Kapitel 5

• Study notes for Ph.D. Macroeconomics Chapter 5 cover:
• Representative Agent Real Business Cycles
• Neoclassical Growth Model
• Heterogenous Agent Models
• Instructor: Philip Jung, TU Dortmund, Winter 2024

Neocalssical RBC/Stochastic Growth Model

• The first section is a digression on the transversality condition and decentralization with equilibrium concepts.
• Goal: Understand uncertainty and the role of shocks in aggregate business cycles.
• Stochastic elements are introduced into recursive Bellman equations.
• Homework involves solving the stochastic model.
• Discussion on calibration issues.
• Dealing with stochasticity requires measure theory.
• Focus on discrete stochastic random variables.
• The Markov Property is essential, where z(t) belongs to a finite and compact set Z = {z1...zn}.
• z(t) follows a first-order Markov process.
• The probability that z(t) = zj is Pr(z(t) = zj | z(0)...z(t − 1)) = Pr(zt = zj | z(t − 1)).
• A generic element of the transition matrix is πjj' = Pr(z(t+1) = zj' | z(t) = zj).
• For any j,j' = 1...N, the sum over j' of πjj' equals 1.
• Instantaneous payoffs are expressed as U(x(t), x(t+1), z(t)).
• x(t) is within a subset of RK, and U maps from X × X × Z to the real numbers.
• The policy correspondence: x(t + 1) ∈ Γ(x(t), z(t)), with Γ mapping from X × Z to X.
• The history of variable z(t) up to date t is zt = (z(0), z(1)...z(t)).
• Zt is the t-fold product of Z.
• Consumption c(t) = č[zt] is, in general, a function of the entire history of realizations of the random variable z.
• The budget set is defined by: k[zt] ≤ z(t)k[zt-1]α + (1 − δ)k[zt-1] − č[zt]
• Maximization problem given k(0), z(0): max Σ βt U(c[zt] | z0)
• Objective function is maximized subject to: k[zt] ≤ z(t)k[zt-1]α + (1 − δ)k[zt-1] − č[zt]
• Stochastic neoclassical growth model considers that output y is given by that output y is given by y(t) = z(t)k(t)^α

Sequential and Recursive Problems

• Value function V(x(0), z(0)) to maximize the expected sum of discounted utilities
• The value function V(x(0), z(0)) is the supremum over sequences {x[zt]} from t=-1 to infinity of the sum from t=0 to infinity
• Sum of discounted utilities βt U(x[zt-1], x[zt], z(t)),
• x[zt] must be in Γ(x[zt-1], z(t)) for all t, and x[z-1] = x(0) is given.
• Expectation is taken over infinite sequences of z.
• The Markov chain assumption dictates next period's z' depends on the current z, but not past values.
• A second-order process would include not only z(t) but also z(t-1) as state variables.
• The recursive representation of the problem is: V(x, z) = sup {U(x, y, z) + βE[V(y, z') | z]} for all x in X, z in Z.
• V maps from X × Z to the real numbers, and constraint is y ∈ Γ(x, z) depends on z.
• The integral representation includes a Lebesgue integral of a function f with respect to the Markov process z. Integral Representation of the above
• $V(x, z) = \sup_{y \in \Gamma(x, z)} {U(x, y, z) + \beta \int [V(y, z') | z] \Pi(z, dz)}$
• The Lebesgue integral contains summation as a special case.
• $V(x, z_j) = \sup_{y \in \Gamma(x, z)} {U(x, y, z) + \beta \sum_{j'=1}^{N} \pi_{jj'} [V(y, z = z_{j'}) | z_j]}$

Theorems, Euler Equations, and Optimality

• Theorems hold if assumptions are exchanged.
• Γ(x, z) is nonempty for all x, z.
• For all x(0) in X, z(0) in Z, x in Φ(x(0), z(0)), the limit as n approaches infinity of the expected sum from t=0 to n of βt U(x[zt-1], x[zt], z(0)) exists and is finite.
• X is a compact subset of RK.
• Γ is nonempty, compact-valued, and continuous.
• XΓ = {(x, y, z) ∈ X × X × Z : y ∈ Γ(x, z)}, and U : XΓ → ℝ is continuous.
• The value function is V(x, z) = U(x, g(x, z), z) + βE[V(g(x, z), z') | z].
• The optimal policy function is y = g(x, z), a function of z.
• Stochastic Euler Equation: DyU(x*, y*, z) + βE[DxV(y*, z') | z] = 0
• The envelope condition is: DxV(x, z) = DxU(x, y*, z)
• The sequential version: DyU(x*[zt-1], x*[zt], zt) + βE[DxV(x*[zt], z(t+1)) | z] = 0 for all zt-1 in Zt-1.
• Stochastic Transversality Condition: lim βt E{DxU(x*[zs+t-1], x*[zs+t], z(t + s)) • x*[zs+t-1] | z(s)} = 0 as t approaches infinity.

