Economic growth and economic development 336

Introduction to Modern Economic Growth in Example 6.5 in the previous chapter: c (t)) = λ (t) r exp (−ρt) u0 (ˆ In particular, this equation can be rewritten as u0 (ˆ c (t)) = βrλ (t), with β = exp (−ρt), and would be almost identical to equation (6.29), except for the presence of λ (t) instead of the derivative of the value function But as we will see below, λ (t) is exactly the derivative of the value function, so that the consumption Euler equations in discrete and continuous time are identical This is of course not surprising, since they capture the same economic phenomenon, in slightly different mathematical formulations The next necessary condition determines the behavior of λ (t) as λ˙ (t) = −r c (t)) = βrλ (t), we can obtain a Now using this condition and differentiating u0 (ˆ differential equation in consumption This differential equation, derived in the next chapter in a somewhat more general context, will be the key consumption Euler equation in continuous time Leaving the derivation of this equation to the next chapter, we can make progress here by simply integrating this condition to obtain λ (t) = λ (0) exp (−rt) Combining this with the first-order condition for consumption yields a straightforward expression for the optimal consumption level at time t: cˆ (t) = u0−1 [Rλ (0) exp ((ρ − r) t)] , where u0−1 [·] is the inverse function of the marginal utility u0 It exists and is strictly decreasing in view of the fact that u is strictly concave This equation therefore implies that when ρ = r, so that the discount factor and the rate of return on assets are equal, the individual will have a constant consumption profile When ρ > r, the argument of u0−1 is increasing over time, so consumption must be declining This reflects the fact that the individual discounts the future more heavily than the rate of return, thus wishes to have a front-loaded consumption profile In contrast, when ρ < r, the opposite reasoning applies and the individual chooses a back-loaded consumption profile These are of course identical to the conclusions we reached in 322