In a model of growth and R&D, output is produced via a production func- tion: Y(t) = [(1 -ax)K(t)] [A(t) (1 - aL) I-a where new knowledge is produced in an R&D sector: A(t) = BlaKK(t)] [aLL]" A(t) 1-e As usual, a dot over a variable denotes a derivative with respect to time, e.g. A(t) = dA(t)/dt. The fractions of capital and labor in the R&D sector, denoted by ag and oz, are assumed to be constant. o, y, 0, and B are also positive constants; o and & are less than one. Finally, L is constant, for simplicity. Capital accumulation is simply: K ( t ) = sY (t) (i) Show that the growth rate of K can be written as: K(t) A(t) 11-a K(t) CK K ( t ) for some constant CR. Find CK. (Hint: Combine the first and third preceding equations.) (ii) Likewise, show that the growth rate of knowledge is A(t) K(t) A(t) A(t ) for a constant CA. Find CA. (iii) Recall that in a balanced growth path the growth rates of K and A must be constant. Then the ratio K(t)/A(t) must be constant as well: explain why. Use this fact to find K(t)/A(t) along the balanced growth path. (Hint: Recall that if K(t)/A(t) is constant, K and A must grow at the same rate.) (iv) What is the growth rate of output per capita along a balanced growth path? How does it change if B increases permanently? (Hint: Express the growth rate of output per capita in terms of CA and Ca only, then check how an increase in B affects either of those terms.) Explain your intuition. (It may help to note that, in our formulation, knowledge accumulation has constant returns to scale in K and A.)