Stochastic Neoclassical Growth Model

• Stochastic neoclassical growth model with endogenous hours worked expressed through the planner's problem
• Planner's problem involves maximizing : $V = \max_{{C_t, h_t, k_{t+1}}{t=0}^{\infty}} E\left[\sum{t=0}^{\infty} \beta^t u(C_t, 1-h_t) | I_t\right]$
• That has constraints : $C_t + k_{t+1} \leq e^{z_t} k_t^{\alpha} h_t^{1-\alpha} + (1-\delta)k_t$
• $Z_{t+1} = \rho z_t + \epsilon_t$, $\epsilon_t \sim N(0, \sigma_{\epsilon}^2)$
• $k_{t+j} \geq 0$, $k_0$ predeterminded
• $zt$ is an exogenous productivity shock with i.i.d. normal increments
• $I_t$ denotes the information set available to the planner
• FOC and transversality conditions include optimality conditions relating consumption, labor, capital accumulation, and the stochastic process.

Endogenous Growth Model and Transversality

• Endogenous growth model (planner) solves the following problem (V)
• $V = \sum_{t=0}^{\infty} \beta^t \frac{C_t^{1-\sigma}}{1-\sigma}$
• $C_t + k_{t+1} = Ak_t$
• New element is the production function does not have decreasing returns to scale
• It breaks down due to the Inada conditions
• $V = \sum_{t=0}^{\infty} \beta^t \frac{C_t^{1-\sigma}}{1-\sigma}$
• $= \frac{C_0^{1-\sigma}}{1-\sigma} + \beta \left(\frac{C_1}{C_0}\right)^{1-\sigma} \frac{C_0^{1-\sigma}}{1-\sigma} + \beta^2 \left(\frac{C_2}{C_1}\right)^{1-\sigma} \left(\frac{C_1}{C_0}\right)^{1-\sigma} \frac{C_0^{1-\sigma}}{1-\sigma} + .. $
• Simple FOC: transversality condition needed.

Decentralization: Firms and Government Problems

• Firm's Problem: Firms maximize profits given production function and factor prices.
• Constant returns to scale leads to an undetermined optimal firm size
• Capital to labor ratio is determined, leading to aggregate competitive firm behavior.
• The government faces a balanced budget: $W_t h_t \mathcal{T}{l,t} + (r_t - \delta) \mathcal{T}{c,t}k_t = g_t$ , $g_t = G$
• Government Policy: Goverment has fixed sequence of governmental expenditure
• Optimal policy rule, called Ramsey problem when two policy rule applies
• With only one such equation can only derives one policy rule $T_{c,t} = \overline{T_c}$

Decentralization: Household Problem

• Household maximizes utility subject to budget constraints.
• Taxes: $w_t h_t^i (1 - \mathcal{T}{l,t}) + (r_t - \delta) (1 - \mathcal{T}{c,t}) k_t^i -\mathcal{K}_{t+1}$
• Individual household: $c_t^i$ is consumption, $h_t$ is hours worked, and $k_t$ is capital.
• Fiscal Policy: Goverment uses instruments $T_{l,t}$ and $T_{c,t}$

Competitive Equilibrium Conditions

• All individuals must have the same initial endowment
• Consumer optimize
• Solve the firms problem
• Markets clear is feasible
• Goverment budget clear

Recursive Competitive Equilibrium

• A recursive competitive equilibrium includes consumption, capital, and labor supply functions.
• Also aggregate per capita versions, price functions for wages and interest rates, and a governmental rule.
• Individuals optimize and that prices should be consistent with firm optimization.
• The little k big K " notation specifies that individuals care principally for their own individual capital stock holding.

