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<!DOCTYPE html>
<html lang="en" xml:lang="en" >
<head>
<title></title>
<meta charset="utf-8" />
<meta name="generator" content="TeX4ht (https://tug.org/tex4ht/)" />
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<link rel="stylesheet" type="text/css" href="EquityPremiumPuzzle.css" />
<meta name="src" content="EquityPremiumPuzzle.tex">
</head><body
>
<!--l. 6--><p class="noindent" ><img
src="EquityPremiumPuzzle0x.svg" alt="© " class="math";align="absmiddle"><span
class="ecrm-0600">October 30, 2023, </span><a
href="https://www.econ2.jhu.edu/people/ccarroll/" ><span
class="ecrm-0600">Christopher D. Carroll</span></a> <a
href="https://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes" ><span
class="ecrm-0600">EquityPremiumPuzzle</span></a>
<div
class="centerline"> <span
class="ecrm-2074">The Equity Premium Puzzle and the Riskfree Rate</span> </div>
<!--l. 15--><p class="indent" > <a
href="#XmehraPrescottPuzzle"><span
class="ecrm-1440">Mehra and Prescott</span></a><span
class="ecrm-1440"> (</span><a
href="#XmehraPrescottPuzzle"><span
class="ecrm-1440">1985</span></a><span
class="ecrm-1440">) consider a representative agent solving the joint</span>
<span
class="ecrm-1440">consumption and portfolio allocation problem:</span>
<table
class="equation"><tr><td><a
id="x1-2r1"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle1x.svg" alt=" [ ]
∑∞
v(mt ) = max u (ct) + 𝔼t βnu (ct+n )
{ct,ςt}
n=1
s.t.
mt+1 = (mt − ct)Rt+1 + yt+1
Rt+1 = ςtRt+1 + (1 − ςt)R
" class="math-display" ></div>
</td></tr></table>
<!--l. 21--><p class="nopar" >
<span
class="ecrm-1440">where</span> <img
src="EquityPremiumPuzzle2x.svg" alt="R " class="math";align="absmiddle"> <span
class="ecrm-1440">denotes the return on a perfectly riskless asset and</span> <img
src="EquityPremiumPuzzle3x.svg" alt="Rt+1 " class="math";align="absmiddle"> <span
class="ecrm-1440">denotes the</span>
<span
class="ecrm-1440">return on equities (the risky asset) held between periods</span> <img
src="EquityPremiumPuzzle4x.svg" alt="t " class="math";align="absmiddle"> <span
class="ecrm-1440">and</span> <img
src="EquityPremiumPuzzle5x.svg" alt="t + 1 " class="math";align="absmiddle"><span
class="ecrm-1440">;</span> <img
src="EquityPremiumPuzzle6x.svg" alt="ςt " class="math";align="absmiddle"> <span
class="ecrm-1440">is</span>
<span
class="ecrm-1440">the share of end-of-period savings invested in the risky asset;</span> <img
src="EquityPremiumPuzzle7x.svg" alt="Rt+1 " class="math";align="absmiddle"> <span
class="ecrm-1440">is the</span>
<span
class="ecrm-1440">portfolio-weighted rate of return; and</span> <img
src="EquityPremiumPuzzle8x.svg" alt="yt+1 " class="math";align="absmiddle"> <span
class="ecrm-1440">is noncapital income in period</span>
<img
src="EquityPremiumPuzzle9x.svg" alt="t + 1 " class="math";align="absmiddle"><span
class="ecrm-1440">.</span>
<!--l. 24--><p class="indent" > <span
class="ecrm-1440">As usual, the objective can be rewritten in recursive form:</span>
<table
class="equation"><tr><td><a
id="x1-3r1"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle10x.svg" alt=" ⌊ ( ) ⌋
v (mt ) = max u (ct) + β 𝔼t ⌈v ( [ςtRt+1--+ (1-−-ςt)R-](mt − ct) + yt+1 ) ⌉
{ct,ςt} ◟ ◝◜ ◞
Rt+1
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(1)</span></td></tr></table>
<!--l. 27--><p class="nopar" >
<!--l. 29--><p class="indent" > <span
class="ecrm-1440">The first order condition with respect to</span> <img
src="EquityPremiumPuzzle11x.svg" alt="ct " class="math";align="absmiddle"> <span
class="ecrm-1440">is</span>
<table
class="equation"><tr><td><a
id="x1-4r2"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle12x.svg" alt=" ′ ′
u (ct) = β 𝔼t[Rt+1v (mt+1 )].
