases and then decreases.
To the extent that price-level uncertainty depends on the particular mon-
etary policy regime chosen by a country and one of the goals of the policy-
maker is to ensure macroeconomic stability, the results of this paper have
important policy implications. More specifically, if an inflation targeting
regime has different implications for the uncertainty about the nominal price
compared to a price-level targeting regime, then our results have relevant
implications for the choice of these two regimes.
The plan of the paper is as follows. In the next section we describe the the-
oretical framework. Section 3 characterizes the long-term financial contract
3
and shows that such a contract is not time-consistent. Section 4 characterizes
the renegotiation-proof contract and Section 5 discusses the relationship be-
tween the monetary regime and the degree of indexation. Section 6 presents
additional properties of the model numerically and Section 7 concludes.
2 The model
Consider a continuum of risk-neutral entrepreneurs with utility E0P∞
t=0 βtct,
where βis the discount factor and ctis consumption.
Entrepreneurs generate cash revenues s=pzkθ, where pis the nominal
price level, zis an unobservable idiosyncratic productivity shock and kis
a publicly observed input of capital that fully depreciates after production.
We assume that the idiosyncratic productivity shock is iid and log-normally
distributed, that is z∼LN (µz, σ2
z). The price level is also iid and log-
normally distributed, that is, p∼LN(µp, σ2
p). For later reference we denote
by ˜zand ˜pthe logarithm of these two variables. Given the log-normality
assumption, the logarithms of zand pare normally distributed, that is,
˜z∼N(µz, σ2
z) and ˜p∼N(µp, σ2
p).
Entrepreneurs finance investments by signing optimal contracts with ‘com-
petitive’ risk-neutral intermediaries. We will also refer to intermediaries as
investors. Given the interest rate r, the market discount rate is denoted by
δ= 1/(1 + r). We assume that β≤δ, that is, the entrepreneur’s discount
rate is at least as large as the market interest rate.
The central feature of the model is the particular information structure
where the aggregate prices is observed with delay as in Jovanovic & Ueda
(1997). There are two stages in each period and the price level is observed
only in the second stage.
In the first stage the cash revenues s=pzkθare realized. The en-
trepreneur is the first to observe sbut this is not sufficient to infer the
value of zbecause the general price pis unknown at this stage.
The fact that the entrepreneur is the first to observe the revenues gives the
opportunity to divert the cash revenues for consumption purposes without
being detected by the investor (consumption is also not observable). There-
fore, there is an information asymmetry between the entrepreneur and the
investor which is typical in investment models with moral hazard such as
Clementi & Hopenhayn (2006), Gertler (1992) and Quadrini (2004).
In the second stage the general price pbecomes known. Although the
observation of pallows the entrepreneur to infer the value of z, the investor
4
can infer the true value of zonly if the entrepreneur chooses not to divert
the revenues in the first stage.
The actual consumption purchased in the second stage with the diverted
revenues will depend on the price p, which is only revealed in the second stage.
Therefore, when the revenues are diverted, the entrepreneur is uncertain
about the real value of the diverted cash. As we will see, this is the key
feature of the model creating the conditions for the renegotiation of the
optimal long-term contract.
3 The long-term contract
In this section we characterize the optimal long-term contract, that is, the
contract that the parties commit not to renegotiate, consensually, in later
periods. We will then show that this contract is not free from renegotiation
given the particular information structure where the nominal aggregate price
is observed with delay. The renegotiation-proof contract will be characterized
in the next section.
The long-term contract is characterized by maximizing the value for the
investor subject to a value promised to the entrepreneur. We will write the
optimization problem recursively. Assuming that the idiosyncratic produc-
tivity is not persistent, the only ‘individual’ state for the contract at the end
of period is the utility qpromised to the entrepreneur. This is the end-of-
period utility after consumption.
The contract chooses the current investment, k, the next period con-
sumption, c0=g(z, p), and the next period continuation utility, q0=h(z, p),
where zand pare the productivity and the aggregate price for the next pe-
riod. For the contract to be optimal we have to allow the choice of next
period consumption and continuation utility to be contingent on all possible
information that become available (directly or indirectly) in the next period,
that is, zand p.
The maximization problem is subject to two constraints. First, the
utility promised to the entrepreneur must be delivered (promise-keeping).
The contract can choose different combinations of next period consumption
c0=g(z, p) and next period continuation utility q0=h(z, p), but the ex-
pected value must be equal to the utility promised from the previous period,
that is,
q=βEhg(z, p) + h(z, p)i.
5
Second, the entrepreneur must not have an incentive to divert, for any
possible realization of the revenues s(incentive-compatibility). This requires
that the value received when reporting the true sis not smaller than the
value of reporting a smaller sand keeping the difference. If the entrepreneur
reports ˆs, the real value of the diverted revenues is φ(s−ˆs)/p, where φ < 1
is a parameter that captures the efficiency in diversion. Smaller values of
φimply lower the gains from diversion. We interpret φas a proxy for the
characteristics of the of financial markets (lower values of φcharacterize more
developed financial markets).
At the moment of choosing whether to divert the revenues, the nominal p
is not known. Therefore, what matters is the expected value E[φ(s−ˆs)/p |s],
which is conditional on the observation of s. Using the definition of the
revenue function, this can also be written as E[φ(z−ˆz)kθ|s]. Thus, for
incentive-compatibility we have to impose the following constraint:
Eg(z, p) + h(z, p)s≥Eφ(z−ˆz)kθ+g(ˆz, p) + h(ˆz , p)s
for all zand ˆz, with ˆz < z, where zis the true value of productivity and ˆzis
the value that the investor will infer in the second stage if the entrepreneur
diverts the revenues s−ˆs.
