A. E. Scorcu

M. Castellani

English

Research Papers in Economics

  1
 
 
 
 
A Note on Multiplicative Uncertainty and Partisan Policies
* 
 
 
 
Massimiliano Castellani
# and Antonello E. Scorcu
§ 
 
 
# Dipartimento di Scienze Economiche 
Università di Bologna 
Piazza Scaravilli 2 
40125 Bologna, Italy 
fax ++39 051 221968 
castellani@rimini.unibo.it 
 
§ Dipartimento di Scienze Economiche 
Università di Bologna 
Strada Maggiore 45 
40125 Bologna, Italy 
fax ++39 051 2092664 
antonello.scorcu@unibo.it 
 
 
 
December 2003  
 
 
Abstract 
In this paper we consider the effects of «multiplicative» uncertainty 
about the structure of the economy in the standard partisan model. An 
increase in the uncertainty decreases of the inflation rate pursued by 
the liberal policymaker, but increases the inflation rate pursued by the 
conservative policymaker. For certain configurations of the 
parameters, an increase in the uncertainty reduces the expected loss of 
both parties. 
 
JEL classification: E31, E61  
Keywords: multiplicative uncertainty, partisan parties, inflation 
  
 
 
 
 
                                                           
* We would like to thank A. Bucci, G. Candela, R. Cellini and P. Figini for helpful comments on a previous version of 
the paper. The usual disclaimer applies. 
   2
  
 
 
 
What is the optimal discretionary monetary policy in the partisan model when there is 
uncertainty about the structure of the economy?  
In the partisan model augmented with «multiplicative» uncertainty, an increase in the degree 
of uncertainty leads to converging monetary policies because the inflation rates pursued, when in 
office, by a liberal and a conservative policymaker are, respectively, reduced and increased. 
Moreover, for certain configurations of the parameters, the expected loss in presence of uncertainty 
is lower than in the deterministic case.  
 
1. The model 
Let us consider the standard partisan model (see Alesina et al., 1997) in which a liberal (left-wing) 
policymaker L and a conservative (right-wing) policymaker R attach different relative weights to 
output (y) and inflation (π). When in office, policymaker i = L, R, minimizes the expected value of 
the loss: 
    L i = ½ (y – y*)
2 + θi (πi)
2/2     θi > 0      (1) 
with respect to πi, which is the only instrument at his disposal
1. y* is the desired output level for 
both policymakers, and the ideological differences between the two are expressed solely by θR > θL:  
the conservative policymaker R is relatively more concerned with inflation than output, whereas the 
liberal policymaker L is relatively more concerned with output than inflation.  
The economy is described by an expectation-augmented Phillips curve: 
     y   =   y
n +  b(π – π
e)         (2) 
                                                           
1 We neglect the complications arising from the distinction between the rate of inflation and the rate of money growth 
or from the imperfect control of the inflation rate.    3
here π is the inflation rate, π
e the inflationary expectation, y
n > 0 the equilibrium level of output and 
b > 0 is the slope of the (short-run) Phillips curve. As usual in these models, in order to create an 
incentive towards surprise inflation, we assume y* > y
n. 
The only departure from the textbook partisan model is the existence of uncertainty upon the 
structure of the economy; the parameter b is therefore a i.i.d. random variable: 
     b   ∼ (β, σ
2)           ( 3 )  
with β > 0. Whereas the value of β is known from the beginning to both policymakers and private 
agents, the actual value of coefficient b is revealed only after inflationary expectations and the 
monetary policy have been set. Therefore, the policymaker in office has no informational advantage 
upon the private sector in predicting b, nor any strategic advantage to exploit. The uncertainty about 
b is of the «multiplicative» type and its effects have been analyzed in a large body of literature since 
Brainard (1967)
2. 
Inflationary expectations are rational and are described by: 
     π
e = pπR + (1-p)πL        (4) 
(1-p) and p are the exogenous, common knowledge, probabilities of policymakers L and R of being 
elected, respectively; πL and πR are the discretionary inflation rates set by L and R. Because of the 
previous assumption upon the preferences of the policymakers, it will be easy to show that πL > πR.  
At the beginning of each period, inflationary expectations π
e are set; then election occurs, the 
winner enters the office and sets the inflation rate; finally the actual slope of the Phillips curve, b, is 
revealed.  
 
