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 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