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1 Working Paper # Infinite Horizon CCAPM with Stochastic Taxation and Monetary Policy Revised from the Center for Risk Management Research Working Paper Konstantin Magin, University of California, Berkeley April 21, 2018 University of California Berkeley

2 Infinite Horizon CCAPM with Stochastic Taxation and Monetary Policy Revised from the Center for Risk Management Research Working Paper Konstantin Magin April 21, 2018 Abstract This paper derives the infinite horizon CCAPM with heterogeneous agents, stochastic dividend taxation and monetary policy. I find that under reasonable assumptions on assets dividends and probability distributions of the future dividend taxes and consumption, the model implies the constant price/after-tax dividend ratios. I also obtain that the higher current and expected dividend tax rates imply lower current asset prices. Finally, contrary to popular belief, monetary policy is neutral, in the long run, with respect to the real equilibrium asset prices. Keywords: Stochastic Taxation, GEI, Equity Premium, Complete Markets, Comparative Statics, Risk Aversion, CCAPM, Property Rights, Money Supply JEL Classification: D5; D9; E13; G12; H20. This research was supported by a grant from the Center for Risk Management Research. I am very grateful to Bob Anderson for his kind support, encouragement and insightful conversations. The University of California at Berkeley, The Center for Risk Management Research magin@berkeley.edu 1

3 1. INTRODUCTION Unlike monetary policy, the stochastic nature of taxation and its effects on equilibrium asset prices and allocations have received surprisingly little attention in the literature. Yet taxes are part of individuals and corporate budget constraints. Moreover, changes in various tax rates are driven by an ever-changing political balance of power and the direction of those changes seems to be highly unpredictable and therefore stochastic. Thus, it seems entirely appropriate to regard future taxation as both important and stochastic. The most recent research on the topic primarily focuses on insecure property rights in the finite horizon GEI model. Magin (2016 studies the comparative statics of asset prices with respect to various current and future tax rates in the finite horizon GEI model. He finds that under reasonable assumptions, an increase in the current or future stochastic dividend or endowment tax rates reduces current asset prices. Magin (2015 finds that under reasonable assumptions, in the finite horizon GEI model, FM equilibria exist for all stochastic tax rates, except for a closed set of measure zero. The research conducted so far on the role of stochastic taxation in the infinite horizon GEI model relies on the CCAPM with identical agents and focuses primarily on resolving the so-called Equity Premium Puzzle. 1 Magin (2015 develops a version of the CCAPM with insecure property rights. He finds that the current expected equity premium can be reconciled with a coeffi cient of relative risk aversion of 3.76, thus resolving a substantial part of the Equity Premium Puzzle in general stocks. Edelstein and Magin (2013 and (2017 use the CCAPM with insecure property rights developed in Magin (2015 to address the Equity Premium Puzzle in REITs. They find that the current expected equity premium in REITs can be reconciled with a coeffi cient of relative risk aversion of , thus resolving a substantial part of the Equity Premium Puzzle for securitized real estate. Sialm (2009, in an excellent empirical paper, demonstrates that aggregate stock valuation levels are related to measures of the aggregate personal tax burden on equity securities. Sialm (2006 develops a generalized version of the Lucas (1978 dynamic general equilibrium tree model of production economy (risky assets being in positive supply with identical agents and a flat consumption tax that follows a two-state Markov chain. The model is used to analyze the ef- 1 See DeLong and Magin (2009 for a literature review of the Equity Premium Puzzle. 2

4 fects of a flat consumption tax on asset prices. He finds that under plausible conditions, investors require higher term and equity premia as compensation for the risk introduced by tax changes. This paper derives the infinite horizon CCAPM with heterogeneous agents, stochastic dividend taxation and monetary policy and is a natural development of the emerging literature on the effects of insecure property rights on equilibrium asset prices and allocations. I find that under reasonable assumptions on assets dividends and probability distributions of the future dividend taxes and consumption, the model implies the constant price/after-tax dividend ratios. I also obtain that the higher current and expected dividend tax rates imply lower current asset prices. Finally, contrary to popular belief, monetary policy is neutral, in the long run, with respect to the real equilibrium asset prices. The paper is organized as follows. Section 2 derives and analyzes the infinite horizon CCAPM with stochastic taxation. Section 3 derives and analyzes the infinite horizon CCAPM with stochastic taxation and monetary policy. Section 4 concludes. 2. The CCAPM with Stochastic Dividend Taxation We start our analysis by stating the following lemma: LEMMA (Rubinstein (1976: Let random variables x and y be bivariate normally distributed with expectations (E x, E y (µ x, µ y and variance-covariance matrix ( σ 2 V x ρσ x σ y ρσ x σ y σ 2 y, where ρ COV x, y V ARxV ARy is the correlation coeffi cient between random variables x and y. That is, random variables x and y have joint density function 3

