Log-Linearization Methods in OLG Models with an Application to - PowerPoint PPT Presentation

Log-Linearization Methods in OLG Models with an Application to Social Security Richard W. Evans Jeremy Perdue & Kerk L. Phillips

Overview • Build an OLG model with relatively short periods Model has: • demographic dynamics from exogenous birth rates, death rates and immigration o aggregate productivity shocks, but no idiosyncratic shocks to individuals within o cohorts Calibrate the model to the US economy and Social Security system. • Solve the model using linearization techniques commonly applied to • DSGE models. Using Monte Carlo methods, generate forecasts of the balance for the • Social Security trust fund with confidence bands corresponding to the uncertainty associated with business cycle & demographic fluctuations.

Questions & Motivation When will the social security trust fund run out? How do you solve DSGE models that are not stationary? Large amount of current research on countercyclical fiscal policy and reducing national debt. Two biggest problems for U.S. national debt (Soc. Sec. and Medicare/Medicaid) require OLG modeling. Need a solution for a nonstationary OLG model.

The Model – Demographics Cohorts decrease in size over time due to deaths, but also increase in size due to immigration. ′ 𝑂 𝑡+1 = 𝑂 𝑡 𝜍 𝑡+1 + 𝜅 𝑡+1 for 1 ≤ 𝑡 ≤ 𝑇 − 1 (2.1) New births each period are the fertility rate per period for each age cohort times the number of people in the cohort. ′ = 𝑇 𝑂 1 𝑔 𝑡 𝑂 𝑡 (2.2) 𝑡=1

The Model – Households 𝑡 ), interest Each period households receive wage income ( 𝑥𝓂 income[ 1 + 𝑠 − 𝜀 𝑙 𝑡 ], lump-sum transfers ( 𝑈 ), and social security benefits ( 𝑐 𝑡 ). 𝑡 ), and to purchase new capital They use these fund to pay taxes ( 𝜐𝑥𝓂 ′ ( 𝑙 𝑡+1 ) and consumption ( 𝑑 𝑡 ). The household’s problem is written using a Bellman equation: 𝑡+1 Ω ′ , 𝑙 𝑡+1 ′ 𝑊 𝑡 𝑙 𝑡 , Ω = max 𝑙 𝑡+1 ′ 𝑣*𝑑 𝑡 + + 𝛾𝜍 𝑡+1 𝐹*𝑊 + 𝑡 1 − 𝜐 + 1 + 𝑠 − 𝜀 𝑙 𝑡 − 𝑙 𝑡+1 ′ 𝑑 𝑡 = 𝑥𝓂 + 𝑐 𝑡 + 𝑈 (2.3) This problem yields the following Euler equation for a household of age s: 𝑣 𝑑 𝑑 𝑡 = 𝛾𝜍 𝑡+1 𝐹 𝑣 𝑑 *𝑑 𝑡 ′ +(1 + 𝑠 ′ − 𝜀) (2.4)

The Model – Firms Firms hire labor and capital to maximize profits each period. 𝐿,𝑀 𝐿 𝛽 (𝑓 𝑕𝑢+𝑨 𝑀) 1−𝛽 −𝑠𝐿 − 𝑥𝑀 max The solution is characterized by the following three equations. 𝑠 = 𝛽𝑍/𝐿 (2.5) 𝑥 = (1 − 𝛽)𝑍/𝑀 (2.6) 𝑍 = 𝐿 𝛽 (𝑓 𝑕𝑢+𝑨 𝑀) 1−𝛽 (2.7)

The Model – Stochastic Processes Technology is assumed to evolve over time according to the following law of motion. 𝑨 ′ = 𝜔 𝑨 𝑨 + 𝑓 𝑨 ′ ; 𝑓 𝑨′ ~𝑗𝑗𝑒(0, 𝜏 𝑨 2 ) (2.8)

