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CES Lectures

Discounting, Self-Control Problems, Myopia, andTime-Inconsistent Preferences

T. Scott Findley

Department of Economics and FinanceUtah State University

14 November 2013

Lecture 3Optimal Irrational Behavior and Applications

T. Scott F ind ley (2013) Lecture 3 Utah State University 1 / 20

Is it optimal if everyone is (economically) rational?

A Universal “Truth”in Neoclassical Economics

Welfare in competitive (market) equilibria is maximized if...

Output and factor markets are competitive.Individuals are price-takers (due to competition)...

in consumption decisions.in supplying factors of production to firms.

Firms are price-takers (due to competition)...

in purchasing inputs.in selling output.

Output of firms earns its value marginal product.Factors of production earn marginal productivities.Firms pursue profit maximization.Individuals act rationally (maximize utility) given prices.Economy is at a dynamically (Pareto) effi cient allocation.

Embodiment of this “truth”is Fundamental Welfare Theorems.

Optimal Irrational Behavior

BUT, the existence of competitive (market) equilibria...does NOT depend on assumption that individuals are rational.

Rational? ⇒ individuals maximize utility dynamically.aka Life-Cycle Utility Maximizers.

“Optimal Irrational Behavior”concept is somewhat confusing.Feigenbaum, Caliendo, & Gahramanov (2011).Feigenbaum & Caliendo (2010).

Competitive (market) equilibria with irrational households...can yield higher welfare than a fully rational economy.

Irrational? ⇒ individuals maximize “wrong”preferences.Irrational? ⇒ individuals don’t even optimize.

This is NOT a rediscovery of the Golden Rule allocation.Consumption and utility maximized at GR allocation...

constraint: allocations only have to be feasible.

Irrationality leads to higher welfare in competitive equilibria...constraint: allocations must be feasible.constraint: economic behavior interacts through markets.

The Key or Secret to OIB?

The existence of a pecuniary externality.Pecuniary externalities (McKean 1958; Prest & Turvey 1965).

Budget constraint affected via price changes.

Compared to traditional externality ⇒ utility directly affected.

Mechanism: pecuniary externality exists in production economies.Production economy ⇔ General-equilibrium economy.Irrationality cannot yield higher welfare than rationality...

in an endowment (partial-equilibrium) economy.

Process: in competitive markets...no one influences prices.Rational households maximize their own utility...assuming that their actions do NOT affect prices...even though collectively their actions DO affect prices.What if irrational individuals unknowingly coordinate...

on a behavioral rule that saves more than what is rational?

Aggregate saving ↑ ⇒ Aggregate capital ↑ ⇒ National income ↑Consumption possibilities and utility ↑ for everyone.

Is Overconfidence a Macroeconomic Virtue?

Findley and Bagchi (2013)

Dynamic Optimization

Time is continuous and indexed by t.Individual starts work at t = 0, passes away at t = T .receives income flow `(t)w for all t.

Savings account balance is k(t).Individual misperceives r = (1 + z)r∗,.

At any instant t0 ∈ [0, T ], individual solves

exp(−ρ(t− t0))c(t)1−φ

1− φ dt

subject to

dt= rk(t) + `(t)w − c(t), where k(t0) given and k(T ) = 0.

c(t) is control variable, ρ is discount rate, φ is the IEIS.Planned variables/parameters: r, c(t), and k(t).Actual variables/parameters: r∗, c∗(t), and k∗(t).k(t0) = k∗(t0), since the latter is history-dependent and observable.

with g ≡ (r − ρ)/φ, Maximum Principle yields planned path

c(t) =

exp(gt)

k∗(t0) exp(−rt0) +

`(v)w exp(−rv)dv

(g − r)

exp((g − r)T )− exp((g − r)t0),

Individual is naive; behaves as if future selves will follow plans.Individual persistently abandons and revises plans.k∗(t) did not grow as quickly as had planned.

