Econ 204A, Introduction to
Part 3:
Optimal Growth in Continuous
Time
Henning Bohn*
Economic agents facing intertemporal decision problems are at the heart of most modern macro models. Optimal growth examines these choices in a representative agent context. Depending on the applications, such models are sometimes presented with time viewed as continuous, and sometimes with time divided into discrete chunks. Macroeconomists must understand both. Continuous time tends to be more straightforward in deterministic models and for deriving qualitative insights, whereas discrete time is more natural in stochastic models and easier to calibrate numerically. We will first do continuous time and defer discrete time until later in the course (time permitting; like continued in 204B).
The class will follow RomerŐs ch.2A regarding substantive topics, covering optimization, phase diagrams, balanced growth, and fiscal policy applications. However, Romer does not cover Hamiltonians and the Maximum Principle, which is the standard approach to optimal control – a technique worth learning. Technical presentation will therefore differ from Romer. A good technical reference is Barro/Sala-i-MartinŐs technical appendix A.3. Barro/Sala-i-MartinŐs also have excellent substantive coverage in ch.2, but probably more than you want to read; I will try to extract the main ideas without extra required readings.
The continuous time setting can be interpreted as a limiting case of a discrete time model as one subdivides each period into smaller and smaller time intervals. Conversely, one can think of discrete time models as approximations that divide chunks of time into discrete periods. The continuous-time setup highlights the fact that consumption and investment are flow variables that can be adjusted continually, while capital is a stock variable that adjusts gradually. From a technical perspective, optimal consumption and investment decisions with continuous time can be described by differential equations that are often easier to solve than the difference equation that one encounters in discrete-time models.
Fiscal policy is an important application of dynamic modeling. Governments provide public goods and services and infrastructure, all financed by taxes. We can examine the economic impact of these activities by including government spending and taxes in the model. We will follow RomerŐs ch.2.7 and ch.11.1-11.2.
Money is a subject often ignored in general equilibrium models. If you are schooled in traditional undergraduate macro or read newspapers full of comments about Federal Reserve policy, the omission of money should strike you as a significant shortcoming. So, how does money fit into macroeconomic models? To give at least a partial answer, will spend one class on money. We will briefly survey some alternative ways of modeling money—money-in-utility models, cash in advance models, transactions cost models, and (as a preview) overlapping-generations—and then focus on the Sidrauski model. The paper is a classic, a clean presentation of monetary theory that fits nicely into continuous-time analysis and provides another application of optimal control methods.