The Model - Government The government accumulates a balance over time on a trust fund. 𝐼 ′ = 𝐼 + 𝑡 𝑆−1 𝑇 − 𝑂 𝑡 𝜐𝑥𝓂 𝑂 𝑡 𝑐 𝑡 (2.10) 𝑡=𝐹 𝑡=𝑆 AIME evolves over ages 𝐹 to 𝑆 according to: 𝜍 𝑡+1 𝑡 for 𝐹 ≤ 𝑡 ≤ 𝑆 − 1 ′ 𝑡−𝐹−1 1 𝑏 𝑡+1 = 𝑡−𝐹 𝑏 𝑡 + 𝑡−𝐹 𝑥𝓂 (2.13) 𝜍 𝑡+1 +𝜅 𝑡+1 Benefits are assigned when a household retires at age R and are a function of AIME at retirement. 𝑐 𝑆 = 𝜄𝑏 𝑆 (2.11) Once set at retirement benefits remain constant until death. 𝜍 𝑡+1 ′ 𝑐 𝑡+1 = 𝜍 𝑡+1 +𝜅 𝑡+1 𝑐 𝑡 for 𝑡 > 𝑆 (2.12)

The Model – Bequests We model redistribution of the capital of deceased households over the current population, by assuming an equal share for each household regardless of age. 𝑇 𝑂 𝑡 1−𝜍 𝑡 𝑙 𝑡 𝑈′ = 𝑡=1 (2.9) 𝑇 ′ 𝑂 𝑡 𝑡=1

The Model – Market Clearing The capital and labor market clearing conditions are given by: 𝑇 𝐿 = 𝑂 𝑡 𝑙 𝑡 + 𝐼 (2.14) 𝑡=1 𝑡 𝑇 𝑀 = 𝑂 𝑡 𝓂 (2.15) 𝑡=1 There is also a goods market clearing condition 𝑇 𝑍 + 1 − 𝜀 𝐿 = 𝑑 𝑡 + 𝐿′ , 𝑡=1 but it is redundant by Walras Law.

The Model - Stationarizing We must transform the non-stationary variables to stationary ones, denoted with a carat (^). Some per capita variables, such as consumption and wages, will grow at the long-run rate of g . ≡ 𝑦/𝑓 𝑕𝑢 for 𝑦 ∈ (*𝑙 𝑡 + 𝑡=2 𝑇 𝑆 𝑇 𝑇 𝑦 , *𝑏 𝑡 + 𝑡=𝐹 , *𝑐 𝑡 + 𝑡=𝑆 , *𝑑 𝑡 + 𝑡=1 , 𝑥) To transform cohort populations we need to remove a unit root, which we do by dividing by the total population, N . 𝑇 𝑦 ≡ 𝑦/𝑂 for 𝑦 ∈ (*𝑂 𝑡 + 𝑡=1 , 𝑀) Some aggregate variables grow at the rate 𝑕 and also have a unit root. ≡ 𝑦/(𝑂𝑓 𝑕𝑢 ) for 𝑦 ∈ (𝑍, 𝐿, 𝐼) 𝑦

Calibration S maximum age in periods E period workers enter the labor force R period workers retire 𝑡 + 𝑡=1 𝑇 *𝓂 effective labor by age 𝑇 *𝑔 𝑡 + 𝑡=1 average fertility rates by age 𝑇 *𝜅 𝑡 + 𝑡=2 average immigration rates by age 𝑇 *𝜍 𝑡 + 𝑡=2 average survival rates by age 𝜐 payroll tax rate 𝜀 capital depreciation rate 𝛾 subjective discount factor g growth rate of technology 𝛿 coefficient of relative risk aversion 𝛽 capital share in GDP 𝜄 pension benefits as percent of AIME In addition we have parameters governing the stochastic processes 𝑇 𝜔 𝑨 , *𝜔 𝑔𝑡 , 𝜔 𝜅𝑡 , 𝜔 𝜍𝑡 + 𝑡=1 autocorrelations 2 , 𝜏 𝜅𝑡 2 + 𝑡=1 2 , *𝜏 2 , 𝜏 𝑇 𝜏 𝑨 variances 𝜍𝑡 𝑔𝑡