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(2)</span></td></tr></table>
<!--l. 32--><p class="nopar" >
<span
class="ecrm-1440">and, taking</span> <img
src="EquityPremiumPuzzle13x.svg" alt="m " class="math";align="absmiddle"> <span
class="ecrm-1440">and</span> <img
src="EquityPremiumPuzzle14x.svg" alt="c " class="math";align="absmiddle"> <span
class="ecrm-1440">as given, the FOC with respect to</span> <img
src="EquityPremiumPuzzle15x.svg" alt="ςt " class="math";align="absmiddle"> <span
class="ecrm-1440">is</span>
<table
class="equation"><tr><td><a
id="x1-5r3"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle16x.svg" alt=" ′
𝔼t [(Rt+1 − R)v (mt+1 )(mt − ct)] = 0
′
𝔼t [(Rt+1 − R)v (mt+1 ) ] = 0.
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(3)</span></td></tr></table>
<!--l. 38--><p class="nopar" >
<!--l. 40--><p class="indent" > <span
class="ecrm-1440">But the usual logic of the </span><a
href="https://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes/consumption/Envelope" ><span
class="ectt-1440">Envelope</span></a> <span
class="ecrm-1440">theorem tells us that</span>
<table
class="equation"><tr><td><a
id="x1-6r4"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle17x.svg" alt=" ′ ′
u (ct+1 ) = v (mt+1 ),
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(4)</span></td></tr></table>
<!--l. 43--><p class="nopar" >
<span
class="ecrm-1440">so, substituting (</span><a
href="#x1-6r4"><span
class="ecrm-1440">4</span><!--tex4ht:ref: eq:envelope --></a><span
class="ecrm-1440">) into (</span><a
href="#x1-4r2"><span
class="ecrm-1440">2</span><!--tex4ht:ref: eq:cfoc --></a><span
class="ecrm-1440">) and (</span><a
href="#x1-5r3"><span
class="ecrm-1440">3</span><!--tex4ht:ref: eq:gamfoc --></a><span
class="ecrm-1440">), the above FOC’s reduce to</span>
<table
class="equation"><tr><td><a
id="x1-7r5"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle18x.svg" alt="u ′(c ) = 𝔼 [βR u′(c )],
t t t+1 t+1
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(5)</span></td></tr></table>
<!--l. 47--><p class="nopar" >
<span
class="ecrm-1440">and</span>
<table
class="equation"><tr><td><a
id="x1-8r6"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle19x.svg" alt=" ′
𝔼t [(Rt+1 − R )u (ct+1 )] = 0.
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(6)</span></td></tr></table>
<!--l. 51--><p class="nopar" >
<!--l. 54--><p class="indent" > <span
class="ecrm-1440">Now assume CRRA utility,</span> <img
src="EquityPremiumPuzzle20x.svg" alt="u(c) = c1− ρ∕(1 − ρ) " class="math";align="absmiddle"> <span
class="ecrm-1440">and divide both sides by</span>
<img
src="EquityPremiumPuzzle21x.svg" alt=" − ρ
ct " class="math";align="absmiddle"> <span
class="ecrm-1440">to get</span>
<table
class="equation"><tr><td><a
id="x1-9r7"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle22x.svg" alt="𝔼 [(c ∕c )− ρ(R − R )] = 0.
