Prof. Dr. Thomas Steger Advanced Macroeconomics | Lecture| SS 2013

Dynamic Optimization (June 25, 2013)

 No-Ponzi-Game condition  Method of Lagrange Multipliers  Dynamic Programming  Control Theory

Institut für Theoretische Volkswirtschaftslehre Makroökonomik

Dynamic Optimization

Institut für Theoretische Volkswirtschaftslehre Makroökonomik

The No-Ponzi-Game condition  Finite life (time may be continuous or discrete)  Everyone must repay his/her debt, i.e. leave the scene without debt at terminal point in time.  No-Ponzi-Game condition (NPGC) represents an equilibrium constraint that is imposed on every agent.

 Infinite life (time is continuous)  Assume Mr. Ponzi (and his dynasty) wishes to increase consumption today by x€. Consumption expenditures are being financed by borrowing money. Debt repayment as well as interest payments are being financed by increasing indebtedness further. Debt then evolves according to t

0

1

2

…

debt

x€

x(1+r)€

x(1+r)2€

…

 Noting that d(t)=‐a(t) the above NPGC may be stated as

Charles Ponzi (1910) Charles Ponzi became known 1920s as a swindler in for his money making scheme. He promised clients huge profits by buying discounted postal reply coupons in other countries and redeeming them at face value in the US as a form of arbitrage. In reality, Ponzi was paying early investors using the investments of later investors. This type of scheme is now known as a "Ponzi scheme". (Wikipedia, June 3rd 2013)

 If Mr. Ponzi increases consumption by x€, financed by employing his innovative financing scheme, debt evolves according to d(t)=ertx€ such that the present value of debt would remain positive, which is excluded 2

Dynamic Optimization

Institut für Theoretische Volkswirtschaftslehre Makroökonomik

The Method of Lagrange Multipliers (1)  Consider the problem of maximizing an intertemporal objective function extending over three periods

where ct denotes the control variable, xt the state variable, and 0