The paper finds that sufficient ambiguity leads to the uniqueness of equilibrium in macroeconomic coordination games. The results have a Keynesian flavour: sufficient optimism gives rise to a Pareto-optimal equilibrium; and sufficient pessimism results in a Pareto-inferior equilibrium. This analysis is applied to a "Big Push" model from the economic growth literature.Ambiguity, Strategic Complementary, Coordination Games, Optimism, "Big Push".

David Kelsey

Wei Pang

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 1 
  
How productive is optimism? the Impact of ambiguity on the "big push" 
  
 
 
David Kelsey  
University of Exeter 
Wei Pang  
Kingston University
Abstract 
The paper finds that sufficient ambiguity leads to the uniqueness of equilibrium in macroeconomic coordination games. 
The results have a Keynesian flavour: sufficient optimism gives rise to a Pareto-optimal equilibrium; and sufficient 
pessimism results in a Pareto-inferior equilibrium. This analysis is applied to a "Big Push" model from the economic 
growth literature.
We are particularly grateful to Jim. L. Ford, Dieter Balkenborg, Lynne Evans, Jacques Dreze, Steven Durlauf, David Canning, John Fender, 
Joachim Stibora for helpful discussion and comments. We would also like to acknowledge helpful comments from seminar participants at 
University of Birmingham, Exeter University, Hong Kong University, IFPTI/Cornell conference 2005, the Royal Economic Society Easter 
School, European Economic Association conference 2003, CORE Summer School 2007. 
Citation: David Kelsey and Wei Pang, (2010) ''How productive is optimism? the Impact of ambiguity on the "big push"'', Economics Bulletin, 
Vol. 30 no.1 pp. 855-865. 
Submitted: Feb 05 2010.   Published: March 25, 2010. 
 