5 Then and f(x, y 1 2πσ xσ y 1 ρ 2 e a a 1 2(1 ρ 2 (x µ x 2 σ 2 x e x f(x, ydxdy e µ x + σ2 x 2 2 y 2ρ (x µ x (y µ y + (y µy2 σxσy σ 2 y. N( a+µ y σ y e y f(x, ydxdy e µ y + σ 2 N( a+µ x + ρσ y σ x + ρσ x, a e x+y f(x, ydxdy e µ x +µ y + (σ 2 x +2ρσxσy+σ2 y 2 N( a+µ x σ x + ρσ y + σ x. We will first use the Lemma to derive the infinite horizon CCAPM with stochastic taxation: THEOREM 1: Let K be the set of financial assets and I be the set of agents. Consider an economy with K < financial assets and I < agents, where the total supply of each asset is equal to 1 and agents maximize their utility function U i (c i, G E b T i (u i (+T + v i (G t+t i I, T 0 where u i is a CRRA utility function, such that u i (c c1 λ i 1 λ i, subject to +T + n z i kt+t +1 q kt+t n z i kt+t (q kt+t + (1 τ d t+t d kt+t (i, T I {0,..., }, where λ i is the coeffi cient of relative risk aversion, is the consumption of an agent i I at period t, z i kt is the number of shares of an asset k K held by an agent i I at period t, q kt is the price per share of an asset k K at period t, d kt is the dividend per share of an asset k K at period t, τ d t is the stochastic dividend tax rate at period t, G t is the government spending at period t given by 4

6 G t+t n τ d t+t d kt+t. The Transversality Condition (TC holds for all assets and agents, i.e., lim E b T i u i (+T q T u i ( kt+t 0 (k, i K I, where {+T } T 1 Assume further is the equilibrium consumption for an agent i I. c ti+t (1 τ d t+t d kt+t (τ d t+t (1 τ d t+t d kt+t (τ d t+t (k, i, T K I {0,..., }, i.e., all after-tax dividends are growing at the same rate and individuals consumption is growing at the same rate as total dividends. 2 ( (1 τ d E b i t+t d kt+t +1 (τ d t+t +1 1 λi e µ (1 τ d t d kt+t (τ d t+t c σ2 c < 1 3 (k, i, T and with K I {0,..., }, ln(b i ( +T +1 +T 1 λ i N(µ c, σ c (i, T I {0,..., } COV ln (+T1, ln (+T2 0 ( T 1, T 2 {1,..., } {1,..., }, T 1 T 2. Then q kt (τ d t, µ c, σ 2 e µ c σ2 c c (1 τ d 1 e µ c σ2 t d kt (τ d t k K. (1 c 2 Since our modified CCAPM describes a production economy, the total dividend d kt+t represents total output, i.e. GDP. 3 See Magin (2014 for calculations. 5

7 COROLLARY 2: Assume in addition that Suppose assumptions of the above Theorem hold. µ c σ2 c τ d t 0. Then and q kt τ d t e µ c σ2 c 1 e µ c σ2 c d kt (1 τ d τ d t d kt (τ d t k K t E qkt, 1 τ d t 1 + E d kt, 1 τ d t. PROOF: See Appendix. Poterba (2004 estimated that E dkt, 1 τ d t 1.14, E qkt, 1 τ d t 2.14, This is also consistent with Magin (2017 who using a finite horizon CCAPM with stochastic taxation, estimated that for S&P 500 holders COROLLARY 3: Then P E kt (τ d t, µ c, σ 2 c E qkt, 1 τ d t Suppose assumptions of the above Theorem hold. q kt EP S kt e µ c σ2 c 1 e µ c σ2 c (1 τ d t P O kt (τ d t, 4 Both findings are consistent with the current data we observe. A mere expectation of 15 20% corporate income tax cut generated roughly 20 30% of the US stock market appreciation across the board since November of