t t+1 t t+1
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(7)</span></td></tr></table>
<!--l. 58--><p class="nopar" >
<span
class="ecrm-1440">and the consumption ratio can of course be rewritten as</span>
<div class="align"><img
src="EquityPremiumPuzzle23x.svg" alt="pict" ><a
id="x1-10r8"></a></div>
<!--l. 70--><p class="noindent" ><span
class="ecrm-1440">Now use the fact </span><a
href="https://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes/MathFactsList/TaylorOne" ><span
class="ectt-1440">TaylorOne</span></a><span
class="ecrm-1440">:</span>
<blockquote class="quote">
<ul class="itemize1">
<li class="itemize"><span
class="ecrm-1440">If</span> <img
src="EquityPremiumPuzzle24x.svg" alt="z " class="math";align="absmiddle"> <span
class="ecrm-1440">is small,</span> <img
src="EquityPremiumPuzzle25x.svg" alt="(1 + z)λ ≈ 1 + λz " class="math";align="absmiddle"></li></ul>
</blockquote> <span
class="ecrm-1440">to</span>
<span
class="ecrm-1440">approximate equation (</span><a
href="#x1-9r7"><span
class="ecrm-1440">7</span><!--tex4ht:ref: eq:ShareEulernew --></a><span
class="ecrm-1440">)</span>
<table
class="equation"><tr><td><a
id="x1-11r9"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle26x.svg" alt="𝔼 [(1 − ρΔ log c )(R − R )] ≈ 0.
t t+1 t+1
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(9)</span></td></tr></table>
<!--l. 74--><p class="nopar" >
<!--l. 76--><p class="noindent" ><span
class="ecrm-1440">Using one more fact,</span>
<blockquote class="quote">
<ul class="itemize1">
<li class="itemize"><img
src="EquityPremiumPuzzle27x.svg" alt="𝔼[xy ] = 𝔼 [x ]𝔼 [y ] + cov (x,y ), " class="math";align="absmiddle"></li></ul>
</blockquote>
<!--l. 82--><p class="noindent" ><span
class="ecrm-1440">we get</span>
<table
class="equation"><tr><td><a
id="x1-12r10"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle28x.svg" alt="(1 − ρ 𝔼t[Δ log ct+1])(R − 𝔼t [Rt+1 ]) + covt (− ρΔ log ct+1,− Rt+1 ) ≈ 0
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(10)</span></td></tr></table>
<!--l. 85--><p class="nopar" >
<span
class="ecrm-1440">or</span>
<table
class="equation"><tr><td><a
id="x1-13r11"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle29x.svg" alt=" ρcovt-(Δ--log--ct+1,-Rt+1--)
𝔼t [Rt+1 ] − R ≈
1 − ρ 𝔼t[Δ log ct+1 ]
≈ ρcovt (Δ log ct+1, Rt+1 )
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(11)</span></td></tr></table>
<!--l. 90--><p class="nopar" >
<span
class="ecrm-1440">where the last approximation holds because</span> <img
src="EquityPremiumPuzzle30x.svg" alt="𝔼 [Δ log c ]
t t+1 " class="math";align="absmiddle"> <span
class="ecrm-1440">is small. (See the</span>
<span
class="ecrm-1440">appendix for a derivation of the portfolio share when next period’s consumption</span>
<span
class="ecrm-1440">function is known).</span>
<!--l. 96--><p class="noindent" ><span
class="ecbx-1440">The Equity Premium Puzzle</span>
<!--l. 98--><p class="indent" > <span
class="ecrm-1440">Because this expression must hold at all</span> <img
src="EquityPremiumPuzzle31x.svg" alt="t " class="math";align="absmiddle"><span
class="ecrm-1440">, we can check it empirically by</span>
<span
class="ecrm-1440">calculating empirical estimates of the two components and assuming</span>
<span
class="ecrm-1440">that the sample averages correspond to the representative agent’s</span>
<span
class="ecrm-1440">expectations. That is, if we have data for periods</span> <img
src="EquityPremiumPuzzle32x.svg" alt="1 ...n " class="math";align="absmiddle"><span
class="ecrm-1440">, we assume</span>
<span
class="ecrm-1440">that the unconditional expectations correspond to the sample means,</span>
<img
src="EquityPremiumPuzzle33x.svg" alt=" ∑n
𝔼 [R ] = (1∕n ) s=1 Rs " class="math";align="absmiddle"><span
class="ecrm-1440">;</span> <img
src="EquityPremiumPuzzle34x.svg" alt=" ∑n
𝔼 [Δ log c] = (1 ∕n ) s=1 Δ log cs " class="math";align="absmiddle"><span
class="ecrm-1440">; and</span>
<img
src="EquityPremiumPuzzle35x.svg" alt=" ∑
cov (Δ log c,R ) = (1∕n ) n (Δ log c − 𝔼 [Δ log c])(R − 𝔼 [R ]).