Calibration E = 16 age of entry into the labor force in years R = 65 age of retirement in years 𝜀 = .10 capital depreciation rate per year 𝛾 = .98 subjective discount factor per year g = .005 growth rate of technology per year 𝛿 = 1.0 coefficient of relative risk aversion 𝛽 = .33 capital share in GDP 𝜄 = .20 pension benefits as percent of AIME 𝜐 = .0392 payroll tax rate 𝜔 𝑨 = .90 autocorrelation of technology (per year) 2 = .0004 𝜏 𝑨 variance of technology

Calibration Average monthly OASDI only benefit to earnings ratio 2001 18.66% 2002 18.75% 2003 18.34% 2004 18.20% 2005 18.30% 2006 18.42% 2007 18.58% 2008 18.56% 2009 19.56% average 18.60%

Calibration Fit a polynomial to effective labor by age 1.4 1.2 1 0.8 data fitted 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 100

Calibration Fit a polynomial to immigration rates by age 7.00% 6.00% 5.00% 4.00% data fitted 3.00% 2.00% 1.00% 0.00% 0 10 20 30 40 50 60 70 80 90 100

Calibration Fit a polynomial to fertility rates by age 140 120 100 80 data fitted 60 40 20 0 0 10 20 30 40 50 60

Calibration Fit a polynomial to death hazard rates by age (log scale) 1.0000 0.1000 data 0.0100 fitted 0.0010 0.0001 0 10 20 30 40 50 60 70 80 90 100

Calibration • Implied cumulative survival rates by age 1 0.9 0.8 0.7 0.6 data 0.5 fitted 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100

Steady State - Aggregate α 0.33 𝐿 0.9379 γ 1 𝐼 0.0000 𝑍 𝑕 * 0.005 0.8380 δ * 0.1 𝐷 0.2784 β * 0.98 𝐽 0.2180 θ 0.2 𝑀 0.7927 S 50 𝑠 * 0.1379 𝑥 0.7082 ψ z .9 𝑈 0.0242 𝐶 σ z .02 0.0218 𝑜 * 0.0094 τ 0.0389

Steady State - Population 3.50% Steady State… Current Distribution 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 94 98

Steady State - Household 3 After-Tax Wage Income Consumption 2.5 Assets SS Benefits Bequests 2 1.5 1 0.5 0 -0.5 -1 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

Linear Approximation - Example Illustrate the technique first on a simple infintely-lived agent DSGE model. Euler equation 𝑣 𝑑 𝑑 = 𝛾𝐹 𝑣 𝑑 *𝑑′+(1 + 𝑠 ′ − 𝜀) Budget constraint 𝑑 = 𝑥 + 1 + 𝑠 − 𝜀 𝑙 − 𝑙′ Firm’s optimization conditions 𝑠 = 𝛽𝐿 𝛽−1 (𝑓 𝑨 ) 1−𝛽 𝑍 𝑥 = 1 − 𝛽 𝐿 𝛽 𝑓 𝑨 −𝛽 Technology 𝑨 ′ = 𝑂𝑨 + 𝑓 ′ ; 𝑓′~𝑗𝑗𝑒(0, 𝜏 2 ) (5.1)

Linear Approximation - Example First categorize variables into three categories: 1. Exogenous state variables ( z ) 2. Endogenous state variables ( k ) 3. Non-state variables ( c , r , w ) Use a 1 st -order Taylor-series expansion to linearize the log of the Euler equation. The other equations are used as definitions 𝛾𝐹 (𝑑/𝑑′) 𝛿 (1 + 𝑠 ′ − 𝜀) = 1 Write the approximation as: ′′ + 𝐻𝑙 ′ + 𝐼𝑙 + 𝑀𝑨 ′ + 𝑁𝑨 + = 0 𝐹*𝐺𝑙 is the deviation of k from its steady state value Where 𝑙 And 𝑨 is the deviation of z from its steady state value