Although the constraint is imposed for all possible values of ˆz < z, we
can restrict attention to the lowest value ˆz= 0. It can be shown that, if the
incentive compatibility constrain is satisfied for ˆz= 0, then it will also be
satisfied for all other ˆz < z. Using this property, the contractual problem
can be written as:
V(q) = max
k, g(z,p), h(z,p)(−k+δEzkθ−g(z , p) + V(h(z, p)))(1)
subject to
Eg(z, p) + h(z, p)|s≥Eφ zkθ+g(0, p) + h(0, p)|s(2)
q=βEg(z, p) + h(z, p)(3)
g(z, p), h(z, p)≥0.(4)
6
The problem maximizes the value for the investor subject to the value
promised to the entrepreneur. In addition to the incentive-compatibility and
promise-keeping constraints, we also impose the non-negativity of consump-
tion and continuation utility. These are limited liability constraints.
The following proposition characterizes some properties of the optimal
contract.
Proposition 1 The optimal policies for next period consumption and con-
tinuation utility depend only on z, not p.
Proof 1 See Appendix A.
Therefore, the contract is fully indexed to nominal price fluctuations. The
intuition behind this result is simple. What affects the incentive to divert is
the ‘real’ value of the cash revenues. But the real value of revenues depends
on znot p. Although zis not observable when the entrepreneur decides
whether to divert, conditioning the payments on the ex-post inference of z
is sufficient to discipline the entrepreneur. Therefore, we can rewrite the
optimal policies as c0=g(z) and q0=h(z).
It will be convenient to define u(z) = g(z) + h(z) the next period utility
before consumption. Then the optimization problem can be split in two sub-
programs. The first program optimizes over the input of capital and the total
next period reward for the entrepreneur, that is,
V(q) = max
k, u(z)(−k+δE zkθ+W(u(z)))(5)
subject to
Ehu(z)|si≥Ehφ zkθ+u(0) |si
q=βEu(z)
u(z)≥0
7
The second program determines how the total reward for the entrepreneur,
u(z), will be delivered with immediate or future payments, that is,
W(u0) = max
c0, q0−c0+V(q0)(6)
subject to
u0=c0+q0
c0, q0≥0
This program is solved at the end of the period, after observing pand,
indirectly, z.
Proposition 2 There exists qand ¯q, with 0< q < ¯q < ∞, such that V(q)
and W(q)are continuously differentiable, strictly concave for q < ¯q, linear
for q > ¯q, strictly increasing for q < q and strictly decreasing for q > q. The
entrepreneur’s consumption takes the form:
c0=
0if u0<¯q
u0−¯qif u0>¯q
Proof 2 See Appendix B.
The key for understanding these properties is to think of qas the en-
trepreneur’s net worth. Because of incentive compatibility, together with
the limited liability constraint, the input of capital is constrained by the
entrepreneur’s net worth. As the net worth increases, the constraints are
relaxed and more capital can be invested. For very low values of q, the in-
put of capital is so low and the marginal revenue is so high that marginally
increasing qleads to an increase in revenues bigger than the increase in q.
Therefore, the investor would also benefit from raising q. This is no longer
true once the promised value has reached a certain level (q≥q) and the
value function becomes downward sloping.
The concavity property derives from the concavity of the revenue func-
tion. However, once the entrepreneur’s value has become sufficiently large
8
(q > ¯q), the firm is no longer constrained to use a suboptimal input of cap-
ital. Then, further increases in qwill not change kbut they only involve
a redistribution of wealth from the investor to the entrepreneur. The value
function will then become linear.
We should point out that the consumption policy characterized in the
proposition is unique only if β < δ. In the case of β=δ,cand qare not
uniquely determined when u0>¯q. However, it is still the case that c0= 0
and q0=u0when u0≤¯q.
3.1 The long-term contract is not renegotiation-proof
The optimal long-term contract studied in the previous section assumes that
the contractual parties do not renegotiate in future periods even if chang-
ing ex-post the terms of the contract could be beneficial for both of them.
Obviously this is a very strong assumption. What we would like to do in
this section is to show that both parties could benefit from changing the
terms of the contracts in later periods or stages. In other words, the optimal
long-term contract is not free from (consensual) renegotiation.
Consider the optimal policies for the long-term contract c0=g(z) and
q0=h(z). The utility induced by these policies after the observation of s
(and after the choice of diversion) is:
˜u=Ehg(z) + h(z)|si≡f(s)
Now suppose that, after the realization of s, we consider changing the
terms of the contract in a way that improves the investor’s value but does
not harm the entrepreneur. That is, the value received by the entrepreneur is
still ˜u. The change is only for one period and then we revert to the long-term
contract. In doing so we solve the following problem:
e
V(k, s, ˜u) = max
u(z)−k+δEhzkθ+W(u(z)) |si(7)
subject to
˜u=Ehu(z)|si
where W(.) is the value function with commitment defined above.
9
Notice that everything is now conditional on sbecause the problem is
solved after observing the revenues. At this point the agency problem is
no longer an issue in the current period, and therefore, we do not need the
incentive-compatibility constraint. The optimal choice of next period utility
is characterized by the following proposition.
Proposition 3 The optimal policy for the next period utility after the ob-
servation of sdoes not depend on zand it is equal to u(s) = ˜u.
Proof 3 Proposition 2 has established that the value function W(.)is strictly
concave for q < ¯q. Therefore, given the promise-keeping constraint ˜u=
E[u(z)|s], the expected value of W(u(z)) is maximized by choosing the next
period utility to be constant, that is, u(z) = ˜ufor all z. Q.E.D.
This property derives from the concavity of W(.). Because at this stage
the incentive problem has already been solved (the entrepreneur has already
reported the revenues), the expected value of W(u(z)) is maximized by choos-
ing a constant value for the next period utility. Because the optimal u(z)
in the long-term contract depends on z, Proposition 3 establishes that this
contract is not free from renegotiation.
There is also another reason why the optimal long-term contract is not
free from renegotiation, even if there is not a lag in the observation of the
price level. After a sequence of negative shocks, the value of qapproaches the
lower bound of zero. But low values of qalso imply that kapproaches zero.