s=1 s s " class="math";align="absmiddle">
<!--l. 109--><p class="indent" > <span
class="ecrm-1440">The equity premium puzzle is essentially that</span> <img
src="EquityPremiumPuzzle36x.svg" alt="cov( Δ log c, R ) " class="math";align="absmiddle"> <span
class="ecrm-1440">is very small</span>
<span
class="ecrm-1440">(about 0.004) but</span> <img
src="EquityPremiumPuzzle37x.svg" alt="𝔼 [R ] − R " class="math";align="absmiddle"> <span
class="ecrm-1440">is about 0.08 (stocks have earned real returns of</span>
<span
class="ecrm-1440">about 8 percent more than riskless assets over the historical period), which</span>
<span
class="ecrm-1440">means that the only way equation (</span><a
href="#x1-13r11"><span
class="ecrm-1440">11</span><!--tex4ht:ref: eq:eqprem --></a><span
class="ecrm-1440">) can hold is if</span> <img
src="EquityPremiumPuzzle38x.svg" alt="ρ " class="math";align="absmiddle"> <span
class="ecrm-1440">is implausibly large</span>
<span
class="ecrm-1440">(these values imply a value of</span> <img
src="EquityPremiumPuzzle39x.svg" alt="ρ = 20 " class="math";align="absmiddle"><span
class="ecrm-1440">).</span>
<!--l. 116--><p class="indent" > <span
class="ecrm-1440">How do we know what plausible values of</span> <img
src="EquityPremiumPuzzle40x.svg" alt="ρ " class="math";align="absmiddle"> <span
class="ecrm-1440">are? Consider the following. You</span>
<span
class="ecrm-1440">must choose between a gamble in which you consume $50,000 for the rest of your</span>
<span
class="ecrm-1440">life with probability 0.5 and $100,000 with probability 0.5, or consuming some</span>
<span
class="ecrm-1440">amount</span> <img
src="EquityPremiumPuzzle41x.svg" alt="X " class="math";align="absmiddle"> <span
class="ecrm-1440">with certainty. The coefficient of relative risk aversion determines</span>
<span
class="ecrm-1440">the</span> <img
src="EquityPremiumPuzzle42x.svg" alt="X " class="math";align="absmiddle"> <span
class="ecrm-1440">which would make you indifferent between consuming X or being</span>
<span
class="ecrm-1440">exposed to the gamble. For example, if</span> <img
src="EquityPremiumPuzzle43x.svg" alt="ρ = 0 " class="math";align="absmiddle"><span
class="ecrm-1440">, then you have no risk aversion at</span>
<span
class="ecrm-1440">all and you will be indifferent between $75,000 with certainty and the 50/50</span>
<span
class="ecrm-1440">gamble with expected value of $75,000. Here are the values of X associated with</span>
<span
class="ecrm-1440">different values of</span> <img
src="EquityPremiumPuzzle44x.svg" alt="ρ " class="math";align="absmiddle"> <span
class="ecrm-1440">(table taken from Mankiw and Zeldes</span><span
class="ecrm-1440"> </span><a
href="#Xmankiw&zeldes:stockholders"><span
class="ecrm-1440">Mankiw and</span>
<span
class="ecrm-1440">Zeldes</span></a><span
class="ecrm-1440"> (</span><a
href="#Xmankiw&zeldes:stockholders"><span
class="ecrm-1440">1989</span></a><span
class="ecrm-1440">).)</span>
<div class="table">
<!--l. 131--><p class="indent" > <figure class="float"
>
<div class="tabular"> <table id="TBL-1" class="tabular"
><colgroup id="TBL-1-1g"><col
id="TBL-1-1"></colgroup><colgroup id="TBL-1-2g"><col
id="TBL-1-2"></colgroup><tr
class="hline"><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-1-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-1-1"
class="td11"> <img
src="EquityPremiumPuzzle45x.svg" alt="ρ " class="math";align="absmiddle"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-1-2"
class="td11"> <img