8 where P O kt is the payout ratio, EP S kt are earnings per share and P E kt is the price/earnings ratio k K. PROOF: Obvious. the price/earnings ratio P E kt is completely determined by purely subjective µ c, σ 2 c, companies payout policies P O kt (τ d t and government tax policies τ d t. So there is no upper or lower bound on the price/earnings ratio P E kt investors can reliably use in their decision-making process. Using historical numbers instead of subjective estimates for µ c, σ 2 c leads to a form of the Equity Premium Puzzle. Indeed, historically 5, Set we estimate Thus, Hence, E V AR ln( c t+l+1 c t+l ln( c t+l+1 c t+l 0.02, b 1 R f µ c ln( (1 λ i 0.02, σ 2 c (1 λ i µ c σ2 c ln( (1 λ i (1 λ i e µ c σ2 c e ln(0.99+(1 λ i (1 λ i Magin (2015 estimated that for S&P 500 holders So e µc+ 1 2 σ2 c 1 e µ c σ2 c Hence, λ i e ln(0.99+( ( e ln(0.99+( ( P E kt q kt EP S kt Let us now introduce monetary policy into our analysis. 5 Mehra (2003 and Mehra and Prescott (

9 3. The CCAPM with Stochastic Dividend Taxation and Monetary Policy We will now use the Lemma to derive the infinite horizon CCAPM with Stochastic Taxation and monetary policy: THEOREM 2: Let K be the set of financial assets and I be the set of agents. Consider an economy with K < financial assets and I < agents, where the total supply of each asset is equal to 1 and agents maximize their utility function U i (c i, G E b T i (u i (+T + v i (G t+t, T 0 where 0 < b i < 1 and u i is a CRRA utility function, such that u i (c c 1 λ i 1 λ i, subject to +T + n q kt+t z ikt+t +1 + M it+t +1 p t+t n (q kt+t +(1 τ d t+t d kt+t z ikt+t + M it+t p t+t (i, T I {0,..., }, where M it is the quantity of money held by an agent i I at period t, M t i IM it is the total supply of money, G t is the government spending at period t given by ( G t+t τ d t+t Mt+T +1 M t+t d kt+t +. p }{{ t+t } Seigniorage The Transversality Condition (TC holds for all assets and agents, i.e., lim E b T i u i (+T q T u i ( kt+t 0 (k, i K I, where {+T } T 1 is the equilibrium consumption for an agent i I. Assume further c ti+t (1 τ d Mi+T t+t d kt+t +1 M t+t p t+t (1 τ d Mt+1 M t d kt t p t (i, T I {0,..., }, 8

10 i.e., individuals consumption is growing at the same rate as the total output d kt+t (GDP consumed by the private sector, τ d t+t ( Mt+T d kt+t + +1 M t+t p t+t d kt+t g T {0,..., }, i.e., the percentage of the total output government is constant over time, d kt+t (GDP consumed by the d kt+t d kt d kt+t d kt (k, i, T K I {0,..., }, i.e., all dividends are growing at the same rate, ( 1 λi E b i dkt+t +1 d kt+t e µ c σ2 c < 1 6 (k, i, T K I {0,..., } and random variables x t+t ln (1 τ t+t +1, y it+t ln(b i ( c ti+t 1 λ i are bivariate normally distributed with expectations (E ln (1 τ t+t, E ln(b i ( c ti+t 1 λ i (µ τ, µ c and the variance-covariance matrix ( σ 2 V τ 0 0 σ 2 4 (i, T I {0,..., } c with COV ln (+T1, ln (+T2 0 ( T 1, T 2 {1,..., } {1,..., }, T 1 T 2. 4 See Magin (2014 for calculations. 9