Given the structure of the production function, the marginal productivity of
capital will approach infinity. Under these conditions, increasing the value
of q—that is, renegotiating the contract—will also increase the value for the
investor. Essentially, for low values of qthe value function V(q) is increasing
in q, as established in Proposition 2. The proof of this proposition also shows
that, if β < δ, the increasing segment of the value function will be reached
with probability 1 at some future date. Therefore, the long-term contract
will eventually be renegotiated.2
2When β=δ, the renegotiation interval will be reached with a positive probability
only if the current qis smaller than ¯q.
10
4 The renegotiation-proof contract
The implication of Proposition 3 is that a policy that is free from renegoti-
ation would make the promised utility dependent on s, not on z. In other
words, the real payments associated with the renegotiation-proof contract
depend on nominal quantities. This is in contrast to the long-term con-
tract where real payments depend only on real quantities, and therefore, it
is immune from price level fluctuations.
We have also seen from Proposition 2 that the long-term contract is not
free from renegotiation unless we impose a lower bound on q. Therefore, we
will consider the following problem:
V(q) = max
k,u(s)(−k+δE zkθ+W(u(s)))(8)
subject to
u(s)≥φEhzkθ|si+u(0),∀s
q=βEu(s)
u(s)≥u
where W(.) is again defined by (6). In this problem we have imposed that the
future utility can be contingent only on s. Furthermore, we have imposed
that the future utility cannot take a value smaller than u. The value of
uis endogenous and will be determined so that the contract is free from
renegotiation as in Wang (2000) and Quadrini (2004). For the moment,
however, we take uas exogenous and solve Problem (8) as if the parties
commit not to renegotiate.
We establish next a property that will be convenient for the analysis that
follows.
Lemma 1 The incentive-compatibility constraint is always satisfied with equal-
ity.
Proof 1 This follows directly from the concavity of the value function. If the
incentive compatibility constraint is not satisfied with equality, we can find an
11
alternative policy for u(s)that provides the same expected utility (promise-
keeping) but makes next period utility less volatile and allows for a higher
input of capital. The concavity of W(.)implies EW (u(s)) will be higher
under the alternative policy. Q.E.D.
Using this result, we can combine the incentive-compatibility constraint
with the promised-keeping constraint and rewrite the optimization problem
as follows:
V(q) = max
k−k+δEhzkθ+W(u0)i(9)
subject to
u0=φhE(z|s)−¯zikθ+q
β(10)
q
β−φ¯zkθ≥u(11)
where ¯z=Ez is the mean value of productivity.
The first constraint defines the law of motion for the next period utility
while the second insures that this is not smaller than the lower bound u.
Notice that, in deriving the constraints, we have used the result that E[E(z|
s)] = Ez = ¯z. See Appendix C for the derivation of these two equations.
Proposition 4 There exists u > 0such that the solution to problem (9) is
renegotiation-proof.
Proof 4 See Appendix D.
The lower bound uinsures that the utility promised to the entrepreneur
does not reach the region in which the promised utility would be renegotiated
ex-post. This is at the point in which the derivative of the value function is
zero, that is, Vq(q=u) = 0. Therefore, changing the value promised to the
entrepreneur does not bring, on the margin, neither gains nor losses to the
investor.
12
4.1 First order conditions
Denote by δµ the Lagrange multiplier for constraint (11). The first order
conditions are:
δθkθ−1"¯z(1 −φµ) + φEE(z|s)−¯zWu0#= 1,(12)
Wu0= max nVq0,−1o,(13)
and the envelope condition is:
Vq= δ
β!EWu0+µ(14)
The investment kis determined by equation (12). If the entrepreneur does
not gain from diversion, that is, φ= 0, we have the frictionless optimality
condition for which the discounted expected marginal productivity of capital
must be equal to the marginal cost. When φ > 0 the investment policy is
distorted.
Before continuing, it will be instructive to compare the first order condi-
tions for the renegotiation-proof contract with those for the long-term con-
tract, that is, Problem (1). In this case we obtain:
δθkθ−1"¯z(1 −φµ) + φEz−¯zWu0#= 1 (15)
Wu0= max nVq0,−1o,(16)
which is the same as for the renegotiation-proof contract except that E(z|s)
is replaced with z.
The comparison of conditions (12) and (15) illustrates how the lack of in-
dexation in the renegotiation-proof contract affects the dynamics of the firm.
If there is no price uncertainty, then E(z|s) = z, and the renegotiation-proof
contract is equivalent to the long-term contract. Because Wu0is negative
and decreasing (due to the concavity of W(.)), the term E(z−¯z)Wu0is neg-
ative. So in general, the input of capital is reduced by a higher volatility of
z. Another way to say this is that capital investment is risky for the investor
because a higher krequires a more volatile u0to create the right incentives
(see equation (10)). This is bad because the value of the contract for the
investor is concave in u0.
13
Now consider the extreme case in which the volatility of prices is very
large. In the limit, σp=∞. In this case, E(z|s) = ¯z. This implies that the
term E(E(z|s)−¯z)Wu0= 0. Therefore, high uncertainty of the price level
tends to increase investment. In this sense, the lack of commitment is good
for capital investment when the nominal price uncertainty is high.
With high price uncertainty, the entrepreneur’s (expected) value from
diversion is less dependent on the realization of revenues. In fact, in the
limiting case in which σp=∞, the real value from diversion is just ¯zkθ,
no matter what the realization of revenues is. Therefore, the next period
promised utility u0does not depend neither on znor on s(see again equation
(10)). Capital investment is not risky for the investor because it does not
induce a more volatile u0and it does not discourage investment.
