The balanced growth path of a semi-endogenous version of the Romer model. (Jones, 1995a.) Consider the model of Section 3.5 with two changes. First, existing knowledge contributes less than proportionally to the production of new knowledge, as in Case 1 of the model of Section 3.2: A(t) = BLA(t)A(t)

θ , θ < 1.

Second, population is growing at rate n rather than constant: L(t) = L(0)ent, n > 0.

(Consistent with this, assume that utility is given by equation [2.2] with u (•)

logarithmic.) The analysis of Section 3.2 implies that such a model will exhibit transition dynamics rather than always immediately being on a balanced growth path. This problem therefore asks you not to analyze the full dynamics of the model, but to focus on its properties when it is on a balanced growth path.

Specifically, it looks at situations where the fraction of the labor force engaged in R&D, aL , is constant, and all variables of the model are growing at constant rates.

(Note: You are welcome to assume rather than derive that on a balanced growth path, the wage and consumption per person grow at the same rate as Y/L.)

(a) What are balanced-growth-path values of the growth rates of Y and Y/L and of r as functions of the balanced-growth-path value of the growth rate of A and parameters of the model?

(b) On the balanced growth path, what is the present value of the profits from the discovery of a new idea at time t as a function of L(t), A(t), w(t), and exogenous parameters?

(c) What is A(t)/A(t) on the balanced growth path as a function of aL and exogenous parameters? What is L(t)A(t)

θ−1 on the balanced growth path? (Hint:

Consider equations [3.6] and [3.7] in the case of γ = 1.)

(d) What is aL on the balanced growth path?

(e) Discuss how changes in each of ρ, B, φ, n, and θ affect the balanced-growthpath value of aL . In the cases of ρ, B, and φ, are the effects in the same direction as the effects on LA in the model of Section 3.5?