Actual consumption envelope can be derived

c∗(t) =

exp(gt)

k∗(t) exp(−rt) +

t`(v)w exp(−rv)dv

(g − r)

exp((g − r)T )− exp((g − r)t),

c∗(t) is an implicit function since k∗(t) depends on c∗(t).This can be solved with actual law of motion

dk∗(t)

dt= r∗k∗(t) + `(t)w − c∗(t), for t ∈ [0, T ],

where return actually experienced is r∗, and k∗(0) = 0.

Aggregate quantities can be specified as

Y = KαL1−α,

0`(t)dt,

K = λ

0k∗(t)|r>r∗dt+ (1− λ)

0k∗(t)|r=r∗dt,

where λ and (1− λ) are overconfident rational fractions,

r∗ =∂Y

∂K− δ = αY/K − δ,

w =∂Y

∂L= (1− α)Y/L,

absent population and technology growth.

α is share of capital in national income; δ is depreciation rate.

Labor effi ciency profile is specified as

`(t) = `max exp(−µ(σt− 1)2

), µ ∈ R+, σ ∈ R+, for t ∈ [0, T ],

which is a continuously differentiable, quasi-normal function.

Can calibrate to properties of life-cycle income profile.

Note thatσ−1 = arg max `(t) .

Set σ = 1/25 to match income peaking at age 50.t = 25 in model, since t = 0 corresponds to age 25.

Ratio of peak income to initial income is about 1.4.

Therefore set `max`(0) = 1.4, and pin down µ

µ = ln

[`max`(0)

]= 34%.

Normalize `(0) = 1, which implies `max = exp(µ) = 1.4.

Compensating variations∫ T

0exp(−ρt) [(1 + Π)c∗(t)|nor=r∗ ]

1−φ

1− φ dt =

0exp(−ρt) [c∗(t)|or=r∗ ]

1−φ

1− φ dt

0exp(−ρt) [(1 + Ω)c∗(t)|nor=r∗ ]

1−φ

1− φ dt =

0exp(−ρt)

[c∗(t)|or>r∗

]1−φ1− φ dt

c∗(t)|nor=r∗ denotes consumption of rational individual...in an equilibrium with no overconfidence (all others rational).

c∗(t)|or>r∗ is actual consumption of overconfident individual.c∗(t)|or=r∗ denotes consumption of rational individual...

in an equilibrium with others who are overconfident.

Ω becomes ψ as λ→ 1.

Figure 1. Compensating Variation (%) in Partial Equilibrium

0 40 80 120 160 200 240 280 320 360 400

phi = 0.5

phi = 1

phi = 2

phi = 3

Figure 2. Compensating Variation (%) in General Equlibrium

0 40 80 120 160 200 240 280 320 360 400

phi = 0.5

phi = 1

phi = 2

phi = 3

Figure 1. Compensating Variation (%) in Partial Equilibrium

0 40 80 120 160 200 240 280 320 360 400

phi = 0.5

phi = 1

phi = 2

phi = 3

Figure 2. Compensating Variation (%) in General Equlibrium

0 40 80 120 160 200 240 280 320 360 400

phi = 0.5

phi = 1

phi = 2

phi = 3

Table 1c. Equilibrium Values for φ = 0.5 and ρ = 0.015.z (%) r∗ (%) r (%) w K Y K/Y ψ (%)0 2.42 2.42 1.25 434.9 129.5 3.4 0.0020 2.09 2.51 1.27 456.9 131.8 3.5 0.1940 1.85 2.59 1.29 474.6 133.5 3.6 0.2760 1.65 2.65 1.30 489.3 135.0 3.6 0.2880 1.50 2.70 1.31 501.6 136.1 3.7 0.25100 1.37 2.75 1.32 512.1 137.1 3.7 0.21120 1.27 2.78 1.33 521.2 138.0 3.8 0.15140 1.17 2.82 1.34 529.2 138.7 3.8 0.09160 1.10 2.85 1.34 536.2 139.4 3.8 0.03180 1.03 2.88 1.35 542.5 139.9 3.9 -0.03200 0.97 2.90 1.35 548.1 140.4 3.9 -0.09220 0.91 2.93 1.36 553.2 140.9 3.9 -0.15240 0.87 2.95 1.36 557.8 141.3 3.9 -0.21260 0.82 2.97 1.37 561.9 141.7 4.0 -0.27280 0.78 2.98 1.37 565.8 142.0 4.0 -0.32300 0.75 3.00 1.37 569.3 142.3 4.0 -0.37320 0.72 3.01 1.37 572.5 142.6 4.0 -0.42340 0.69 3.03 1.38 575.5 142.9 4.0 -0.47360 0.66 3.04 1.38 578.3 143.1 4.0 -0.51380 0.64 3.05 1.38 580.9 143.3 4.1 -0.56400 0.61 3.06 1.38 583.3 143.5 4.1 -0.60