4.2 Equilibrium
The equilibrium is characterized by a distribution of firms over the en-
trepreneur’s value q. The support of the distribution is [u, ¯q]. Because of
nominal price fluctuations, the distribution moves over time. Only in the
limiting cases of σp= 0 and σp=∞, the distribution of firms converges to
an invariant distribution. When σp= 0 this happens because there is no ag-
gregate uncertainty. When σp=∞this happens because all firms converge
deterministically to q= ¯qand stay there forever.
Within the distribution, firms move up and down depending on the re-
alization of the idiosyncratic productivity z(and the nominal price level).
The firm moves up in the distribution when it experiences a high value of z
(unless it has already reached q= ¯q), and moves down when the realization
of zis low (unless the firm is at q=u). The idiosyncratic nature of the
productivity insures that at any point in time some of the firms move up and
others move down.
5 Monetary policy regimes and indexation
We can use the results established in the previous section to characterize
how inflation shocks affect the economy under alternative monetary policy
regimes. In this framework, monetary policy regimes are fully captured by
the volatility of the price level, σp. Therefore, we will use the terms ‘monetary
policy regime’ and ‘price level uncertainty’ interchangeably.
14
We are interested in asking the following question: Suppose that there
is a one-time unexpected increase in the price level (inflation shock). How
would this shock impact economies with different degrees of aggregate price
level volatility σp?
The channel through which the monetary regime affects the financial
contract is by changing the expected value of zgiven the observation of s,
that is E[z|s]. This can be clearly seen from the law of motion of next
period utility, equation (10), and from the first order condition (12). As it
is well known from signaling models, the greater the volatility of the signal,
the lower is the information that the signal provides. The assumption that
˜p= log(p) and ˜z= log(z) are normally distributed allows us to show this
point analytically.
Agents start with a prior about the distribution of ˜z, which is the normal
distribution N(µz, σ2
z). They also have a prior about ˜s, which is also normal
N(µz+µp, σ2
z+σ2
p) since ˜s= ˜z+ ˜p. What we want to derive is the posterior
distribution of ˜zafter the observation of ˜s. Because the prior distributions
for both variables are normal, the posterior distribution of ˜zis also normal
with mean:
E(˜z|˜s) = σ2
p
σ2
z+σ2
p
µz+σ2
z
σ2
z+σ2
p
(˜s−µp),(17)
and variance:
V ar(˜z|˜s) = σ2
zσ2
p
σ2
z+σ2
p
.(18)
This derives from the fact that the conditional distribution of normally dis-
tributed variables is also normal.3A formal proof can be found in Greene
(1990, pp. 78-79).
Expression (17) makes clear how the volatility of nominal prices, σp, af-
fects the expectation of zgiven the realization of revenues. In particular,
the contribution of sto the expectation of zdecreases as the volatility of
prices increases. In the limiting case in which σp=∞,E(˜z|˜s) = µz(and
E(z|s) = ¯z). Therefore, the observation of sdoes not provide any informa-
tion about the value of z. Given this, the law of motion for the next period
utility, equation (10), becomes u0=q/β. Hence, the next period utility does
not depend on s. This implies that the renegotiation-proof contract becomes
fully indexed, that is, the values of the contract for the entrepreneur and the
investor do not depend on nominal quantities.
3It can be also be shown that the covariance between ˜zand ˜p,Cov(˜z , ˜p) = σ2
z.
15
The optimality condition for the input of capital becomes:
δθkθ−1¯z(1 −φµ)=1.
Therefore, if a shock to pdoes not affect the next period utility, it will
not affect the next period input of capital either. We then conclude that
in the limiting case of σp=∞, nominal price shocks do not have any real
consequences for the economy. The inflation neutrality, however, holds only
in the limiting case of σp=∞, as stated in the next proposition.
Proposition 5 Consider a one-time unexpected increase in price ∆p. The
impact of the shock on the next period promised utilities strictly decreases in
σpand converges to zero as σp→ ∞.
Proof 5 See Appendix E.
The intuition behind this property is simple. When σp= 0, agents inter-
pret an increase in nominal revenues induced by the change in the price level
as deriving from a productivity increase, not a price level increase. Therefore,
the utility promised to the entrepreneur has to increase in order to prevent
diversion. But in doing so, the promised utilities will increase on average for
the whole population. Essentially, the inflation shock redistributes wealth
from investors to entrepreneurs. As the entrepreneurs become wealthier, the
incentive-compatibility constraints in the next period are relaxed and this
allows for higher aggregate investment. For higher values of σp, however,
increases in revenues induced by nominal price shocks are interpreted less
as change in z. As a result, the next period utilities will increase less on
average.
This result suggests that economies with very volatile price level are less
vulnerable than economies with more stable monetary regimes to the same
price level shock. However, this does not mean that economies with more
volatile price level display lower volatility overall because they experience
larger shocks on average. Ultimately, how the contribution of different mon-
etary policy regimes affect the business cycle is a quantitative question. But
a-priori we cannot say whether countries with more volatile inflation experi-
ence greater or lower macroeconomic instability. This point will be illustrated
numerically in the next section.
16
6 Numerical analysis
This section further characterizes the properties of the economy numerically
with a parameterized version of the model. Although we do not conduct a
formal calibration exercise, the quantitative analysis allows us to illustrate
additional properties that cannot be established analytically but seem to be
robust to alternative parametrization values.
The model period is a year and the discount factor of the entrepreneur is
β= 0.95. The gross real-revenue is given by zAkθ. The scale parameter Ais
such that the optimal capital input is normalized to k= 1. The idiosyncratic
productivity zis log normally distributed with parameters µz= 0.125 and
σz= 0.5. The decreasing return to scale parameter θis set to 0.85.
The market discount rate is set to δ= 0.96, which is higher than the
entrepreneur discount factor. The parameter φgoverns the degree of financial
frictions (ie, the return from diversion) and it is set to φ= 1. This means that
the entrepreneur is able to keep the whole hidden cash-flow. The general price
level is log normally distributed with parameters µp= 0.01 and σp= 0.02.
