Economic growth and economic development 353

Introduction to Modern Economic Growth
will change the current flow return plus the value of the stock by the amount Hx ,
but it will also affect the value of the stock by the amount λ˙ (t). The Maximum
Principle states that this gain should be equal to the depreciation in the value of the
stock, −λ˙ (t), since, otherwise, it would be possible to change the x (t) and increase

the value of H (t, x (t) , y (t)).

The second and complementary intuition for the Maximum Principle comes from
the HJB equation (7.37) in Theorem 7.10. In particular, let us consider an exponentially discounted problem like those discussed in greater detail in Section 7.5
below, so that f (t, x (t) , y (t)) = exp (−ρt) f (x (t) , y (t)). In this case, clearly
V (t, x (t)) = exp (−ρt) V (x (t)), and moreover, by definition,
h
i
∂V (t, x (t))
= exp (−ρt) V˙ (x (t)) − ρV (x (t)) .
∂t

Using these observations and the fact that Vx (t, (x (t))) = λ (t), the HamiltonJacobi-Bellman equation takes the “stationary” form
ρV (ˆ
x (t)) = f (ˆ
x (t) , yˆ (t)) + λ (t) g (t, xˆ (t) , yˆ (t)) + V˙ (ˆ
x (t)) .
This is a very common equation in dynamic economic analysis and can be interpreted
as a “no-arbitrage asset value equation”. We can think of V (x) as the value of an
asset traded in the stock market and ρ as the required rate of return for (a large
number of) investors. When will investors be happy to hold this asset? Loosely
speaking, they will do so when the asset pays out at least the required rate of
return. In contrast, if the asset pays out more than the required rate of return,
there would be excess demand for it from the investors until its value adjusts so
that its rate of return becomes equal to the required rate of return. Therefore, we

can think of the return on this asset in “equilibrium” being equal to the required
rate of return, ρ. The return on the assets come from two sources: first, “dividends,”
that is current returns paid out to investors. In the current context, we can think of
this as f (ˆ
x (t) , yˆ (t)) + λ (t) g (t, xˆ (t) , yˆ (t)) (with an argument similar to the above
discussion). If this dividend were constant and equal to d, and there were no other
returns, then we would naturally have that V (x) = d/ρ or
ρV (x) = d.
339