Table 2a. Equilibrium Values for φ = 1 and ρ = 0.035.

z (%) r∗ (%) r (%) w K Y K/Y ψ (%)0 5.12 5.12 1.10 305.2 114.4 2.7 0.0020 4.51 5.41 1.13 328.5 117.4 2.8 0.8240 4.05 5.67 1.15 347.9 119.8 2.9 1.2660 3.69 5.91 1.17 364.4 121.7 3.0 1.4980 3.40 6.12 1.19 378.8 123.4 3.1 1.57100 3.16 6.32 1.20 391.4 124.8 3.1 1.58120 2.96 6.51 1.21 402.7 126.1 3.2 1.52140 2.78 6.68 1.23 412.8 127.2 3.2 1.43160 2.63 6.84 1.23 422.0 128.2 3.3 1.32180 2.50 6.99 1.24 430.3 129.0 3.3 1.19200 2.38 7.13 1.25 437.9 129.8 3.4 1.05220 2.27 7.26 1.26 445.0 130.6 3.4 0.91240 2.17 7.39 1.26 451.5 131.2 3.4 0.76260 2.08 7.50 1.27 457.6 131.8 3.5 0.61280 2.00 7.62 1.28 463.2 132.4 3.5 0.46300 1.93 7.73 1.28 468.5 132.9 3.5 0.31320 1.86 7.83 1.29 473.4 133.4 3.5 0.17340 1.80 7.93 1.29 478.1 133.9 3.6 0.02360 1.74 8.02 1.29 482.4 134.3 3.6 -0.12380 1.69 8.11 1.30 486.6 134.7 3.6 -0.25400 1.64 8.20 1.30 490.5 135.1 3.6 -0.39

Table 6. Compensating Variation (%) for Mixed Economy Assumption OC

λ (%)z (%) 10 20 30 40 50 60 70 80 9010 -0.09 -0.04 0.02 0.08 0.14 0.20 0.27 0.34 0.4020 -0.42 -0.30 -0.18 -0.06 0.07 0.21 0.36 0.51 0.6630 -0.89 -0.73 -0.54 -0.35 -0.14 0.08 0.32 0.56 0.8140 -1.48 -1.25 -1.01 -0.74 -0.45 -0.14 0.19 0.53 0.8950 -2.13 -1.85 -1.54 -1.20 -0.83 -0.43 -0.01 0.44 0.9160 -2.82 -2.50 -2.13 -1.71 -1.26 -0.77 -0.25 0.30 0.8870 -3.54 -3.17 -2.74 -2.26 -1.73 -1.15 -0.53 0.13 0.8280 -4.27 -3.86 -3.37 -2.82 -2.21 -1.54 -0.82 -0.06 0.7490 -5.00 -4.55 -4.02 -3.40 -2.71 -1.96 -1.14 -0.27 0.64100 -5.72 -5.24 -4.66 -3.98 -3.22 -2.38 -1.47 -0.50 0.52110 -6.44 -5.92 -5.30 -4.56 -3.73 -2.81 -1.80 -0.73 0.39120 -7.13 -6.60 -5.93 -5.14 -4.24 -3.24 -2.14 -0.97 0.25130 -7.81 -7.26 -6.55 -5.72 -4.75 -3.67 -2.48 -1.22 0.11140 -8.48 -7.90 -7.17 -6.28 -5.25 -4.09 -2.83 -1.47 -0.04150 -9.12 -8.53 -7.77 -6.84 -5.75 -4.52 -3.17 -1.72 -0.19160 -9.74 -9.14 -8.35 -7.38 -6.24 -4.94 -3.51 -1.97 -0.35170 -10.35 -9.74 -8.92 -7.91 -6.72 -5.35 -3.84 -2.22 -0.51180 -10.93 -10.31 -9.48 -8.44 -7.19 -5.76 -4.18 -2.46 -0.66190 -11.50 -10.87 -10.02 -8.95 -7.65 -6.16 -4.50 -2.71 -0.82200 -12.04 -11.42 -10.55 -9.45 -8.11 -6.56 -4.83 -2.95 -0.98