We will also report the results for alternative values of σp. For the description
of the solution technique see Appendix F.
6.1 Some steady state properties
Assuming that the economy experiences a long sequence of prices equal to the
mean value Ep =eµp+σ2
p/2= ¯p, the economy would converge to a stationary
equilibrium. With some abuse of terminology, we will refer to this stationary
equilibrium as ‘steady state’. Notice that, even if the realized prices are
always the same, agents do not know this in advance, and therefore, they
assume that the price level is stochastic and form expectations accordingly.
Panel (a) of Figure 1 reports the decision rule for investment as a function
of the entrepreneur’s value qin the steady state. Investment kis an increas-
ing function of q. For very high values of q, the capital input is no longer
constrained, and therefore, investment kreaches the optimal scale which is
normalized to one.
Panel (b) plots the distribution of firms over their size kin the steady
state. As Panel (a) shows, some firms will ultimately reach the highest size.
Even if some of them will be pushed back after a negative productivity shock,
there is always a significant mass of firms in this size class.
17
0 5 10 15 20 25 30
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1(a) Investment Decision
Entrepreneur’s Value (q)
0.35 0.38 0.43 0.48 0.54 0.61 0.68 0.77 0.86 1.00
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35 (b) Invariant Distirbution
Firm Size (Capital)
Figure 1: Investment Decision Rule and Firm Invariant Distribution
18
6.2 Degree of indexation
The central feature of the model is that the degree of indexation depends
on nominal price uncertainty. If financial contracts were fully indexed, then
a price shock would not affect the values that the entrepreneur and the in-
vestor receive from the contract. On the other hand, if contracts were not
indexed, a price shock would generate a redistribution of wealth. For exam-
ple, if entrepreneurs borrow with standard debt contracts that are nominally
denominated (instead of using the optimal contracts characterized here), an
unexpected increase in the price level redistributes wealth from the investor
(lender) to the entrepreneur. Therefore, a natural way to measure the de-
gree of indexation is the elasticity of next period entrepreneur’s value—the
promised utility u0—with respect to a nominal price shock.
Essentially, the next period value of the contract for the entrepreneur
is the net worth of the firm. With an elasticity of zero, the financial con-
tract would be fully indexed because the net worth is insulated from inflation
shocks. If the elasticity is different from zero, the financial contract is im-
perfectly indexed.
Figure 2 plots the elasticity as a function of the current value of the firm
(current promised utility q). The elasticity is computed for a positive 25
percent shock to the price level.
The first feature shown by the figure is that the optimal contract is not
fully indexed: for any size of firms, a positive inflation shock redistributes
wealth to the firm while a negative shock redistributes wealth to the investor
(lender). The second feature is that the degree of indexation increases with
the size of the firm. Therefore, smaller and more constrained firms are more
vulnerable to inflation shocks. Because the next period entrepreneur’s value
affects next period investment, this also means that the investment of smaller
firms is more vulnerable to inflation shocks.
Table 1 presents the overall degree of indexation in an economy with low
nominal price uncertainty (σp= 0.02) and with high nominal price uncer-
tainty (σp= 1.5). In this experiment, the degree of indexation is given by the
elasticity of the aggregate next period value of entrepreneurs, computed by
aggregating over the whole distribution of firms. The elasticity is computed
by considering separately a positive and a negative 25 percent shock to the
price level.
As can be seen from the table, the degree of indexation increases with
price uncertainty. For example, when σp= 0.02, the elasticity is 0.67 while
19
0.5 1 1.5 2 2.5
0.4
0.5
0.6
0.7
0.8
0.9
1Degree of Indexation (elasticity of U’ with repect to P)
Entrepreneur’s value (q)
Figure 2: Degree of Indexation as a Function of Entrepreneur’s Value (q)
20
Table 1: Degree of Indexation for Different Price Level Uncertainty
Positive Price Negative Price
Level Shock Level Shock
Low Price Level Uncertainty 0.667 0.839
High Price Level Uncertainty 0.011 0.045
it is only 0.01 when σp= 1.5. The result that the degree of indexation is
higher in economies with high nominal price uncertainty is consistent with
the experiences of several countries such as Brazil and Argentina where price
uncertainty has been high and indexation widely diffuse.
Table 1 also shows that the degree of indexation is asymmetric. Specifi-
cally, the elasticity of firms’ value is higher after a negative price level shock
than a positive shock. The asymmetry stems from the fact that a negative
price level shock not only tightens the financial constraints of smaller firms
(with q < q) but also pushes a larger fraction of unconstrained firms (those
with q=q) to become constrained. Put differently, while a positive inflation
shock affects large firms by a smaller margin (since they are operating at the
optimal scale), a negative shock decreases the scale of large firms.
6.3 Aggregate investment, output and price level uncertainty
Table 2 presents aggregate capital and output for low and high price level
uncertainty economies. The table highlights that the stock of capital is bigger
when price level uncertainty is higher.
This finding derives from the characteristics of the contractual frictions.
When the price level is very volatile, the observation of the nominal revenues
by the firm in the first stage of the period does not provide enough informa-
tion about the actual value of the productivity z. The signal becomes noisier
and the information content of the signal is smaller. This implies that the
incentive to divert is not affected significantly by the observation of revenues.
Because of this, the value of the contract for the entrepreneur is less volatile
and the distribution of firms over kis more concentrated around the optimal
investment.
This can be further understood by recalling that the conditional expec-
tation of zdoes not depend on swhen σp=∞. Specifically, when σp=∞,
we have that E[z|s] = z. Then from equation (10) we can see that the next
21
Table 2: Aggregate Capital and Output for Different Price Level Uncertainty
Capital Output
Low Price-Level Uncertainty 0.644 0.835
High Price-Level Uncertainty 0.963 1.187
period promised utility is u0=q/β. This implies that qevolves deterministi-
cally and, eventually, all firms will reach ¯qpermanently. On the other hand,
when σp= 0, there will be a non-degenerate distribution of firms with only
few operating at the optimal scale. The average stock of capital will then be
smaller.