2. Policy activism and convergence under uncertainty 
In the derivation of the main result of the paper, it is convenient to consider the case in which the 
two policymakers expect the same β, but have different expectations about the variability of b, 
                                                           
2 The case of an additive random disturbance can be analysed by the use of the Certainty Equivalence Principle and will 
not be considered.   4
possibly because of different evaluations about the importance of the short-run effects of the 
monetary policy. The variances σ
2
L and σ
2
R are expected by policymakers L and R, respectively. 
Hence the differences in the two policymakers concern both the loss functions (θR > θL) and the 
structure of the economy (σ
2
L ≠ σ
2
R).  
The problem for policymaker R is the minimization of: 
   E [ L R]= ½ E[(y – y*)
2 + θR πR
2]   =        ( 5 )  
= ½ [(y
n)
2 +  (β
2 + σ
2
R)(πR – π
e)
2 +  2y
nβ(πR – π
e) - 2 y*
 β(πR – π
e) -2 y*y
n + (y*)
2 + θR πR
2] 
and from the first order condition dE[LR]/dπR = 0 we get: 
   ( β
2 + σ
2
R)(πR – π
e) + θR πR + β(y
n – y*) = 0         (6) 
For policymaker L the problem is the minimization of: 
   E [ L L] = ½ E[(y – y*)
2 + θL πL
2]          ( 7 )  
and the relevant first order condition is: 
   ( β
2 + σ
2
L)(π L – π
e) + θL πL + β(y
n – y*) = 0         (8) 
Substituting the inflationary expectation (4) in equations (6) and (8), we obtain πR, πL: 
   πR = h[θL+p(β
2+σ
2
L) + (1-p)(β
2+σ
2
R)]        (9) 
   πL = h[θR+p(β
2+σ
2
L) + (1-p)(β
2+σ
2
R)]        (10) 
where h = β(y* – y
n)/ {θLθR + θRp(β
2+σ
2
L)+ θL(1-p) (β
2+σ
2
R)} > 0.  
Note that πL > πR because θR > θL, that is, policymaker L pursues a more expansionary 
policy than policymaker R. Two different inflationary effects can be recognised in the optimal 
choice of πi: a direct effect given by the inflationary incentives of policymaker i, and an indirect 
effect, given by the inflationary incentives of the other policymaker, that influences the choices of i 
through π
e. 
Expressions (9) and (10)
 can be simplified in the case σ
2
R = σ
2
L: 
  π*R =  h*(θL+ β
2 + σ
2)            ( 1 1 )  
  π*L =  h*(θR + β
2 + σ
2)            ( 1 2 )    5
where h* = β(y* – y
n)/{θLθR + (β
2+σ
2) [θR p + θL(1-p)]}. 
However, the effect of a change in σ
2 is different for π*L and π*R; straightforward algebra 
gives us the following results: 
   d π*R/dσ
2 = k* [(1-p)θL(θR - θL)]  >  0       (13) 
   d π*L/dσ
2 = k* [pθR(θL - θR)]  <  0       (14) 
where k* = β(y* – y
n)/{θLθR + (β
2+σ
2) [θRp + θL(1-p)]}
2 > 0.  
An increase in uncertainty implies some convergence in monetary policies, as it reduces the 
discretionary inflation rate of the (activist) policymaker L, and increases the discretionary inflation 
rate of the conservative policymaker R. This result qualifies the traditional conclusion of Friedman 
(1953) and Brainard (1967) of an attenuated optimal policy in the case of parameter uncertainty
3; in 
fact, policymaker R reaction is stronger when uncertainty is higher. Obviously, under uncertainty 
the policymaker R inflation rate is always higher than in that the deterministic case; the opposite 
results holds for policymaker L. 
However, the attenuating effect induced by the multiplicative uncertainty holds for the 
average (and expected) inflation rate E(π) = π
e = pπR + (1-p)πL: 
dE(π)/dσ
2 = -p(1-p)k* (θR - θL)   <   0         ( 1 5 )  
The previous results can be explained if we consider separately the effects of a change in σ
2
i 
on πi, as computed from equations (9) and (10): 
  d πR/dσ
2
R = k(1-p)(θR - θL)[θL + p(β
2+σ
2
L)]  >  0      (16) 
  d πR/dσ
2
L = kp(1-p)(θL - θR)(β
2+σ
2
R)  <  0       (17) 
  d πL/dσ
2
L = kp(θR - θL)[θR + (1-p)(β
2+σ
2
R)]  <  0      (18) 
  d π L/dσ
2
R = kp(1-p) (θR - θL) (β
2+σ
2
L)  >  0       (19) 
where k = β(y* – y
n)/{θLθR + θRp(β
2+σ
2
L)+ θL(1-p) (β
2+σ
2
R)}
2 > 0.  
                                                           
3 In the original Brainard (1967) framework a similar result emerges with specific values of the covariances between 
parameters. Cf. Turnovsky (1977).   6
In equation (16), the increase in the uncertainty perceived by R leads to an increase of the 
inflation rate engineered by the policymaker R, when in office. An increase in σ
2
R weakens the 
direct inflationary incentive on πR, whereas the indirect inflation increasing effect becomes more 
important; because πL > πR, this implies dπR/dσ
2
R > 0. The indirect effect eventually predominates 
for σ
2
R→∞; in fact from equation (9) we obtain: 
   L i m   πR = Lim πL =β(y* – y
n)/θL        (20) 
σ
2
R→∞        σ
2
R→∞         
 
