Introduction to Modern Economic Growth Proof We have V (t0 , x0 ) ≡ = Z ∞ t0 Z t1 f (t, xˆ (t) , yˆ (t)) dt f (t, xˆ (t) , yˆ (t)) dt + t0 Z ∞ f (t, xˆ (t) , yˆ (t)) dt t1 R∞ The proof is completed if V (t1 , xˆ (t1 )) = t1 f (t, xˆ (t) , yˆ (t)) dt By definition R∞ V (t1 , xˆ (t1 )) ≥ t1 f (t, x (t) , y (t)) dt for all admissible (x (t) , y (t)) Thus this R∞ equality can only fail if V (t1 , xˆ (t1 )) > t1 f (t, xˆ (t) , yˆ (t)) dt To obtain a con- tradiction, suppose that this is the case Then there must exist an admissible pair R∞ x (t) , y˜ (t)) with x˜ (t1 ) = xˆ (t1 ) such that t1 f (t, x˜ (t) , y˜ (t)) dt > from t1 onwards, (˜ R∞ f (t, xˆ (t) , yˆ (t)) dt Then construct the pair (x (t) , y (t)) such that (x (t) , y (t)) = t1 x (t) , y˜ (t)) for all t ≥ t1 Since (ˆ x (t) , yˆ (t)) for all t ∈ [t0 , t1 ] and (x (t) , y (t)) = (˜ (˜ x (t) , y˜ (t)) is admissible from t1 onwards with x˜ (t1 ) = xˆ (t1 ), (x (t) , y (t)) is admis- sible, and moreover, Z Z ∞ f (t, x (t) , y (t)) dt = t0 = > = t1 t0 Z t1 t0 Z t1 Zt0∞ f (t, x (t) , y (t)) dt + f (t, xˆ (t) , yˆ (t)) dt + f (t, xˆ (t) , yˆ (t)) dt + Z ∞ t1 Z ∞ t1 Z ∞ f (t, x (t) , y (t)) dt f (t, x˜ (t) , y˜ (t)) dt f (t, xˆ (t) , yˆ (t)) dt t1 f (t, xˆ (t) , yˆ (t)) dt t0 = V (t0 , x0 ) , which contradicts (7.32) establishing that V (t1 , xˆ (t1 )) = thus (7.33) R∞ t1 f (t, xˆ (t) , y (t)) dt and Ô Two features in this version of the Principle of Optimality are noteworthy First, in contrast to the similar equation in the previous chapter, it may appear that there is no discounting in (7.33) This is not the case, since the discounting is embedded in the instantaneous payoff function f , and is thus implicit in V (t1 , xˆ (t1 )) Second, this lemma may appear to contradict our discussion of “time consistency” in the previous chapter, since the lemma is stated without additional assumptions that ensure time consistency The important point here is that in the time consistency 333