Agents With Heterogenous Information

• Agents care about aggregate capital
• Important to predict or calculate wages and interest rates. Need to understand:
  • law of motion
  • Aggregate prices

Heterogenous Agents and Self Insurance: Consumption and Savings

• Introduces market incompleteness and will study the basic consumption and saving problems Savings problems include:
  • Bewley (1977)
  • Huggett (1993): The risk free rate in Heterogeneous-Agent,incomplete insurance economies, JEDC
  • Aiyagari (1994): Uninsured Idiosyncratic Risk and Aggregate Saving, QJE
  • Sargent/Ljungquist Chapter 18

Savings Problem Formulation

• Asset Grid: Asset holdings are constrained to a particular grid, A = [-a0, a1, ...an].
• Labor income evolves according to an m-state Markov chain with transition matrix P.
• Maximize consumer policies and utility

Recursive Representation and Solution

• Bellman Equation: Includes states of the Markov chain and asset grid points.
• Solution: Provides a value function v(a, s) and a policy function a' = g(a, s).
• The A incorporates the lower and upper limits and the equations hold for each h, i
• The solution Matlab ( see Sargent and Ljungquist,chapter 4 for a simple matrix notation)

Wealth Distribution in Heterogenous Agent Models

• Idiosyncratic shocks cause asset holdings to diverge among initially identical individuals.
• To tracking of the distribution is crucial to understand economy dynamics
• The unconditional distribution given by At(a, s) = Prob(at = a, st = s),
• Law of Motion: Exogenous Markov Transition P, endogenous policy function a' = g(a, s)
• It includes the transition of individual measures over asset holdings.

Stationary Distribution Properties

• Stationary distribution characterized by: At+1(a'', s'') = At(a, s) It also captures what fraction of the time is spent on a per agent bases.

Pure Credit Economy (Huggett)

• The economy is simplified to an endowment economy where labor is constant.
• Individual can borrow and lend.
• Hugget specified that there is a lower bound on assets that can be held

Stationary Equilibrium (Huggett Economy)

• the interest rate r a policy function and invariant distrobution, with loan market clearing
• Given r the policy and probability fuction solve the problem

Computation

• Solve E(a) = 0 with iterate until convergence.
• It may or may not converge

Aiyagari Framework: Introducing Capital

• Model features are characterized by a stationary distribution and policy function.
• Equilibrium Factors include productively derived with competitive results

Aiyagari Equilibrium Conditions

• Prices satisfy: $w = \frac{\partial F(K,N)}{\partial N}$ and $\overline{r}(K_t) = \frac{\partial F(K,N)}{\partial K}$
• Polices satisfy Household optimization
• Invariant distribution is induced by the policy rule and Law of Motion
• Cross section must be consistent with what agents hold

Stationary Equilbrium: Computation

• Fixed point problem
• Guess K0 and price Solve individual policies and impose consistency is equilbrium

Heterogenous Agent Model Properties

• Need the borrowing contraint where B(1+r) <1 otherwise consumer will simply keep borrowing
• Euler equations and optimal polices are computed

Borrower

• Consumption equal amount of labor income after paying debt

Natural Borrowing Limits

• Imposing that consumption is less than zero
• At must be the amount the a debt that is in a region that can be payed
• It is not known however
• Taking average would make the model loose the value of contraints

Risk free Natural Borrowing Limits

• Holding 1 allows to hold
• Replace St with the worst possible state

Can beta be equal to 1+r?

• Consider Euler equation to check possible beta values
• Show at beta < 1 is is the euler equation is true

Moving Forward

• Embed krusell and smith into busincess cycle settings

Krusell Smith: Business Cycles in Heterogenous Agent Model

• Disturbance z and income
• It is subject the transition rules

Problem with Current Frame Work

• Disruption occurs when shock affect distributions
• All aggregate variables must be disturbed

Issues with Distribution

• Prices follow distribution of the law to motion, where expectation depend on cross section

General Critique of Models

• Benjamin Moll (2024): The Trouble with Rational Expectations in Heterogeneous Agent Models: A Challenge for Macroeconomics
• Argument: Assumes that equilibrium values are unrealsitic"

The main 3 criteria

• 1 simplification of numerical solutions
• 2 cosistency with empirical evidence, i.e. survey literature on expectation (see Bachmann handbook chapter)
• 3 immunity to Lucas critique

Way of Solving temporary equilibrium

• temporary equilibrium the same way that the expected value takes place Rational Exp must be model consistence