474 V. Maksimovic and G. Phillips
operate is similarly obtained askL=pd 2 β−r. Notice thatkH>kL, so that a type H firm
uses more capacity than the low-quality firm at every price level.
If both H and L firms are active in the industry and the price of capacity exceeds its
salvage value, the market price of the output isp=a−bn(λkH+d( 1 −λ)kL).We
determine the price of capacity by equating the demand for capacity by each type of
firm to the total number of capacity units available, either on the secondary market or
as supplied by manufacturers, so that
σ+ρr=λn (A.1)p−r
2 β+( 1 −λ)npd−r
2 β,
where the total amount of capacity employed by the industry isK=σ+ρr. The first
term on the right hand side of the equation is the demand for capacity by theλnhigh-
quality firms. The second term is the demand for capacity by the( 1 −λ)nlow-quality
firms. Solving equation(1)for the opportunity cost of capacity yields
r= (A.2)p(λ+d( 1 −λ))
n+ 2 βρ−
2 βσ
n+ 2 βρ.
Substituting the expression for the rental cost of capital(A.2)into the expressions for
the desired capacity by high- and low-quality firms, we obtain
kH=σ
n+ 2 βρ+
( 1 −d)( 1 −λ)n+ 2 βρ
2 w(n+ 2 βρ)p,kL=σ
n+βρ−
( 1 −d)λn− 2 βρd
2 w(n+ 2 βρ)p.The derivative of the ratio(kH/kL)with respect to the output price,p,is
2 ( 1 −d)(n+ 2 ρ)βK
( 2 wσ+( 2 βdρ−( 1 −d)λn)p)^2> 0.
The last expression shows that a positive price shock (increase inp) increases the ratio
kH/kL. Thus, positive price shocks are associated with higher growth of high-quality
firms relative to low-quality firms. Since positive demand shocks toaat timet= 1
translate into increases inp, it is straightforward, but messy, to show that the same
relation obtains for the ratiokH/kLanda.^41
Remark 2.Consider a multiperiod generalization of the above industry equilibrium
in which the model is repeated over a sequence of dates, with the demand intercept
achanging over time. Positive (negative) innovations in a will cause more productive
firms to engage in purchases of new capacity and purchases from other firms (divest)
and less productive firms to divest (acquire) capacity.
(^41) The analysis presented here assumes an interior equilibrium. A full analysis would take into account the
exit and entry of entrepreneurs.