src="EquityPremiumPuzzle46x.svg" alt="X " class="math";align="absmiddle"> </td>
</tr><tr
class="hline"><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-2-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-2-1"
class="td11"> 1 </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-2-2"
class="td11"> 70,711 </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-1-3-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-3-1"
class="td11"> 3 </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-3-2"
class="td11"> 63,246 </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-1-4-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-4-1"
class="td11"> 5 </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-4-2"
class="td11"> 58,565 </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-1-5-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-5-1"
class="td11"> 10 </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-5-2"
class="td11"> 53,991 </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-1-6-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-6-1"
class="td11"> 20 </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-6-2"
class="td11"> 51,858 </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-1-7-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-7-1"
class="td11"> 30 </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-7-2"
class="td11"> 51,209 </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-1-8-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-8-1"
class="td11"> <img
src="EquityPremiumPuzzle47x.svg" alt="∞ " class="math";align="absmiddle"> </td><td style="white-space:nowrap; text-align:center;" id="TBL-1-8-2"
class="td11"> 50,000 </td>
</tr><tr
class="hline"><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-9-"><td style="white-space:nowrap; text-align:center;" id="TBL-1-9-1"
class="td11"> </td></tr></table></div>
</figure>
</div>
<!--l. 166--><p class="noindent" ><span
class="ecbx-1440">The Riskfree Rate Puzzle</span>
<span
class="ecrm-1440">Rewrite the consumption Euler equation (</span><a
href="#x1-7r5"><span
class="ecrm-1440">5</span><!--tex4ht:ref: eq:ceuler --></a><span
class="ecrm-1440">) as</span>
<table
class="equation"><tr><td><a
id="x1-14r12"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle48x.svg" alt=" ′ ′
u (ct) = 𝔼t [β(R + ςt[Rt+1 − R])u (ct+1 )]
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(12)</span></td></tr></table>
<span
class="ecrm-1440">and note that from (</span><a
href="#x1-9r7"><span
class="ecrm-1440">7</span><!--tex4ht:ref: eq:ShareEulernew --></a><span
class="ecrm-1440">) we know that</span> <img
src="EquityPremiumPuzzle49x.svg" alt=" ′
𝔼t[β ςt(Rt+1 − R )u (ct+1 )] = 0 " class="math";align="absmiddle"> <span
class="ecrm-1440">so that</span>
<span
class="ecrm-1440">(</span><a
href="#x1-14r12"><span
class="ecrm-1440">12</span><!--tex4ht:ref: eq:newgam --></a><span
class="ecrm-1440">) reduces to the ordinary Euler equation</span>
<table
class="equation"><tr><td><a
id="x1-15r13"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle50x.svg" alt="u ′(c ) = 𝔼 [βRu ′(c )]
t t t+1
1 = βR 𝔼t[(ct+1 ∕ct)− ρ ].
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(13)</span></td></tr></table>
<span
class="ecrm-1440">Using the same ‘facts’ and approximations as above, we get the standard</span>
<span
class="ecrm-1440">approximation to the Euler equation,</span>
<table
class="equation"><tr><td><a
id="x1-16r14"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle51x.svg" alt="Δ log c ≈ (1∕ρ )(r − 𝜗 ).