11 Then q kt e µ τ σ2 τ e µ c σ2 c 1 e µ c σ2 c 4. Conclusion d kt (τ d t k K. This paper derives the infinite horizon CCAPM with heterogeneous agents, stochastic dividend taxation and monetary policy and is a natural development of the emerging literature on the effects of insecure property rights on equilibrium asset prices and allocations. I find that under reasonable assumptions on assets dividends and probability distributions of the future dividend taxes and consumption, the model implies the constant price/after-tax dividend ratios. I also obtain that the higher current and expected dividend tax rates imply lower current asset prices. Finally, contrary to popular belief, monetary policy is neutral, in the long run, with respect to the real equilibrium asset prices. Appendix PROOF OF THEOREM 1: We have that +T n z i kt+t (q kt+t + (1 τ d t+t d kt+t (τ d t+t n z i kt+t +1 q kt+t (i, T I {0,..., }. Substituting budget constraint into the utility function, we obtain U i (c i, G E b T i T 0 u i( z i kt+t (q kt+t + (1 τ d t+t d kt+t (τ d t+t z i kt+t +1 q kt+t + v i (G t+t }{{} +T Differentiating U i with respect to z i kt+1 and setting U i z i kt+1 to 0, we obtain q kt u i ( + b i E u i (+1 ( q kt+1 + (1 τ d t+1d kt+1 (τ d t

12 ( λi q kt E b i cti+t ( qkt+1 + (1 τ d t+1 d kt+1 (τ d t+1 (k, i K I. By repeated substitution and using the Transversality Condition, we get q kt E b T i T 1 ( cti+t λi (1 τ d t+t d kt+t (τ d t+t By assumption of the Theorem, we have that Hence, c ti+t (1 τ d t+t d kt+t (τ d t+t c ti+t (1 τ d t+t d kt+t (τ d t+t (1 τ d t+t d kt+t (τ d t+t (k, i, T K I {0,..., }. (k, i K I. (1 (k, i, T K I {0,..., }. Substituting the expression for c ti+t into the previous equation (1, we obtain ( q kt E b T (1 τ d i t+t d kt+t (τ d t+t λi (1 τ (1 τ d d t T 1 d kt(τ d t t+t d kt+t (τ d t+t (k, i K I. q kt E b T i T 1 ( (1 τ d t+t d kt+t (τ d t+t 1 λi (1 τ d t d kt (τ d t (k, i K I. Moreover, ( ( (1 τ d ln b i t+t d kt+t +1 (τ d t+t +1 1 λi ln(b (1 τ d t d kt+t (τ d t+t i ( c ti+t +1 +T 1 λ i N(µ c, σ c (k, i, T K I {0,..., }. Also, 11

13 ( b T (1 τ i d t+t d kt+t (τ d 1 λi t+t T 1 T 0 ( (1 τ d b i t+t d kt+t +1 (τ d t+t +1 1 λi (1 τ d t d kt+t (τ d t+t ( k, i, T K I {1,..., }. Taking logarithms of both sides, we obtain ( ( ln b T (1 τ i d t+t d kt+t (τ d 1 λi t+t (1 τ d t d kt(τ d t T 1 ( ln (1 τ d b i t+t d kt+t +1 (τ d t+t +1 1 λi (1 τ d t+t d kt+t (τ d t+t T 0 T 1 ( ( (1 τ d ln b i t+t +1 d kt+t +1 (τ d t+t +1 1 λi (1 τ d t+t d kt+t (τ d t+t T 0 ( k, i, T K I {1,..., }. Hence, T 0 ( ( ln b T i (1 τ d t+t d kt+t (τ d t+t 1 λi T 1 ( ( (1 τ d ln b i t+t +1 d kt+t +1 (τ d t+t +1 1 λi (1 τ d t+t d kt+t (τ d t+t ( k, i, T K I {1,..., }. Clearly, ( ( E ln b T i T 1 E T 0 (1 τ d t+t d kt+t (τ d t+t 1 λi ( ( (1 τ d ln b i t+t +1 d kt+t +1 (τ d t+t +1 1 λi T µ (1 τ d t+t d kt+t (τ d t+t c ( k, i, T K I {1,..., } and, since by assumption of the Theorem COV ln (+T1, ln (+T2 0 ( T 1, T 2 {1,..., } {1,..., }, T 1 T 2. 12

