Continuous-Time Methods in Macroeconomics, Lecture notes of Macroeconomics

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Continuous-Time Methods in Macroeconomics

Jes´us Fern´andez-Villaverde^1 and Galo Nu˜no^2 October 15, 2021 (^1) University of Pennsylvania

(^2) Banco de Espa˜na

Motivation

• Many interesting questions in macroeconomics require:
  1. Nonlinear techniques. Examples: How do financial crises arise? Why do countries or firms default? When do firms invest in large, lumpy projects? Why do individuals decide to migrate?
  2. Heterogeneous agents. Examples: What mechanisms account for changes in income and wealth inequality? Is there a trade-off between inequality and economic growth? How does inequality affect monetary and fiscal policy? What are the consequences of entry-exit in models of industry dynamics?
  3. Many state variables. Examples: Discrete node models, corporate finance models, rich life-cycle models, models where parameters are quasi-states.
• Often, all three elements come together. Example: heterogeneous agents models with nominal frictions and many assets.

Our goal

• Move to the “feasible” region of the Big-O complexity chart.
• This is relevant both for time and memory complexity.
• In particular, we want to find ways to keep the “curse of dimensionality” under control.

Taming the “curse of dimensionality”

• Three strategies:
  1. Better numerical algorithms (i.e., continuous-time methods, deep learning).
  2. Better software implementations (i.e., robust OS, modern programming languages, functional programming, flexible data structures, advances in massive parallelization).
  3. Better hardware designs (i.e., GPUs, AI accelerators, FPGAs).
• Some of these techniques are relatively new in economics or, at least, less familiar to many researchers.
• A complete treatment of the material would require at least a whole semester.
• In this class, we will focus on better numerical algorithms: continuous-time methods and deep learning.

Why continuous time? I

• Long and illustrious tradition in finance: classical results by Merton and others.
• However, less used in macroeconomics (except in growth and neoclassical investment theories).
• Why?
  1. Economic data comes in discrete intervals: most time-series is in discrete time.
  2. Arrival of dynamic programming in the early 1970s.
  3. Stochastic calculus has some entry cost (notice: in growth theory, you can often skip stochastic calculus because you deal with deterministic models).
• Recent “boom” of continuous-time methods in business cycle research and related areas: Stokey (2009), Brunnermeier and Sannikov (2014), Ahn et al. (2017), ...

Why continuous time? II

• Itˆo’s Lemma allow us to substitute the integrals of discrete time for derivatives in continuous time). Bellman equation: V (x) = max α

u (α, x) + β

V (x′) p(dx|α, x)

vs. Hamilton-Jacobi-Bellman equation:

ρVt (x) = ∂V ∂t

• max α

u (α, x) +

∑^ N

n=

μnt (x, α) ∂V ∂xn

+^1

∑^ N

n 1 ,n 2 =

σ^2 t (x, α)

n 1 ,n 2

∂^2 V

∂xn 1 ∂xn 2

• Why is this so important? Integrals depend on typical sets and typical sets are hard to characterize: the average member of a population with many dimensions (the “Asimov data set”) is an outlier.
• Check: https://mc-stan.org/users/documentation/case-studies/curse-dims.html.

Why deep learning?

• Neural networks are compositional while traditional functional approximation methods are additive.

Compare: y = f (x) ∼= g NN^ (x; θ) = θ 0 +

∑^ M

m=

θmφ

θ 0 ,m +

∑^ N

n=

θn,mxn

with a standard projection:

y = f (x) ∼= g CP^ (x; θ) = θ 0 +

∑^ M

m=

θmφm (x)

where φm is, for example, a Chebyshev polynomial.

• This crucial difference allows neural networks to break the “curse of dimensionality.”
• Furthermore, better hardware and software.

Course outline

1. Dynamic programming in continuous time.
2. Deep learning and reinforcement learning.
3. Heterogeneous agent models.
4. Optimal policy with heterogeneous agent models.

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