This finding may appear to conflict with the fact that countries with
monetary policy regimes that feature greater price level uncertainty are also
countries with lower output per-capita. However, it is also plausible to as-
sume that in these countries the contractual frictions, captured by the pa-
rameter φ, are higher than in rich countries. As we will see later, more severe
contractual frictions may offset the impact of greater price level uncertainty
on capital accumulation.
6.4 Impulse responses of different firms
The impulse responses to a nominal price shock is computed assuming that
the economy is in the steady state when the shock hits. As before, we define
a steady state as the limiting equilibrium to which the economy converges
after the realization of a long sequence of prices equal to the mean value
Ep =eµp+σ2
p/2= ¯p.
Starting from this equilibrium, we assume that the economy is hit by a
one-time price level shock. After the shock, future realizations of prevert
to the mean value ¯pand the economy converges again to the steady state.
Notice that, even if the price stays constant before and after the shock, agents
assume that prices are stochastic and form expectations accordingly.
We start examining the response of different size classes of firms. In
particular, we concentrate on two groups: (i) firms that are currently at
q=q; and (ii) firms that are at q < q. We label the first group ‘large firms’
and the second group ‘small firms’. Figures 3 and 4 plot the responses for
the investment and relative fraction of these two groups of firms.
Panel (a) of Figure 3 shows that a one-time price level increase has no
22
−20 0 20 40 60
0.9
0.95
1
1.05
1.1 (a) Low Uncertainty, Price Change = 0.25
Time
−20 0 20 40 60
0.9
0.95
1
1.05
1.1 (b) Low Uncertainty, Price Change = −0.25
Time
−20 0 20 40 60
0.9
0.95
1
1.05
1.1 (c) High Uncertainty, Price Change = 0.25
Time
−20 0 20 40 60
0.9
0.95
1
1.05
1.1 (d) High Uncertainty, Price Change = −0.25
Time
Small Firms
Large Firms
Average Firm Size
Figure 3: Average Firm Size Over Time After Positive and Negative Price
Level Shocks of Equal Magnitude for Different Price Level Uncertainty
23
−20 0 20 40 60
0.6
0.8
1
1.2
1.4
1.6 (a) Low Uncertainty, Price Change = 0.25
Time
−20 0 20 40 60
0.6
0.8
1
1.2
1.4
1.6 (b) Low Uncertainty, Price Change = −0.25
Time
−20 0 20 40 60
0.6
0.8
1
1.2
1.4
1.6 (c) High Uncertainty, Price Change = 0.25
Time
−20 0 20 40 60
0.6
0.8
1
1.2
1.4
1.6 (d) High Uncertainty, Price Change = −0.25
Time
Fraction of Large Firms
Figure 4: Fraction of Large Firms Over Time After Positive and Negative
Price Level Shocks of Equal Magnitude for Different Different Price Level
Uncertainty
24
effect on the average investment of large firms, that is, firms that keep q= ¯q.
But, the same shock has a positive effect on the average size of small firms,
that is, small firms expand. Large firms are not affected by a positive price
level shock because they are already at the optimal scale.
We now contrast the effects of a positive price level shock when the price
level uncertainty is high (σp= 1.5) and low (σp= 0.02). The average firm
size of large firms is not affected by the shock independently of the nominal
price uncertainty. On the contrary, the response of the average size of small
firms does depend on the nominal price uncertainty. In particular, we see
that it rises only slightly when the price uncertainty is high. This is because
the average size of small firms was initially close to the optimal scale in the
economy with high price level uncertainty.
Figure 4 also shows that the fraction of large firms increases after the
positive shock when the price uncertainty is low. This is due to the fact
that a positive shock relaxes the financial constraints of small firms and, as
result, the average size of small firms rises. Contrary to the economy with
low price uncertainty, the increase in the number of large firms is small in
the economy with high price uncertainty. This stems from the fact that most
firms are large and operating close to the optimal scale when the nominal
price uncertainty is high.
6.5 Impulse responses for the aggregate economy
Figure 5 presents the dynamics of aggregate capital after a one-time change
in the price level when the nominal price uncertainty is low (ie., σp= 0.02)
and high (σp= 1.5). It can be seen from Panel (a) that capital increases
after a positive price level shock. The maximum increase in capital happens
in the same period that the shock occurs and slowly converges to the initial
level. Although the shock is temporary, the effect is persistent.
Panel (c) presents the effects of the same increase in the price level on
capital accumulation when the price uncertainty is high. Comparing Panels
(a) and (c), one can observe that a positive price level shock has a small effect
on capital when the price uncertainty is high. This is due to the fact that
the degree of indexation is higher in the economy with high price uncertainty
and that most firms operate at or close to the optimal input of capital.
Figure 5 suggests that countries with a monetary policy regime that is
characterized by a low nominal price uncertainty is more vulnerable than
countries with greater price uncertainty to the same nominal price shock.
25
−20 0 20 40 60 80 100
0.9
0.95
1
1.05
1.1 (a) Low Uncertainty, Price Change = 0.25
Time
−20 0 20 40 60 80 100
0.9
0.95
1
1.05
1.1 (b) Low Uncertainty, Price Change = −0.25
Time
−20 0 20 40 60 80 100
0.9
0.95
1
1.05
1.1 (c) High Uncertainty, Price Change = 0.25
Time
−20 0 20 40 60 80 100
0.9
0.95
1
1.05
1.1 (d) High Uncertainty, Price Change = −0.25
Time
Aggregate Capital
Figure 5: Aggregate Capital Over Time After Positive and Negative Price
Level Shocks of Equal Magnitude for Different σp
26
Table 3: Volatility of Investment and Output for Different Price Level Un-
certainty
Standard Deviation Standard Deviation
Capital Output
Price-Level Uncertainty (σp)
σp= 0.02 0.008 0.009
σp= 0.20 0.073 0.082
σp= 1.50 0.134 0.147
σp= 1.70 0.120 0.130
However, countries with greater price uncertainty experience on average
larger shocks. This leads to the following question: Are economies with
low price uncertainty more unstable that economies with high price uncer-
tainty? To answer this question, we conduct a simulation exercise for several
economies that differ only in the volatility of the price level, σp. Each econ-
omy is simulated for 20,000 periods. We report the standard deviation of
investment and output in Table 3.