14 we have that ( ( V AR ln b T i T 1 V AR T 0 (1 τ d t+t d kt+t (τ d t+t 1 λi ( ( (1 τ d ln b i t+t +1 d kt+t +1 (τ d t+t +1 1 λi T σ 2 (1 τ d t+t d kt+t (τ d t+t c ( k, i, T K I {1,..., }. ( ( ln b T (1 τ i d t+t d kt+t (τ d 1 λi t+t N(T µ c, T σ c ( k, i, T K I {1,..., }. Fix an arbitrary ( k, i, T K I {0,..., }. Let x ln ( b T i ( (1 τ d t+t d kt+t (τ d t+t 1 λi. Using from the previous Lemma, we obtain ( E b T (1 τ i d t+t d kt+t (τ d 1 λi t+t E e x e µ x + σ2 x 2 e T µ c T σ2 c ( k, i, T K I {1,..., }. Thus, ( E b T i (1 τ d t+t d kt+t (τ d t+t 1 λi e T µ c T σ2 c ( k, i, T K I {1,..., }. Hence, summing over T {1,..., }, we obtain ( E b T (1 τ d i t+t d 1 λi kt+t ( E b T (1 τ d (1 τ d t T 1 d kt(τ d t i t+t d 1 λi kt+t (1 τ d t T 1 d kt(τ d t e T µ c T σ2 c (k, i K I. T 1 13

15 Taking into consideration that by assumption ( (1 τ d E b i t+t d 1 λi kt+t +1 e µ (1 τ d t d kt+t (τ d t c σ2 c < 1 (k, i, T K I {0,..., } and summing over T {1,..., }, we obtain q kt E b T i T 1 So e T µ c T σ2 c e µ c σ2 c T 1 ( (1 τ d t+t d kt+t (1 τ d t d kt q kt (τ d t, µ c, σ 2 c PROOF OF COROLLARY 1: equation (1, we obtain ln q kt ln e µ c σ2 c 1 e µ c σ2 c 1 e µ c σ2 c. 1 λi (1 τ d t d kt Taking derivatives of both sides we obtain eµ c σ2 c (1 τ d 1 e µ c σ2 c t d kt (τ d t. eµ c σ2 c (1 τ d 1 e µ c σ2 c t d kt (τ d t k K. q kt τ d t Taking logarithms of both sides of + ln(1 τ d t + ln d kt (τ d t k K. 1 q kt 1 1 τ d t + 1 d kt d kt τ d t. So q kt τ d t 1 (1 q kt τ d t 1 1 d kt d kt τ d t (1 τ d t. Hence, q kt τ d t 1 q kt / (1 τ d t τ d t 1 1 τ d t d kt d kt τ d t / (1 τ d t τ d t 1 1 τ d t. 14

16 E qkt, 1 τ d t 1 + E d kt, 1 τ d t. PROOF OF THEOREM 2: We have that +T (q kt+t + (1 τ d t+t d kt+t z ikt+t n ( q kt+t z ikt+t +1 Mit+T +1 M it+t p t+t (i, T I {0,..., }. Substituting budget constraint into the utility function, we obtain E b T i T 0 U i (c i, G u i( z ikt+t (p ( kt+t + (1 τ d Mit+T +1 M it+t t+t d kt+t z ikt+t +1 p kt+t p t+t }{{} +T Differentiating U i with respect to z ikt+1 and setting U i z ikt+1 q kt u i ( + b i E u i (+1 ( q kt+1 + (1 τ d t+1d kt+1 0 to 0, we obtain (k, i K I. ( λi q kt E b i cti+t ( qkt+1 + (1 τ d t+1 d kt+1 (k, i K I. By repeated substitution and using the Transversality Condition, we get q kt E b T i T 1 ( cti+t λi (1 τ d t+t d kt+t By assumption of the Theorem, we have that c ti+t (1 τ d Mi+T t+t d kt+t +1 M t+t p t+t (1 τ d Mt+1 M t d kt t (k, i K I. (2 p t (i, T I {0,..., }. 15

