Introduction to Modern Economic Growth The model of expanding product variety was first suggested by Judd (1985), but in the context of an exogenous growth model The endogenous growth models with expanding product variety is presented in Grossman and Helpman (1991a,b) The treatment here is somewhat different from that in Grossman and Helpman, especially because we used the ideal price index as a numeraire, rather than Grossman and Helpman’s choice of total expenditure as numeraire 13.7 Exercises Exercise 13.1 This exercise asks you to derive (13.8) from (13.7) (1) Rewrite (13.7) at time t as: ∙ Z s ¸ Z t+∆t exp − r (τ ) dτ (χ(ν, s) − ψ) x(ν, s)ds V (ν, t) = t t ∙ Z s ¸ Z ∞ + exp − r (τ ) dτ [χ(ν, s)x(ν, s) − ψx(ν, s)] ds t+∆t t+∆t which is just an identity for any ∆t Interpret this equation and relate this to the Principle of Optimality (2) Show that for small ∆t, this can be written as V (ν, t) = ∆t · (χ(ν, t) − ψ) x(ν, t) + exp (r (t) ∆t) V (ν, t + ∆t) + o (∆t) , and thus derive the equation ∆t·(χ(ν, t) − ψ) x(ν, t)+exp (r (t) · ∆t) V (ν, t+∆t)−exp (r (t) · 0) V (ν, t)+o (∆t) = 0, where, recall that, exp (r (t) · 0) = Interpret this equation and the sig- nificance of the term o (∆t) (3) Now divide both sides by ∆t and take the limit ∆t → 0, to obtain exp (r (t) · ∆t) V (ν, t + ∆t) − exp (r (t) · 0) V (ν, t) = ∆t→0 ∆t (4) When the value function is differentiable in its time argument, the previous (χ(ν, t) − ψ) x(ν, t) + lim equations is equivalent to ¯ ∂ (exp (r (t) · ∆t) V (ν, t + ∆t)) ¯¯ = (χ(ν, t) − ψ) x(ν, t) + ¯ ∂t ∆t=0 Now derive (13.8) (5) Provide an economic intuition for the equation (13.8) 601