Before discussing the results, it is useful to describe intuitively how the
volatility of investment and output changes when σpincreases. There are
two opposing effects of σpon the volatility of investment and output. On the
one hand, a high σpreduces the volatility of investment since the economy is
more indexed. On the other, a higher σpimplies that on average the economy
experiences larger price shocks.
Table 3 shows that these two opposing forces lead to a non monotone re-
lation between the nominal price uncertainty and the volatility of investment
and output. For low or moderate values of σp, the volatility of investment
increases with σp. This means that the fact that the economy experiences
larger shocks dominates the lower elasticity to each shock (greater indexa-
tion). However, for high values of σp, the volatility of investment decreases
with σp, implying that higher indexation more than offsets the increase in
the magnitude of the price shocks. Recall from the previous analysis that, in
the limit with σp=∞, the economy is fully indexed and the real economy is
muted from nominal price shocks.
27
6.6 Price-level uncertainty and financial development
In this section we discuss how the interaction between the nominal price un-
certainty and the degree of financial development affects the level and the
volatility of the real economy. In our model the degree of financial develop-
ment is captured by the parameter φ. A high value of φcorresponds to a
less developed financial system since firms can gain more from the diversion
of resources.
In the previous experiments, φwas set to one. In this section we will
compare the previous results with an alternative economy where φ= 0.5. We
think of the economy with φ= 0.5 as an economy with a ‘more developed
financial system’. The standard deviations of aggregate capital and output
are reported in Table 4.
As expected, investment is lower when financial markets are less devel-
oped. This is because when φis high, financial constraints are tighter and, as
result, investment is lower on average. We can also see that investment, for
a given price level uncertainty (i.e., monetary policy regime), is more volatile
in the economy with a less developed financial system.
How can we interpret these results? We know that some low income
countries experience very high volatility of inflation. As we have seen in
Table 2, our model predicts that these countries should have a higher stock
of capital (after controlling for the technology level of these countries). At
the same time, these countries are also likely to face more severe contractual
frictions which, according to our model, induce a lower stock of capital. If
the impact of financial development dominates the impact of greater price
uncertainty, the model would still predict a lower stock of capital for poorer
countries as the data seem to suggest.
7 Conclusion
In this paper we have studied a model with repeated moral hazard where
financial contracts are not fully indexed to inflation because, as in Jovanovic
& Ueda (1997), the nominal price level is observed with delay.
Nominal indexation is endogenously determined in the model and it is
different for different types of firms. In particular, we find that small, more
constrained firms are more vulnerable to unexpected inflation, that is, they
are constrained to sign contracts with a lower degree of nominal indexation.
As a result, the impact of inflation shocks on aggregate investment and output
28
Table 4: Standard deviation of investment and aggregate investment for
different degree of financial development and price-level uncertainty.
More developed Less developed
financial system financial system
(φ= 0.50) (φ= 1.00)
Low Price Level Uncertainty (σp= 0.02)
Aggregate Capital 0.803 0.644
Standard Deviation Capital 0.006 0.008
Moderate Price-Level Uncertainty (σp= 0.20)
Aggregate Capital 0.812 0.658
Standard Deviation Capital 0.050 0.073
High Price-Level Uncertainty (σp= 1.5)
Aggregate Capital 0.984 0.963
Standard Deviation Capital 0.092 0.134
Extreme Price-Level Uncertainty (σp= 1.70)
Aggregate Capital 0.986 0.955
Standard Deviation Capital 0.085 0.130
derives predominantly from the response of constrained firms.
Another finding is that the overall degree of nominal indexation increases
with the degree of price uncertainty. An implication of this is that economies
with higher price uncertainty are less vulnerable to a given inflation shock,
that is, investment and output respond less. However, this does not imply
that these economies display lower overall volatility: even if the response to
a given shock is smaller, the economy experiences larger shocks on average.
This paper has important policy implications if price-level uncertainty
depends, to some extent, on the monetary policy regime chosen by a coun-
try. This is because the economic outcomes under different monetary policy
regimes can change when the extent of nominal indexation is endogenous.
This may be an important consideration when assessing the relative mer-
its of alternative monetary policy regimes. In particular, when comparing
inflation-targeting, where the price-level uncertainty is expected to be high,
against price-level targeting which should lead to lower price-level uncer-
tainty.
29
Appendix
A Proof of Proposition 1 (preliminary)
To simplify the proof we make a change of variables in Problem (1). Define
y=kθ. After substituting k=y1
θ, the optimization problem becomes:
V(q) = max
y, g(z,p), h(z,p)(−y1
θ+δEzy −g(z , p) + V(h(z, p)))(19)
subject to
Eg(z, p) + h(z, p)|s≥Eφ zy +g(0, p) + h(0, p)|s(20)
q=βEg(z, p) + h(z, p)(21)
g(z, p), h(z, p)≥0.(22)
The change of variables is useful because it makes the incentive-compatibility
constraint linear in all the decision variables. In this way it is easier to show
that this is a well defined concave problem.
We can verify that Problem (19) satisfies the Blackwell conditions for
a contraction mapping. Therefore, there is a unique fix point V∗. The
mapping preserves concavity. This implies that the fixed point for V∗is
concave, although not necessarily strictly concave.
Consider a particular solution S1≡ {y1, g1(z, p), h1(z, p)}, where the next
period consumption and continuation utility are dependent on both zand p.