17 Also, by assumption of the Theorem, we have that ( τ d Mt+T +1 M t+t t+t d kt+t + p t+t }{{} Seigniorage }{{} Toyal Government Revenue d kt+t }{{} Total Output g T {1,..., }. c ti+t (1 τ d Mi+T t+t d kt+t +1 M t+t p t+t (1 τ d Mt+1 M t d kt t p t d kt+t g d kt+t d kt g d kt d kt+t d kt (i, T I {0,..., }. In addition, by assumption of the Theorem, we have that d kt+t d kt d kt+t d kt k K. Hence, c ti+t d kt+t d kt (k, i K I. Substituting the expression for c ti+t into the previous equation (2, we obtain ( λi q kt E b T i dkt+t d kt (1 τ d t+t d kt+t (k, i K I. T 1 ( 1 λi dkt+t q kt E b T i (1 τ d t+t T 1 d }{{ kt }{{} d kt (k, i K I. } e y e x t+t it+t 16

18 Then and so ( 1 λi dkt+t q kt E b T i (1 τ d t+t T 1 d }{{ kt }{{} d kt } e y e x t+t it+t ( 1 λi dkt+t E T 1 bt i (1 τ d t+t d }{{ kt }{{} d kt (k, i K I } e y e x t+t it+t ( 1 λi dkt+t q kt E T 1 bt i (1 τ d t+t d }{{ kt }{{} d kt (k, i K I. } e y e x t+t it+t We also know from the previous Lemma that if random variables x t+t and y yit+t are bivariate normally distributed with expectations (E ln (1 τ t+t, E }{{} ln(b i( c ti+t 1 λ i (µ τ, µ c x t+t }{{} y yit+t and the variance-covariance matrix ( σ 2 V τ 0 0 σ 2 c, then by we obtain ( 1 λi dkt+t E bt i (1 τ d t+t d }{{ kt }{{} } e y e x t+t it+t e µ τ σ2 τ e T µ c T σ2 c 17

19 (k, i, T K I {0,..., }. ( 1 λi dkt+t q kt E T 1 bt i (1 τ d t+t d }{{ kt }{{} d kt } e y e x t+t it+t e µ τ σ2 τ e T µ c T σ2 c d kt (k, i K I. T 1 Hence, q kt e µ τ σ2 τ e T µ c T σ2 c T 1 d kt k K. Taking into consideration that by assumption ( 1 λi E b i dkt+t +1 d kt+t e µ c σ2 c < 1 (k, i, T K I {0,..., } and summing over T {1,..., }, we obtain e T µ c T σ2 c e µ c σ2 c T 1 q kt e µ τ σ2 τ 1 e µ c σ2 c. e µ c σ2 c d 1 e µ c σ2 kt k K. c References DeLong, J. B. and K. Magin. The U.S. Equity Return Premium: Past, Present, and Future. Journal of Economic Perspectives, 2009, 23:1, Edelstein, R. and K. Magin. Using the CCAPM with Stochastic Taxation and Money Supply to Examine US REITs Pricing Bubbles. The Journal of Real Estate Research, 2017, Vol. 39:4. Edelstein, R. and K. Magin. The Equity Risk Premium for Securitized Real Estate The Case for U.S. Real Estate Investment Trusts. The Journal of Real Estate Research, 2013, Vol. 35:4. 18

20 Lucas, R. E. Asset Prices in an Exchange Economy. Econometrica, 1978, 46:6, Magin, K. Comparative Statics in Finite Horizon Finance Economies with Stochastic Taxation, The Center for Risk Management Research Working Paper Magin, K. Comparative Statics of Equilibria with Respect to Stochastic Tax Rates, The Center for Risk Management Research Working Paper Magin.pdf Magin, K. Generic Existence of Equilibria in Finite Horizon Finance Economies with Stochastic Taxation, The Center for Risk Management Research Working Paper Magin, K. Equity Risk Premium and Insecure Property Rights. Economic Theory Bulletin, 2015, Vol. 3:2, pp DOI /s Poterba, J. Taxation and Corporate Payout Policy. NBER Working Paper 10321, 2004, Rubinstein, M. The Valuation of Uncertain Income Streams and the Pricing of Options. Bell J. Econ. 7, , Sialm, C. Tax Changes and Asset Pricing. American Economic Review, 2009, 99:4, Sialm, C. Stochastic Taxation and Asset Pricing in Dynamic General Equilibrium. Journal of Economic Dynamics and Control, 2006, 30,