     1 Introduction
Keynes (1937) argued that uncertainty plays a vital role in macroeconomics. Investment
is subject to waves of expectations driven by optimism and pessimism. Sudden shifts in
psychological forces can produce booms and slumps. The description of the 2001 ￿nancial
crisis in Argentina in DeLong (2005), could serve as a modern illustration:
￿When it [the currency board] collapsed, Argentina￿ s consolidated debt-to-
GDP ratio was about 50%. That is not an unsustainable debt load. And the
Argentinian government was managing to run a primary surplus. If there had
been con￿dence in Argentina￿ s ￿scal future ￿ con￿dence that no ￿nancial
crisis was on the horizon￿ then interest rates would have been much lower,
and the primary surplus would have generated only a moderate general de￿cit.
With low interest rates, Argentina￿ s prospects for growth would have been rel-
atively good. With good growth prospects and a relatively moderate overall
government budget de￿cit, there would be no reason to fear that ￿scal policy is
unsustainable. Only the fact that a crisis was expected pushed interest rates
up to the level where investment was strangled, growth impossible, the overall
budget de￿cit large, and a crisis inevitable￿ .
We propose one way to model Keynes￿ideas by studying the in￿ uence of ambiguity on
macroeconomics. It is plausible that ambiguity would a⁄ect macroeconomic coordination
since it is intrinsically di¢ cult to assign a probability of success when success requires
simultaneous investment across many sectors. Consider, for instance, how many investors
made incorrect predictions about the dotcom boom of the 1990￿ s.
The ￿Big Push￿concerns the development of an economy from a low activity state to
1a higher level equilibrium.1 Threshold e⁄ects and non-linearity imply that simultaneous
industrialisation of many sectors can be self-sustaining even if no sector can pro￿tably
invest on its own.
Our model is a macroeconomic coordination game with strategic complementarity. In
other words an increase in one player￿ s strategy will give others an incentive to increase
their strategy as well, see Cooper & John (1988). Strategic complementarities can give rise
to Pareto-ranked multiple equilibria. In the ￿Big Push￿model, there is one equilibrium
where no ￿rm industrialises and another where all industrialise. We ￿nd that su¢ cient
ambiguity will lead to a unique equilibrium. Optimism can lead an economy to a Pareto-
superior equilibrium, while pessimism causes it to become ￿stuck￿in an ine¢ cient state.
Ambiguity is de￿ned as situations where beliefs cannot be quanti￿ed by subjective
probabilities without doubt. In the ￿Big Push￿model, whether or not industrialisation is
pro￿table depends on whether enough other ￿rms industrialise. Firms have beliefs about
the likelihood of industrialisation but may not feel con￿dent in them. This can be caused
by either limited access to information or because there is no reliable information (Frisch
& Baron (1988)). The level of con￿dence may cause a ￿rm to either increase or decrease
their belief in success therefore in￿ uencing their investment behaviour2.
2 Ambiguity
Here, we explain our model of ambiguity in games. Consider a symmetric game ￿ =
￿
I;(Si;ui)i2I
￿
with two players I = f1;2g; where player i has a closed, bounded and
convex strategy set Si ￿ R: Both players have the same payo⁄ function ui (si;s￿i).3
1See Rosenstein-Rodan (1943) and Murphy, Shleifer & Vishny (1989)
2Empirical evidence suggests that individuals can be either ambiguity averse or ambiguity loving.
(see Ellsberg (1961), Heath & Tversky (1991) ).
3For consideration of asymmetric equilibrium under ambiguity, see Eichberger & Kelsey (2000).
2Players are assumed to follow the axioms of Chateauneuf, Eichberger & Grant (2007),
(henceforth CEG), which is a special case of Choquet Expected Utility (CEU), see Schmei-
dler (1989). These imply that the beliefs of player i may be represented by a neo-additive
capacity on S￿i.
De￿nition 2.1 A neo-additive capacity, is a function ￿ : P (S￿i) ! R de￿ned by,
￿ (?) = 0;￿ (S￿i) = 1 and v (E) = ￿￿ + (1 ￿ ￿)￿ (E) for ? $ E $ S￿i, where 0 6 ￿ 6
1;0 6 ￿ 6 1 and ￿ is an additive probability on S￿i:
CEG represent preferences as a weighted average of the minimum, the maximum and
the mean of ui.
De￿nition 2.2 The Choquet expected value of ui (si;s￿i) with respect to a neo-additive
capacity v is de￿ned as: Vi(si) =
R
ui (si;s￿i)dv (s￿i) = ￿ (1 ￿ ￿) ￿ ui(si;s￿i) + ￿￿ ￿
ui (si;s￿i) + (1 ￿ ￿)E￿ui(si;s￿i); where ui (si;s￿i) = mins￿i2S￿i ui(si;s￿i); ui (a(s)) =
maxs￿i2S￿i ui (si;s￿i); and E￿ui (si;s￿i) denotes the expected value of utility with respect
to ￿:
These preferences may be interpreted as representing a situation where agents have
beliefs represented by probabilities ￿. However they may have some doubts about these
beliefs represented by ￿: Agents respond to ambiguity in part with optimism (represented
by ￿ ) by over-weighting the best outcome and in part with pessimism (represented by
1 ￿ ￿ ).
We de￿ne the support of a neo-additive capacity to be equal to the support of the
additive belief on which it is based.4
De￿nition 2.3 The support, suppv; of neo-additive capacity v = ￿￿ + (1 ￿ ￿)￿ is
de￿ned by suppv = supp￿:
4For a justi￿cation of this support notion see Eichberger & Kelsey (2002).
3We use an equilibrium concept which takes ￿ and ￿ as exogenous.5
De￿nition 2.4 An equilibrium under ambiguity (EUA) is a belief system (v￿
1;:::;v￿
I)
where v￿
i is a neo-additive capacity on S￿i, if for all i 2 I suppv￿
i ￿ ￿j6=iRj
￿
v￿
j
￿
; where
Ri (vi) =
￿
^ si 2 Si :
R
ui (^ si;s￿i)dvi (s￿i) >
R
ui (si;s￿i)dvi (s￿i);8si 2 Si
￿
denotes the
best response of player i given his(her) beliefs vi: An equilibrium is pure if suppv￿
i contains
a single strategy pro￿le, otherwise we say that it is mixed.
We con￿ne our attention to symmetric pure equilibria. A standard Nash equilibrium
can be regarded as a special case where there is no ambiguity i.e. ￿ = 0.
3 The Big Push
We consider a ￿Big Push￿model with one time period and a continuum of production
sectors, indexed by i 2 [0;1].6 There are 2 types of ￿rms in each sector: a competitive
fringe with constant returns to scale converting one unit of input into one unit of output
and a monopolist able to access two technologies, a low and a high productivity technol-
ogy with marginal costs ￿h and ￿l respectively, where ￿h > ￿l > 1. The low technology
has no ￿xed costs and represents ￿ non-industrialisation￿ . In contrast, the high technol-
ogy represents ￿ industrialisation￿and incurs a ￿xed cost, I, which is assumed to be the
expenditure on intermediate goods. There is a representative consumer who owns all
claims to labour income, pro￿ts, and sales of the intermediate goods. (S)he is a price
taker in consumption and in the competitive labour market and supplies inelastically N
units of labour. Labour demand in each sector is denoted by ni. Taking the wage rate
as the numeraire, the total wage bill is
R 1
0 ni = N. Labour is the only variable cost of
5For further discussion of the equilibrium concept see Dow & Werlang (1994) and Eichberger & Kelsey
(2000).
6The model is adapted from Shleifer & Vishny (1988) and Fatas & Metrick (1997)
4production.
Production in sector i is denoted by qi and the price of output by pi: Expenditure
is ￿I when a fraction ￿ of ￿rms adopt the high technology. Denoting production pro￿t
as ￿; total income (expenditure) can be written as Y = N + ￿ + ￿I: Assuming utility
is Cobb-Douglas, U = exp
hR 1
0 lnqidi
i
; and the budget constraint is
R 1
0 piqidi = Y; (s)he
will spend equal amounts on each commodity. Thus revenue piqi is constant and equal
to Y in each market.
Pro￿t is maximized at pi = 1; where qi = Y . The maximum pro￿ts of ￿rms adopting
high and low technology are respectively, ￿l
i = alY and ￿h
i = ahY ￿ I, where 0 < ah :=
￿h￿1
￿h < 1 and 0 < al :=
￿l￿1
￿l < 1.
If a fraction ￿ of ￿rms industrialise, we have Y = N
1￿￿ah￿(1￿￿)al; and ￿rm i will
industrialise if,
￿
h
i ￿ ￿
l
i = (ah ￿ al) ￿
N
1 ￿ ￿ah ￿ (1 ￿ ￿)al
￿ I > 0: (1)
Equation (1) shows that strategic complementarity multiple equilibria are present.
Formally, for a ￿xed cost of industrialisation I, and (ah ￿ al) ￿ N
1￿alj￿=0 < I < (ah ￿ al) ￿
N
1￿ahj￿=1; we have at least two Pareto-ranked equilibria.7
3.1 The Role of Ambiguity
If ￿rms view the decisions of others as ambiguous, they will perceive their own demand
curve to be ambiguous. Demand is highest (resp. lowest) when all (resp. no) ￿rms are
