Economic growth and economic development 394

Introduction to Modern Economic Growth and the transition equation 8.7.. Notice that the transversality condition is written in terms of the current-value costate variable, which is more

Introduction to Modern Economic Growth and the transition equation (8.7)

Notice that the transversality condition is written in terms of the current-value costate variable, which is more convenient given the rest of the necessary conditions Moreover, as discussed in the previous chapter, for any µ (t) > 0, ˆH (a, c, µ) is a concave function of (a, c) The first necessary condition (and equation (8.13) below),

in turn, imply that µ (t) > 0 for all t Therefore, Theorem 7.15 implies that these conditions are sufficient for a solution

We can next rearrange the second condition to obtain:

µ (t) =− (r (t) − ρ) , which states that the multiplier changes depending on whether the rate of return

on assets is currently greater than or less than the discount rate of the household Next, the first necessary condition above implies that

To make more progress, let us differentiate this with respect to time and divide by

µ (t), which yields

u00(c (t)) c (t)

u0(c (t))

˙c (t)

c (t) =

˙µ (t)

µ (t). Substituting this into (8.12), we obtain another form of the famous consumer Euler equation:

c (t) =

1

εu(c(t))(r (t)− ρ) where

00(c (t)) c (t)

u0(c (t))

is the elasticity of the marginal utility u0(c(t)) This equation is closely related to the consumer Euler equation we derived in the context of the discrete time problem, equation (6.30), as well as to the consumer Euler equation in continuous time with constant interest rates in Example 7.1 in the previous chapter As with equation (6.30), it states that consumption will grow over time when the discount rate is less than the rate of return on assets It also specifies the speed at which consumption

380