Ceris-CNR, W.P. N° 6/1997
complements (so that ∂v (t ) / ∂pt > 0 ). The optimality condition for the firm’s output
price is then given by the following Euler equation:





(1 − τ )q (t ) + µ  p
t

t





t

−

∂c(t ) ∂h(t )  
−

∂q t
∂q t  

(5)

 ∂h(t + 1) ∂i (t + 1)

− E t βt ,t +1 (1 − τ t +1 )µt
q (t ) + µt pt ]bt  = 0
+
[
∂q t
∂ ( pt q t )



Equation (5) states that along the optimal path marginal benefits and marginal costs of
changing the output price must offset each other. First of all, note that a price change
affects the firm’s production level via the product demand as given in (1). We allow for
strategic considerations in that the firm’s price affects demand also via the impact upon
the rivals’ price (see (4)). According to (5) costs include the marginal production cost
and the marginal cost of adjusting output today triggered by a price change; benefits
comprise marginal revenue, the saving in adjustment cost due to not having to adjust
output tomorrow, and the lower interest payments due to spreading a given amount of
debt on a larger firm's size (recall that this effect is negative). Let us now define the
following variable:
 ∂D( t ) ∂v t 

 ∂v t ∂p t 

λt = 

∂D( t )
∂p t

(6)

In view of the assumptions made about the components of (4) λt is non positive.6 In the
case of a monopolistic firm λ t = 0 . More generally the size of λt will depend on the
size of the strategic effect compared to that of the direct effect, the former depending in
turn on the degree of product differentiation, measured by ∂D( t ) ∂v t , and on the
tightness of price competition, measured by ∂vt ∂pt .

6

Clearly,

µ

in (4) and

λ

in (6) are related, as

µt = (1 + λt )(∂D(t ) / ∂pt ) .

9