Unsurprisingly, the inflation rate in (20) corresponds to the case p = 0. 
On the other hand, as shown in equation (17), an increase in σ
2
L makes more important the 
direct effect, which tends to lower πR. When σ
2
L→ ∞ the indirect effect vanishes and the behavior 
of R is the same to the one in which p = 1. From equation (10) we obtain: 
  L i m   πR = Lim πL = β(y* – y
n)/θR         (21) 
σ
2
L→∞        σ
2
L→∞         
 
  Because the direct effect in equation (16) is stronger than the indirect effect in equation  
(17), the overall result is the positive relationship between uncertainty and inflation shown in 
equation (13). The same reasoning can be applied to πL. 
The effect of an increase in the uncertainty in the structure of the economy tends to reduce 
the differences in the monetary policies of partisan policymakers
4:  
  d ( πL - πR)/dσ
2  <   0             ( 2 2 )  
Unsurprisingly, when the uncertainty in the economy increases indefinitely (σ
2
L = σ
2
R = σ
2 → ∞) 
the attenuation effect predominates and the differences between the partisan policies disappear: 
  L i m   πL = Lim πR = β(y* – y
n)/[(1-p)θL+pθR].        (23) 
σ
2→∞        σ
2→∞ 
 
 
   7
3. Inflation variance and expected losses 
By substituting equations (11) and (12) into (4), (2) and (1) defines the (ex-ante) loss function of 
party i in terms of the parameters p, θL, θR and σ
2. Obviously, even if the inflation variance is not a 
policy instrument, because of the different effects it exerts on π*L and π*R, there is a link between 
σ
2 and the ex-ante loss E[Li] in equation (5). Provided that policymakers behave optimally (with πi 
given by equations (11) and (12)), in certain circumstances dE[Li]/dσ
2 < 0. In these cases 
policymaker i  is better off in an uncertain rather than in a deterministic world, and his expected loss 
is lower the higher the degree of uncertainty on the structure of the economy.  
  In fact, straightforward but tedious algebra leads, for both policymakers, to the following 
result: 
  d E [ L i]/dσ
2 = (β
2+σ
2)( θR-θL)p
2 + 2[θLθR + θL(β
2+σ
2)]p - [θLθR + (β
2+σ
2)]   (24) 
It is easily shown that dE[Li]/dσ
2 < 0 for p = 0, whereas dE[Li]/dσ
2 > 0 for p = 1 and p = ½. Hence, 
when the liberal policymaker is very likely to win the elections, an increase in uncertainty reduces 
the losses of both policymakers, because it lowers π*L (equation (14)). The result is reversed if the 
conservative policymaker is more likely to win. 
 
4. Conclusions 
In this note we have introduced a multiplicative uncertainty effect on the structure of the 
economy in the basic partisan model.  
We have shown that in the basic partisan model, an increase in the uncertainty leads: i) to an 
increase of the optimal inflation rate pursued by the conservative policymaker; ii) to a decrease of 
the liberal policymaker inflation rate; iii) to a convergence in the discretionary monetary polices 
pursued by the two policymaker.  
                                                                                                                                                                                                 
4 On the contrary, in Schultz (2002) an increase in the uncertainty leads to more extreme partisan policies.   8
The first result, in particular, differs from the conclusion, popularized by Friedman (1953) 
and Brainard (1967) of a more prudent behavior in the case of increased uncertainty.  
Moreover, for certain configurations of the parameters, a rise in the degree of uncertainty 
reduces the expected losses of both policymakers. 
   9
 
References 
 
 
Alesina A., Roubini N. and Cohen G.D., 1997, Political Cycles and the Macroeconomy, MIT 
Press, Cambridge. 
 
Ball L. and Cecchetti S. G., 1991, Wage Indexation and Discretionary Monetary Policy, 
American Economic Review, 81, 1310-19. 
 
Brainard W.C., 1967, Uncertainty and the Effectiveness of Policy, American Economic Review, 
57, 411-25. 
 
Friedman, M., 1953, The Effect of Full-Employment Policy on Economic Stability, in: Essays 
in Positive Economics, Chicago University Press, Chicago. 
 
Schultz C., 2002, Policy biases with voters' uncertainty about the economy and the government, 
European Economic Review, 46, 486-506. 
 
Turnovsky S.J., 1977, Macroeconomic analysis and stabilization policy, Cambridge University 
Press, Cambridge. 
 
 

A Note on Multiplicative Uncertainty and Partisan Policies

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