Note. K/Y ranges from 2.7 to 3.3 across the parameter space.

Table 7. Compensating Variation (%) for Mixed Economy Assumption Rational

λ (%)z (%) 10 20 30 40 50 60 70 80 90 10010 0.05 0.11 0.17 0.23 0.29 0.35 0.41 0.48 0.55 –20 0.10 0.21 0.33 0.45 0.58 0.72 0.86 1.01 1.16 –30 0.15 0.31 0.49 0.68 0.89 1.10 1.33 1.56 1.81 –40 0.19 0.41 0.65 0.91 1.19 1.48 1.80 2.13 2.48 –50 0.23 0.50 0.80 1.13 1.48 1.86 2.27 2.70 3.14 –60 0.27 0.59 0.94 1.34 1.77 2.24 2.74 3.26 3.81 –70 0.31 0.67 1.08 1.55 2.06 2.61 3.19 3.81 4.46 –80 0.34 0.75 1.22 1.75 2.33 2.96 3.64 4.35 5.09 –90 0.37 0.82 1.35 1.94 2.60 3.31 4.07 4.88 5.71 –100 0.40 0.89 1.47 2.13 2.86 3.65 4.50 5.39 6.31 –110 0.43 0.96 1.59 2.31 3.11 3.98 4.91 5.89 6.90 –120 0.45 1.02 1.70 2.48 3.35 4.30 5.31 6.37 7.47 –130 0.48 1.08 1.81 2.65 3.58 4.60 5.69 6.84 8.02 –140 0.50 1.14 1.91 2.81 3.81 4.90 6.07 7.29 8.55 –150 0.52 1.20 2.01 2.96 4.03 5.19 6.43 7.73 9.07 –160 0.54 1.25 2.11 3.11 4.24 5.47 6.79 8.16 9.57 –170 0.56 1.30 2.20 3.25 4.44 5.74 7.13 8.58 10.05 –180 0.58 1.35 2.29 3.39 4.64 6.01 7.46 8.98 10.52 –190 0.60 1.39 2.37 3.53 4.83 6.26 7.79 9.37 10.98 –200 0.61 1.43 2.45 3.65 5.02 6.51 8.10 9.75 11.42 –

Note. K/Y ranges from 2.7 to 3.3 across the parameter space.

Summary

Can irrational individuals benefit from not being rational?

YES, but not always.

Over-confidence ⇒ welfare gains.

Can rational individuals benefit if others not rational?

YES, if others are overconfident about return on saving.

Lesson: it still pays for you to be rational, but...

you will be better off if others are over-optimistic...compared to an equilibrium where everyone is rational.

Implications for public policy?

Should government/education try to temper overconfidence?

Via financial education/literacy.

Maybe not.

Perceptions and Information about Social Security

Findley (2013)

Age is discrete and indexed by t.Individual is young during t = 1 and old during t = 2.T is the length of a period measured in years.

Each period a mass-one continuum of individuals is born.Each cohort comprised of two types of behavior.

Fraction λ misperceives future social security benefits.Fraction 1− λ correctly perceives future benefits.

Period 1: disposable wage either consumed or saved

c1 + S = (1− θ)w.Period 2: savings (+ interest) and true SS benefit b consumed,

c2 = (1 + r)S + b

= (1 + r)S + θw.

where r is the net rate of return on savings.Yet, individual may have misperceptions of future SS benefits,

c2 = (1 + r)S + ψθw,

when in Period 1, where ψ ≥ 0.T. Scott F ind ley (2013) Lecture 3 Utah State University 14 / 20

Focus on competitive equilibria

No need to include indices by calendar time.

