Introduction to Modern Economic Growth The first term must be equal to zero, since otherwise limt→∞ V (t, xˆ (t)) = ∞ or −∞, and the pair (ˆ x (t) , yˆ (t)) cannot be reaching the optimal solution Therefore x (t) , yˆ (t)) = lim exp (−ρt) µ (t) x˙ (t) = lim exp (−ρt) µ (t) g (ˆ (7.56) t→∞ t→∞ Since limt→∞ V (t, xˆ (t)) exists and f and g are weakly monotone, limt→∞ yˆ (t) and limt→∞ xˆ (t) must exist, though they may be infinite (otherwise the limit of V (t, xˆ (t)) would fail to exist) The latter fact also implies that limt→∞ x˙ (t) exists (though it may also be infinite) Moreover, limt→∞ x˙ (t) is nonnegative, since otherwise the condition limt→∞ x (t) ≥ x1 would be violated From (7.52), (7.54) implies that as t → ∞, λ (t) ≡ exp(−ρt)µ(t) → κ for some κ ∈ R+ Suppose first that limt→∞ x˙ (t) = Then limt→∞ xˆ (t) = xˆ∗ ∈ R (i.e., a finite value) This also implies that f (ˆ x (t) , yˆ (t)), g (ˆ x (t) , yˆ (t)) and therefore fy (ˆ x (t) , yˆ (t)) and gy (ˆ x (t) , yˆ (t)) limit to constant values Then from (7.51), we have that as t → ∞, µ (t) → µ∗ ∈ R (i.e., a finite value) This implies that κ = and lim exp (−ρt) µ (t) = 0, (7.57) t→∞ and moreover since limt→∞ xˆ (t) = xˆ∗ ∈ R, (7.55) also follows x (t), where g ∈ R+ , so that xˆ (t) grows at an Suppose now that limt→∞ x˙ (t) = gˆ exponential rate Then substituting this into (7.56) we obtain (7.55) x (t), for any g > 0, so that xˆ (t) grows Next, suppose that < limt→∞ x˙ (t) < gˆ at less than an exponential rate In this case, since x˙ (t) is increasing over time, (7.56) implies that (7.57) must hold and thus again we must have that as t → ∞, λ (t) ≡ exp(−ρt)µ(t) → 0, i.e., κ = (otherwise limt→∞ exp (−ρt) µ (t) x˙ (t) = limt→∞ x˙ (t) > 0, violating (7.56)) and thus limt→∞ µ˙ (t) /µ (t) < ρ Since xˆ (t) grows at less than an exponential rate, we also have limt→∞ exp (−gt) xˆ (t) = for any g > 0, and in particular for g = ρ − limt→∞ µ˙ (t) /µ (t) Consequently, asymptotically µ (t) xˆ (t) grows at a rate lower than ρ and we again obtain (7.55) x (t) for any g < ∞, i.e., xˆ (t) grows at Finally, suppose that limt→∞ x˙ (t) > gˆ more than an exponential rate In this case, for any g > 0, we have that ˙ ≥ g lim exp(−ρt)µ(t)ˆ x(t) ≥ gκ lim xˆ(t) ≥ 0, lim exp(−ρt)µ(t)x(t) t→∞ t→∞ t→∞ 347