Introduction to Modern Economic Growth be a necessary condition for this alternative maximization problem Therefore, the Maximum Principle is implicitly maximizing the sum the original maximand R t1 R t1 f (t, x ˆ (t) , y (t)) dt plus an additional term λ (t) g (t, xˆ (t) , y (t)) dt Under0 standing why this is true provides much of the intuition for the Maximum Principle First recall that V (t, xˆ (t)) is defined in equation (7.33) as the value of starting at time t with state variable xˆ (t) and pursuing the optimal policy from then on We will see in the next subsection, in particular in equation (7.43), that ∂V (t, xˆ (t)) ∂x Therefore, similar to the Lagrange multipliers in the theory of constraint optiλ (t) = mization, λ (t) measures the impact of a small increase in x on the optimal value of the program Consequently, λ (t) is the (shadow) value of relaxing the constraint (7.29) by increasing the value of x (t) at time t.6 Moreover, recall that x˙ (t) = g (t, xˆ (t) , y (t)), so that the second term in the Hamiltonian is equivalent to R t1 λ (t) x˙ (t) dt This is clearly the shadow value of x (t) at time t and the increase in the stock of x (t) at this point Moreover, recall that x (t) is the state variable, thus we can think of it as a “stock” variable in contrast to the control y (t), which corresponds to a “flow” variable Therefore, maximizing (7.40) is equivalent to maximizing instantaneous returns as given by the function f (t, xˆ (t) , y (t)), plus the value of stock of x (t), as given by λ (t), times the increase in the stock, x˙ (t) This implies that the essence of the Maximum Principle is to maximize the flow return plus the value of the current stock of the state variable This stock-flow type maximization has a clear economic logic Let us next turn to the interpreting the costate equation, λ˙ (t) = −Hx (t, xˆ (t) , yˆ (t) , λ (t)) = −fx (t, xˆ (t) , yˆ (t)) − λ (t) gx (t, xˆ (t) , yˆ (t)) This equation is also intuitive Since λ (t) is the value of the stock of the state variable, x (t), λ˙ (t) is the appreciation in this stock variable A small increase in x 6Here I am using the language of “relaxing the constraint” implicitly presuming that a high value of x (t) contributes to increasing the value of the objective function This simplifies terminology, but is not necessary for any of the arguments, since λ (t) can be negative 338