Ch. 8: Conglomerate Firms and Internal Capital Markets 473
assume that all firms in the industry are single-segment firms that produce only in one
industry.
We start by simplifying the firm’s profit function given in equation(2)in the text to
the one industry case
pdjkj−rkj−β(
kj) 2
,
where, for simplicity, we have abstracted from labor costs (so thatα=0 in equation(2)
in the text). The subscriptjrefers to firmj. Recall thatris the market price of a unit
of capacity andβis the standard neoclassical diseconomy of scale. To reduce notation,
we further assume without loss of generality thatdjcan take one of only two values.
Let high productivity, or H firms, produce one unit of that industry’s output per unit
of capacity so that for those firmsdj =dH =1. Let low productivity, or L firms,
produce onlydj=d<1 units of output per unit of capacity. Thus, the profit functions
specialize topkH−rkH−β(kH)^2 for H firms andpdkL−rkL−β(kL)^2 for L firms, after
adjusting the notation to reflect the fact that all the H (L) firms are identical, and where
and the number of capacity units operated by H and L firms iskHandkL, respectively.
Assume that total amount of capacity available to the industry isK=σ+ρr,σ, ρ >
- Thus, we assume that the supply of capacity is not perfectly elastic, reflecting the
addition of new capacity (for high levels ofr) and sales for scrap (for low levels ofr).
Assume that there is an exogenously determined number,n, of entrepreneurs and that
the proportion of entrepreneurs that can operate H firms isλ. To avoid discussion of firm
entry and exit, which would require more notation, also assume that the opportunity cost
of capacity outside the industry is sufficiently low so that it is optimal for all high- and
low-quality firms to operate at the level of demand we are considering.
The time sequence is as follows. There is one period and two dates:t=1, 2. At time
t=1, the entrepreneurs learn the actual realization of the next period’s level of demand
in the industry. A market for capacity opens in which firms can purchase capacity units
at a pricer. The price of capacity,r, adjusts so that supply equals demand for capacity.
At timet=2, the firms realize the cash flows. For simplicity, we assume that capacity
has no salvage value att=2.
To make explicit the role of demand shocks and the distribution of capacity units on
firm growth, we describe the equilibrium in the market for output. The market price that
the customers pay in industry for the output is determined asp=a−bn(λkH+( 1 −
λ)kL), wheren(λkH+( 1 −λ)kL)is the aggregate output anda,bare positive constants.
Remark 1.A positive demand shock causes, productive profit maximizing firms in-
crease in size relative to less productive profit maximizing firms.
Proof of Remark 1.
We obtain the output of type H firms by maximizing the firm’s operating profit,pkH−
rkH−β(kH)^2. Solving forkH, we obtainp 2 −βras the optimal capacity that type H firms
operate at the given opportunity cost,r. The capacity at which the low-quality firms