t+1
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(14)</span></td></tr></table> <span
class="ecrm-1440">The</span>
<span
class="ecrm-1440">‘riskfree rate puzzle’ is that average consumption growth per capita has been</span>
<span
class="ecrm-1440">about 1.5 percent (in the US in the postwar period) while real riskfree interest</span>
<span
class="ecrm-1440">rates have been at most 1 percent. Even if we assume a time preference rate of</span>
<img
src="EquityPremiumPuzzle52x.svg" alt="𝜗 = 0 " class="math";align="absmiddle"> <span
class="ecrm-1440">(no impatience at all, e.g.</span> <img
src="EquityPremiumPuzzle53x.svg" alt="β = 1 " class="math";align="absmiddle"><span
class="ecrm-1440">), the only way this equation can hold</span>
<span
class="ecrm-1440">is if</span> <img
src="EquityPremiumPuzzle54x.svg" alt="ρ " class="math";align="absmiddle"> <span
class="ecrm-1440">is a very small number (maybe even less than one). Of course, this is</span>
<span
class="ecrm-1440">precisely the opposite of the conclusion of the equity premium puzzle, which</span>
<span
class="ecrm-1440">implies the</span> <img
src="EquityPremiumPuzzle55x.svg" alt="ρ " class="math";align="absmiddle"> <span
class="ecrm-1440">must be very large.</span>
<div
class="centerline"> <span
class="ecrm-1728">Appenix</span> </div>
<div
class="centerline"> <span
class="ecbx-1728">The risky portfolio share when</span> <img
src="EquityPremiumPuzzle56x.svg" alt="c
t+1 " class="math";align="absmiddle"> <span
class="ecbx-1728">is Known</span> </div>
<!--l. 177--><p class="indent" > <span
class="ecrm-1440">In either the life cycle version of the model or an infinite horizon model that is</span>
<span
class="ecrm-1440">being solved by time iteration, the consumption function in</span> <img
src="EquityPremiumPuzzle57x.svg" alt="t + 1 " class="math";align="absmiddle"> <span
class="ecrm-1440">will be</span>
<span
class="ecrm-1440">known.</span>
<!--l. 179--><p class="indent" > <span
class="ecrm-1440">Designating that consumption function as</span> <img
src="EquityPremiumPuzzle58x.svg" alt="ct+1(m ) " class="math";align="absmiddle"><span
class="ecrm-1440">, with derivative</span>
<img
src="EquityPremiumPuzzle59x.svg" alt="c′ (m )
t+1 " class="math";align="absmiddle"><span
class="ecrm-1440">, we can derive an approximation to the optimal portfolio share as</span>
<span
class="ecrm-1440">follows.</span>
<!--l. 186--><p class="indent" > <span
class="ecrm-1440">First define</span> <img
src="EquityPremiumPuzzle60x.svg" alt="¯R (ς) = 𝔼 [R ]
t+1 t t+1 " class="math";align="absmiddle"> <span
class="ecrm-1440">and define consumption at the expectation of</span>
<span
class="ecrm-1440">the portfolio return as</span>
<div class="align"><img
src="EquityPremiumPuzzle61x.svg" alt="pict" ></div>
<!--l. 192--><p class="indent" > <span
class="ecrm-1440">For simplicity, we will henceforth assume that</span> <img
src="EquityPremiumPuzzle62x.svg" alt="yt+1 = 1 " class="math";align="absmiddle"><span
class="ecrm-1440">. Results below all go</span>
<span
class="ecrm-1440">through for the case where</span> <img
src="EquityPremiumPuzzle63x.svg" alt="𝔼t [yt+1 ] = 1 " class="math";align="absmiddle"><span
class="ecrm-1440">.</span>
<!--l. 194--><p class="indent" > <span
class="ecrm-1440">Calling the realized equity premium</span> <img
src="EquityPremiumPuzzle64x.svg" alt="φt+1 = (Rt+1 − R ) " class="math";align="absmiddle"><span
class="ecrm-1440">, note that for a</span>
<span
class="ecrm-1440">portfolio share of</span> <img
src="EquityPremiumPuzzle65x.svg" alt="ς " class="math";align="absmiddle"> <span