Individual maximizes

U(c1, c2) =[c1]1−σ

1− σ + β[c2]1−σ

1− σ ,

β = 1/(1 + ρ) is period discount factor, given ρ ≥ 0.σ is inverse elasticity of intertemporal substitution (IEIS).

Savings of the individual is

S = arg max

[(1− θ)w − S]1−σ

1− σ + β[(1 + r)S + ψθw]1−σ

1− σ

=(1− θ)w [β(1 + r)]1/σ − ψθw

1 + r + [β(1 + r)]1/σ,

where SM will denote savings of irrational individual (ψ 6= 1).where SR will denote savings of rational individual (ψ = 1).

Total quantity of labor and aggregate demand for capital are

L = λ+ 1− λ = 1,

K = λSM + (1− λ)SR,

Aggregate output/income is Cobb-Douglas,

Y = KαL1−α = Kα.

α is share of capital in aggregate income.

Factors of production are priced competitively,

r(K) =∂Y

∂K− δ = α

K− δ = αKα−1 − δ,

w(K) =∂Y

∂L= (1− α)

L= (1− α)Kα.

where capital depreciates at the period rate δ.Existence of competitive equilibria: given λ and ψ...

An equilibrium is characterized by a capital stock K...K is the aggregation of savings across consumer types...given prices, r and w.

Factor prices obey marginal conditions, given K.T. Scott F ind ley (2013) Lecture 3 Utah State University 16 / 20

Utility actually experienced in this equilibrium is based on...

true level of benefits received during retirement, even though...behavior when young may have reflected misperceptions.

Some compensating variations, χ and ν,

[(1 + χ)cpp1R]1−σ

1− σ + β[(1 + χ)cpp2R]1−σ

1− σ =[cm1R]1−σ

1− σ + β[cm2R]1−σ

1− σ ,

[(1 + ν)cpp1R]1−σ

1− σ + β[(1 + ν)cpp2R]1−σ

1− σ =[cm1M ]1−σ

1− σ + β[cm2M ]1−σ

1− σ ,

cpp1R, cpp2R is allocation of rational individual in an equilibrium...

in which everyone else also has perfect perceptions.

cm1M , cm2M is allocation of individual with misperceptions.cm1R, cm2R is allocation of rational individual in an equilibrium...

where others have misperceptions about future SS benefits.

Parameter Values for Numerical Examples

θ = 0.106

α = 0.35

βa =1

1 + ρa, where βa and ρa is annual discount factor and rate.

β = (βa)T where T is the number of years in a period.βa ≈ 0.96 for ρa = 0.04, therefore β = (βa)T ≈ 0.31 for T = 30.

If δa is annual rate of capital depreciation, then 1− δ = (1− δa)T

If δa = 0.08, then δ ≈ 0.92 for T = 30.

Compensating Variation (%) in Partial and General Equilibrium

P.E. G.E.ψ (%) η|σ=0.5 η|σ=1 η|σ=2 η|σ=0.5 η|σ=1 η|σ=20 -0.11 -0.22 -0.44 2.73 4.58 6.6120 -0.07 -0.14 -0.29 2.15 3.65 5.3640 -0.04 -0.08 -0.16 1.59 2.72 4.0660 -0.02 -0.04 -0.07 1.05 1.80 2.7380 0.00‡ -0.01 -0.02 0.52 0.89 1.37100 0.00 0.00 0.00 0.00 0.00 0.00120 0.00‡ -0.01 -0.02 -0.50 -0.88 -1.39140 -0.02 -0.04 -0.08 -1.00 -1.75 -2.78160 -0.04 -0.09 -0.18 -1.48 -2.60 -4.17180 -0.08 —0.16 -0.33 -1.94 -3.43 -5.56200 -0.12 -0.25 -0.52 -2.40 -4.26 -6.94

Note: λ = 1, η = ν.

‡ Strictly less than zero at higher decimal places.