Now consider the alternative solution S2≡ {y2, g2(z), h1(z)}, where y2=y1,
g2(z) = Rpg1(z, p)dF (p), h2(z) = Rph1(z, p)dF (p). In the alternative solution,
the next period consumption and continuation utility are contingent only on
z, not p.
We can verify that, if S1satisfies all the constraints to problem (19), then
the constraints are also satisfied by S2. Therefore, S2is a feasible solution.
The next step is to show that S2provides higher value than S1. This follows
directly from the concavity of the value function. Essentially, by choosing
S2we make the next period utility less volatility and increase EV (h(z, p)).
Q.E.D.
30
B Proof of Proposition 2 (preliminary)
In the proof of Proposition 1, we established that the value function is concave
(although not strictly). By verifying the condition of Theorem 9.10 in Stokey,
Lucas, & Prescott (1989), we can also established that the value function is
differentiable.
Consider the incentive-compatibility constraint E[u(z)|s]≥φE(z|s)y+
u(0) and the promise-keeping constraint q=βEu(z). The incentive-compatibility
constraint can be integrated over pto get Eu(z)≥φ¯zy +u(0). Remember
that we have made the change of variable y=kθ. Using this condition with
the promise-keeping constraint we can write:
q=βEu(z)≥βφ¯zy (23)
This says that, as qconverges to zero, y(and therefore k=y1
θ) also
converges to zero. This also implies that the marginal cost of yconverges
to zero (or equivalently, the marginal productivity of capital converges to
infinity). Therefore, starting from a value of qclose to zero, by marginally
increasing qwe can increase the marginal revenue by a large margin, which
makes the value of the contract for the investor higher. Therefore the function
V(q) is increasing for very low values of q.
Define ¯
kas the input of capital for which the expected marginal revenue
is equal to the interest rate, that is, θkθ−1= 1/δ. Obviously, the input of
capital chosen by the contract will never exceed ¯
k.
Now consider a very large q, above the level that makes ¯
kfeasible, that
is, condition (23) is satisfied. Because the contract will never choose a value
of k > ¯
k, further increases in qwill not change the input of capital. This
implies that V(q) (the value for the investor) decreases proportionally to the
increase in q. Therefore, for qabove a certain threshold ¯q, the value function
is linear. The value function being linear for q > q, it is easy to see from
Problem (6) that c0=u0−¯qif β < δ. However, if β=δ, then there are
multiple solutions for c0.
Below the threshold ¯q, however, qdoes constrain k. The strict concavity
of the value function derives from the fact that the revenue function is strictly
concave. The optimal policy for c0then becomes obvious. Q.E.D.
31
C Derivation of equations (10) and (11)
Consider the incentive-compatibility constraint
u(s) = φE(z|s)kθ+u(0).(24)
Integrating over swe get Eu(s) = φE{E(z|s)}kθ+u(0). Because E{E(z|s)}=
¯z, this can also be written as:
Eu(s) = φ¯zkθ+u(0).(25)
Consider now the promise-keeping constraint q=βEu(s). Using equation
(25), this can be written as:
q
β=φ¯zkθ+u(0).(26)
Using this to eliminate u(0) in (24) we get:
u(s) = φhE(z|s)−¯zikθ+q
β,(27)
which is equation (10).
The lower bound on total utility u(s)≥urequires u(0) ≥u. This is
because (s) is increasing in s. From equation (26) we have that u(0) =
q/β −φ¯zkθ. Therefore, the condition u(0) ≥ucan be written as:
q
β−φ¯zkθ≥u, (28)
which is equation (11).
D Proof of Proposition 4 (preliminary)
To be written.
E Proof of Proposition 5 (preliminary)
Consider the law of motion for the next period utility (10) which for conve-
nience we rewrite here:
u0=φhE(z|s)−¯zikθ+q
β(29)
32
The effect of the shock is to increase E(z|s) for each realization of z.
Given the distributional assumptions about zand p, the conditional expec-
tation takes the form:
E(z|˜s) = e
σ2
p
σ2
z+σ2
p
µz+σ2
z
σ2
z+σ2
p
(˜s−µp)+ σ2
zσ2
p
2(σ2
z+σ2
p)
Given a realization of the aggregate log-price ˜pand the idiosyncratic log-
productivity ˜z, the firm observes ˜s= ˜z+ ˜p. We want to compute how
a deviation of the log-price from its mean value µpaffects the conditional
expectation of firms. More specifically, we want to compare the case in which
the observed revenue is ˜s1= ˜z+µpwith the case in which the revenue is ˜s2=
˜z+µp+ ∆. This is done by computing the ratio of conditional expectations
E(z|˜s2)/E(z|˜s1). Using the formula for the conditional expectation written
above we get:
E(z|˜s2)
E(z|˜s1)=e
σ2
z
σ2
z+σ2
p
∆
Therefore, the change in the conditional expectation decreases with σp.
From the law of motion (29) we can then observe that, for each z, the change
in next period utility decreases with σp.Q.E.D.
F Solution method
The solution is based on the iteration of the unknown function Vq=ψ(q).
We create a grid of points for qand guess the value of the function ψ(q) at
each grid point. The values outside the grid are joined with step-wide linear
functions. The detailed steps are as follows:
1. Create a grid for q∈ {q1, ..., qN}.
2. Guess Vi
q=ψ(qi), for i= 1, ..., N .
3. Solve for kand µat each grid point of q:
(a) Check first for the binding solution:
•Solve for kusing (11).
•Solve for µusing (12).
(b) If the µfrom the binding solution is smaller than zero, the solution
must be interior. Then solve for the interior solution:
33
•Set µ= 0.
•Solve for kusing (12).
4. Given the solutions for kand µ, find Wu0using (13). Then update
the guess for the function ψ(q) at each grid point using the envelope
condition (14).
5. Restart from step 3 until convergence in the function ψ(q).
34
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