class="ecrm-1440">the realized return premium will be</span> <img
src="EquityPremiumPuzzle66x.svg" alt="Rt+1 = (Rt+1 − R )ς " class="math";align="absmiddle"><span
class="ecrm-1440">.</span>
<span
class="ecrm-1440">Now, use</span> <img
src="EquityPremiumPuzzle67x.svg" alt="u ′(∙) = ∙− ρ " class="math";align="absmiddle"> <span
class="ecrm-1440">to rewrite the FOC for</span> <img
src="EquityPremiumPuzzle68x.svg" alt="ς " class="math";align="absmiddle"><span
class="ecrm-1440">, equation</span><span
class="ecrm-1440"> </span><span
class="ecrm-1440">(</span><a
href="#x1-8r6"><span
class="ecrm-1440">6</span><!--tex4ht:ref: eq:ShareEuler --></a><span
class="ecrm-1440">)</span><span
class="ecrm-1440">:</span>
<table
class="equation"><tr><td><a
id="x1-17r15"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle69x.svg" alt=" 𝔼t[(ct+1(Rt+1at + 1) )− ρφt+1 ] = 0
𝔼t [(ct+1((Rt+1 − ¯Rt+1 + R¯t+1 )at + 1) )− ρφt+1 ] = 0
¯ − ρ
𝔼t [(ct+1 ((Rt+1 + ς φt+1 )at + 1) ) φt+1 ] = 0
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(15)</span></td></tr></table>
<!--l. 199--><p class="nopar" >
<!--l. 201--><p class="indent" > <span
class="ecrm-1440">Now make a first order Taylor expansion of next period’s consumption around</span>
<img
src="EquityPremiumPuzzle70x.svg" alt="¯ct+1 " class="math";align="absmiddle"> <span
class="ecrm-1440">(you could use a second order expansion for an even more accurate</span>
<span
class="ecrm-1440">approximation), and then use </span><a
href="https://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes/MathFactsList/TaylorOne" ><span
class="ectt-1440">TaylorOne</span></a><span
class="ecrm-1440">:</span>
<table
class="equation"><tr><td><a
id="x1-18r16"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle71x.svg" alt=" 𝔼 [(¯c + ςφ ¯c′ a )− ρφ ] ≈ 0
t t+1 t+1 t+1 t t+1
(¯ct+1 )− ρ 𝔼t [(1 + ¯c− 1 (ςφt+1 ¯c′ at))− ρφt+1 ] ≈ 0
t+1 ′ t+1− 1 − ρ
𝔼t [(1 + (ςφt+1 ¯c t+1at )¯ct+1) φt+1 ] ≈ 0
′ − 1
𝔼t [(1− ρ (ςφt+1 ¯ct+1at )¯ct+1 )φt+1 ] ≈ 0
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(16)</span></td></tr></table>
<!--l. 220--><p class="nopar" >
<span
class="ecrm-1440">where the last step uses</span> <img
src="EquityPremiumPuzzle72x.svg" alt=" δ
(1 + 𝜖) ≈ (1 + δ 𝜖) " class="math";align="absmiddle"><span
class="ecrm-1440">.</span>
<!--l. 223--><p class="indent" > <span
class="ecrm-1440">Now define the proportional MPC out of an additional unit of return</span>
<span
class="ecrm-1440">as</span>
<table
class="equation"><tr><td><a
id="x1-19r17"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle73x.svg" alt="κ = ¯c′ a ¯c − 1
t+1 t t+1
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(17)</span></td></tr></table>
<!--l. 226--><p class="nopar" >
<span
class="ecrm-1440">and calling</span> <img
src="EquityPremiumPuzzle74x.svg" alt="φ ≡ 𝔼t [φt+1 ] " class="math";align="absmiddle"> <span
class="ecrm-1440">and</span> <img
src="EquityPremiumPuzzle75x.svg" alt=" 2 2
𝔼t [(φt+1 − φ ) ] = σ φ " class="math";align="absmiddle"> <span
class="ecrm-1440">(so</span> <img
src="EquityPremiumPuzzle76x.svg" alt=" 2 2 2
𝔼t [φ t+1] = σφ − φ " class="math";align="absmiddle"><span
class="ecrm-1440">)</span>
<span
class="ecrm-1440">substitute this into the foregoing to obtain</span>
<table
class="equation"><tr><td><a
id="x1-20r18"></a>
<div class="math-display" >
<img
src="EquityPremiumPuzzle77x.svg" alt="𝔼 [(1 − ρ (κς φ ))φ ] ≈ 0
t t+1 t+1
φ ≈ ρκ ς 𝔼t[φ2 ]
2 t+1 2
φ ≈ ρκ ς(σ φ − φ )
( )
------φ--------
ρκ (σ2 − φ2 ) ≈ ς
φ
" class="math-display" ></div>
</td><td class="equation-label"><span
class="ecrm-1440">(18)</span></td></tr></table>
