Economic growth and economic development 587

Introduction to Modern Economic Growth
results are identical without this assumption, see Exercise 13.20). Nevertheless, we
refer to the inputs as “machines,” which makes the economic interpretation of the
problem easier.
The term (1 − β) in the denominator is included for notational simplicity. No-

tice that for given N (t), which final good producers take as given, equation (13.2)
exhibits constant returns to scale. Therefore, final good producers are competitive
and are subject to constant returns to scale, justifying our use of the aggregate
production function to represent their production possibilities set.
The budget constraint of the economy at time t is
(13.3)

C (t) + X (t) + Z (t) ≤ Y (t) ,

where X (t) is investment or spending on inputs at time t and Z (t) is expenditure
on R&D at time t, which comes out of the total supply of the final good.
We next need to specify how quantities of machines are created and how the new
machines are invented. Let us first assume that once the blueprint of a particular
input is invented, the research firm can create one unit of that machine at marginal
cost equal to ψ > 0 units of the final good. We also assume the following form for
innovation possibilities frontier, where new machines are created as follows:
(13.4)

N˙ (t) = ηZ (t) ,

where η > 0, and the economy starts with some initial technology stock N (0) > 0.
This implies that greater spending on R&D leads to the invention of new machines.
Throughout, we assume that there is free entry into research, which means that
any individual or firm can spend one unit of the final good at time t in order to
generate a flow rate η of the blueprints of new machines. The firm that discovers

these blueprints receives a fully-enforced perpetual patent on this machine.
There is no aggregate uncertainty in the innovation process. Naturally, there will
be uncertainty at the level of the individual firm, but with many different research
labs undertaking such expenditure, at the aggregate level, equation (13.4) holds
deterministically.
Given the patent structure specified above, a firm that invents a new machine
variety is the sole supplier of that type of machine, say machine of type ν, and sets
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