industrialised, that is, Y h = N
1￿ah and ￿ = 1; (resp. Y l = N
1￿al and ￿ = 0). Therefore, the
7The following two inequalities hold at the same time: ￿h
i (1) ￿ ￿l
i (1) = (ah ￿ al) ￿ N
1￿ahj￿=1 ￿ I > 0
and ￿h
i (0) ￿ ￿l
i (0) = (ah ￿ al) ￿ N
1￿alj￿=0 ￿ I < 0: Meanwhile,
￿h
i (1) ￿ ￿l
i (0) = ah￿N
1￿ahj￿=1 ￿ I ￿ al￿N
1￿alj￿=0 = (ah ￿ al) ￿ N
(1￿ah)(1￿al) ￿ I > (ah ￿ al) ￿ N
1￿ah ￿ I > 0:
5(Choquet) expected value of demand faced by a ￿rm is
V (Y ) = ￿ (1 ￿ ￿) ￿
N
1 ￿ al
+ ￿￿ ￿
N
1 ￿ ah
+ (1 ￿ ￿)
N
1 ￿ ￿ah ￿ (1 ￿ ￿)al
. (2)
The ￿rm will industrialise if:
￿
h
i ￿￿
l
i = (ah￿al)￿
￿
￿ (1 ￿ ￿)
N
1 ￿ al
+ ￿￿
N
1 ￿ ah
+ (1 ￿ ￿)
N
1 ￿ ￿ah ￿ (1 ￿ ￿)al
￿
￿I > 0:
(3)
Next we show that su¢ cient ambiguity will generically lead to an unique equilibrium.8
Proposition 3.1 If there is su¢ cient ambiguity , i.e., ￿ is close to 1; there exists "; such
that if ￿rms are su¢ ciently pessimistic, i.e., 1￿￿ > "; (resp. optimistic i.e., ￿ > 1￿"),
then non-industrialisation (resp. industrialisation) will be the unique equilibrium.
Proof. Suppose, ￿ ! 1; then equation (3) becomes,
￿
h
i ￿ ￿
l
i =
￿
(ah ￿ al) ￿
N
1 ￿ al
￿ I
￿
+ ￿(ah ￿ al)
￿
N
1 ￿ ah
￿
N
1 ￿ al
￿
:
1. If non-industrialisation is the unique best response for ￿rm i, ￿h
i ￿ ￿l
i < 0: This
happens when, 1 ￿ ￿ > " =
(ah￿al)￿ N
1￿ah
￿I
(ah￿al)￿
h
N
1￿ah
￿ N
1￿al
i:
2. Suppose ￿rm i is optimistic, the equilibrium is industrialisation and ￿h
i ￿ ￿l
i > 0:
This happens when ￿ > 1 ￿ " =
I￿(ah￿al)￿ N
1￿al
(ah￿al)￿
h
N
1￿ah
￿ N
1￿al
i:
Given su¢ cient ambiguity, beliefs have the power to determine the equilibrium out-
comes. Pessimism can disarm well intended policy e⁄orts, make ￿ good￿look like ￿ bad￿and
￿ bad￿seem ￿ worse￿ . Hence the economy is trapped in a low level equilibrium. On the other
hand, optimism can amplify the e⁄ects of policies, create and maintain a favourable envi-
8By generically we mean except when ￿ =
I￿(ah￿al)￿ N
1￿al
(ah￿al)￿
h
N
1￿ah ￿ N
1￿al
i:
6Figure 1: The curve ￿￿ depicts the increased pro￿t from industrialisation corresponding to
di⁄erent values of ￿: Optimism shifts the curve ￿￿ upward to ￿￿￿, the perceived scale of
industrialisation now is ￿2 instead of ￿; which indicates the positive value of ￿￿ in the case
of industrialisation; when pessimism prevails, the curve ￿￿ shifts downward to ￿￿1￿￿; the
perceived scale of industrialisation is ￿1 which indicates the negative value of ￿￿:
ronment which further raises optimism. The economy takes o⁄ to reach a self-sustained
￿economics of euphoria￿ .
We may rewrite the decision rule,
￿
h
i ￿ ￿
l
i = (ah ￿ al)
￿
￿ (1 ￿ ￿)
￿
N
1 ￿ al
￿
N
1 ￿ ￿ah ￿ (1 ￿ ￿)al
￿￿
+(ah ￿ al)
￿
￿￿
￿
N
1 ￿ ah
￿
N
1 ￿ ￿ah ￿ (1 ￿ ￿)al
￿
+
N
1 ￿ ￿ah ￿ (1 ￿ ￿)al
￿
￿ I: (4)
Now suppose ￿ is at the marginal level ^ ￿ which makes ￿h
i ￿￿l
i = (ah￿al) N
1￿^ ￿ah￿(1￿^ ￿)al￿I =
0: If ￿rms become more pessimistic and for simplicity we assume that ￿ < 1
2,9 the negative
term dominates the positive one and we have
￿
￿h
i ￿ ￿l
i
￿
j￿=b ￿ < 0. Thus, ￿rm i does not
industrialise. Conversely, if ￿ > 1=2, we have
￿
￿h
i ￿ ￿l
i
￿
j￿=b ￿ > 0; which implies that ￿rm
i industrialises. Intuitively, optimism makes the anticipated scale of industrialisation
larger, which encourages ￿rms to industrialise. This can be shown in ￿gure 1.
9The threshold value of ￿ to capture pessimism is determined by the value of the ￿xed cost I in the
interval of
h
(ah ￿ al) ￿ N
1￿ah; (ah ￿ al) ￿ N
1￿al
i
:
74 Conclusion
Our ￿ndings suggest that in a ￿Big Push￿model, when ambiguity is high the equilibrium
is unique. There are other studies on the selection of multiple equilibria in coordination
games, for instance the global game approach, Morris & Shin (2003) and sunspot equi-
librium theory, Shell & Cass (1983). There is a basic di⁄erence between this paper and
the global game literature. While the global game approach is applied to games with
incomplete information, we investigate coordination games with complete information.
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How productive is optimism? the Impact of ambiguity on the "big push"

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How productive is optimism? the Impact of ambiguity on the "big push"

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