<!--l. 238--><p class="nopar" >
<h3 class="likesectionHead"><a
id="x1-1000"></a><span
class="ecrm-1440">References</span></h3>
<!--l. 5--><p class="noindent" >
<div class="thebibliography">
<p class="bibitem" ><span class="biblabel">
<a
id="Xmankiw&zeldes:stockholders"></a><span class="bibsp"><span
class="ecrm-1440"> </span><span
class="ecrm-1440"> </span><span
class="ecrm-1440"> </span></span></span><span
class="eccc1440-"><span
class="small-caps">M</span><span
class="small-caps">a</span><span
class="small-caps">n</span><span
class="small-caps">k</span><span
class="small-caps">i</span><span
class="small-caps">w</span><span
class="small-caps">,</span> <span
class="small-caps">N</span><span
class="small-caps">.</span></span><span
class="eccc1440-"> <span
class="small-caps">G</span><span
class="small-caps">r</span><span
class="small-caps">e</span><span
class="small-caps">g</span><span
class="small-caps">o</span><span
class="small-caps">r</span><span
class="small-caps">y</span><span
class="small-caps">,</span> <span
class="small-caps">a</span><span
class="small-caps">n</span><span
class="small-caps">d</span> <span
class="small-caps">S</span><span
class="small-caps">t</span><span
class="small-caps">e</span><span
class="small-caps">p</span><span
class="small-caps">h</span><span
class="small-caps">e</span><span
class="small-caps">n</span></span><span
class="eccc1440-"> <span
class="small-caps">P</span><span
class="small-caps">.</span> <span
class="small-caps">Z</span><span
class="small-caps">e</span><span
class="small-caps">l</span><span
class="small-caps">d</span><span
class="small-caps">e</span><span
class="small-caps">s</span> </span><span
class="ecrm-1440">(1989): “The</span>
<span
class="ecrm-1440">Consumption of Stockholders and Non Stockholders,” </span><span
class="ecti-1440">Journal of</span>
<span
class="ecti-1440">Financial Economics</span><span
class="ecrm-1440">, 15, 145–61.</span>
</p>
<p class="bibitem" ><span class="biblabel">
<a
id="XmehraPrescottPuzzle"></a><span class="bibsp"><span
class="ecrm-1440"> </span><span
class="ecrm-1440"> </span><span
class="ecrm-1440"> </span></span></span><span
class="eccc1440-"><span
class="small-caps">M</span><span
class="small-caps">e</span><span
class="small-caps">h</span><span
class="small-caps">r</span><span
class="small-caps">a</span><span
class="small-caps">,</span> <span
class="small-caps">R</span><span
class="small-caps">a</span><span
class="small-caps">j</span><span
class="small-caps">n</span><span
class="small-caps">i</span><span
class="small-caps">s</span><span
class="small-caps">h</span><span
class="small-caps">,</span> <span
class="small-caps">a</span><span
class="small-caps">n</span><span
class="small-caps">d</span> <span
class="small-caps">E</span><span
class="small-caps">d</span><span
class="small-caps">w</span><span
class="small-caps">a</span><span
class="small-caps">r</span><span
class="small-caps">d</span></span><span
class="eccc1440-"> <span
class="small-caps">C</span><span
class="small-caps">.</span> <span
class="small-caps">P</span><span
class="small-caps">r</span><span
class="small-caps">e</span><span
class="small-caps">s</span><span
class="small-caps">c</span><span
class="small-caps">o</span><span
class="small-caps">t</span><span
class="small-caps">t</span> </span><span
class="ecrm-1440">(1985): “The Equity</span>
<span
class="ecrm-1440">Premium: A Puzzle,” </span><span
class="ecti-1440">Journal of Monetary Economics</span><span
class="ecrm-1440">, 15, 145–61.</span>
</p>
</div>
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</html>