General-equilibrium objects for σ = 0.5

ψ (%) r w K Y K/Y η (%)

0 1.882 0.212 0.041 0.326 0.125 2.7320 1.930 0.210 0.040 0.323 0.123 2.1540 1.976 0.208 0.039 0.321 0.121 1.5960 2.022 0.207 0.038 0.318 0.119 1.0580 2.067 0.205 0.037 0.315 0.117 0.52100 2.111 0.203 0.036 0.313 0.116 0.00120 2.155 0.202 0.035 0.310 0.114 -0.50140 2.198 0.200 0.035 0.308 0.112 -1.00160 2.240 0.199 0.034 0.306 0.111 -1.48180 2.282 0.197 0.033 0.304 0.109 -1.94200 2.323 0.196 0.033 0.302 0.108 -2.40

Note: λ = 1, η = ν.

Figure 1. Compensating Variation in Partial and General Equilibrium

General-equilibrium objects for σ = 1

ψ (%) r w K Y K/Y η (%)

0 1.638 0.223 0.047 0.343 0.137 4.5820 1.712 0.219 0.045 0.338 0.133 3.6540 1.787 0.216 0.043 0.332 0.129 2.7260 1.862 0.213 0.041 0.328 0.126 1.8080 1.937 0.210 0.040 0.323 0.123 0.89100 2.012 0.207 0.038 0.319 0.119 0.00120 2.087 0.204 0.037 0.314 0.116 -0.88140 2.162 0.202 0.035 0.310 0.114 -1.75160 2.238 0.199 0.034 0.306 0.111 -2.60180 2.313 0.196 0.033 0.302 0.108 -3.43200 2.388 0.194 0.032 0.298 0.106 -4.26

Note: λ = 1, η = ν.

ψ (%) r w K Y K/Y η (%)

0 1.346 0.238 0.057 0.366 0.155 6.6120 1.445 0.232 0.053 0.358 0.148 5.3640 1.548 0.227 0.050 0.350 0.142 4.0660 1.653 0.222 0.047 0.342 0.136 2.7380 1.763 0.217 0.044 0.334 0.131 1.37100 1.875 0.212 0.041 0.327 0.125 0.00120 1.991 0.208 0.038 0.320 0.120 -1.39140 2.111 0.203 0.036 0.313 0.116 -2.78160 2.234 0.199 0.034 0.306 0.111 -4.17180 2.360 0.195 0.032 0.300 0.107 -5.56200 2.490 0.191 0.030 0.294 0.103 -6.94

Note: λ = 1, η = ν.

ψ (%) r w K Y K/Y η (%)

0 1.638 0.223 0.047 0.343 0.137 4.5820 1.712 0.219 0.045 0.338 0.133 3.6540 1.787 0.216 0.043 0.332 0.129 2.7260 1.862 0.213 0.041 0.328 0.126 1.8080 1.937 0.210 0.040 0.323 0.123 0.89100 2.012 0.207 0.038 0.319 0.119 0.00120 2.087 0.204 0.037 0.314 0.116 -0.88140 2.162 0.202 0.035 0.310 0.114 -1.75160 2.238 0.199 0.034 0.306 0.111 -2.60180 2.313 0.196 0.033 0.302 0.108 -3.43200 2.388 0.194 0.032 0.298 0.106 -4.26

Note: λ = 1, η = ν.

ψ (%) r w K Y K/Y η (%)

0 1.346 0.238 0.057 0.366 0.155 6.6120 1.445 0.232 0.053 0.358 0.148 5.3640 1.548 0.227 0.050 0.350 0.142 4.0660 1.653 0.222 0.047 0.342 0.136 2.7380 1.763 0.217 0.044 0.334 0.131 1.37100 1.875 0.212 0.041 0.327 0.125 0.00120 1.991 0.208 0.038 0.320 0.120 -1.39140 2.111 0.203 0.036 0.313 0.116 -2.78160 2.234 0.199 0.034 0.306 0.111 -4.17180 2.360 0.195 0.032 0.300 0.107 -5.56200 2.490 0.191 0.030 0.294 0.103 -6.94

Note: λ = 1, η = ν.

Table 5. Compensating Variation (η) that equalizes social welfare across equilibria