Introduction to Modern Economic Growth using current borrowings The consequence of this scheme would be that the asset holding of the individual would tend to −∞ as time goes by, clearly violating feasibility at the economy level To understand where this form³ of the no-Ponzi-game condition comes from, ´ RT multiply both sides of (8.9) by exp − r (s) ds to obtain Z t ả Z T ¶ Z T c (t) L(t) exp − r (s) ds dt + exp − r (s) ds A (T ) 0 Z t ả Z T = w (t) L (t) exp − r (s) ds dt + A (0) , 0 then divide everything by L (0) and note that L(t) grows at the rate n, to obtain Z t ả Z T ¶ Z T c (t) exp − (r (s) − n) ds dt + exp − (r (s) − n) ds a (T ) 0 µ Z t ¶ Z T = w (t) exp − (r (s) − n) ds dt + a (0) 0 Now take the limit as T → ∞ and use the no-Ponzi-game condition (8.11) to obtain µ Z t ả Z t ả Z Z c (t) exp − (r (s) − n) ds dt = a (0)+ w (t) exp − (r (s) − n) ds dt, 0 0 which requires the discounted sum of expenditures to be equal to initial income plus the discounted sum of labor income Therefore this equation is a direct extension of (8.9) to infinite horizon This derivation makes it clear that the no-Ponzi-game condition (8.11) essentially ensures that the individual’s lifetime budget constraint holds in infinite horizon 8.2 Characterization of Equilibrium 8.2.1 Definition of Equilibrium We are now in a position to define an equilibrium in this dynamic economy We will provide two definitions, the first is somewhat more formal, while the second definition will be more useful in characterizing the equilibrium below Definition 8.1 A competitive equilibrium of the Ramsey economy consists of paths of consumption, capital stock, wage rates and rental rates of capital, [C (t) , K (t) , w (t) , R (t)]∞ t=0 , such that the representative household maximizes its 378