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 > 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) > 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 573