2.1 The basic finance economy without taxes
Payoff space I model an endowment economy with financial assets. The economy exists at dates
\(t=0\), when decisions are made and initial consumption takes place, and at
\(t=1\), when payoffs are paid out and consumed. I add to the model of Kruschwitz and Löffler (
2009) consumption at
\(t=0\). I denote
\({\mathbf {X}}_{r}\) as an
\(N\times S\) matrix of tradeable, risky and elementary payoffs, in which
N is the number of payoffs and
S the number of possible states at
\(t=1\). With elementary or basic payoffs, I mean non-redundant payoffs. Non-redundant, in turn, means that any single elementary payoff cannot be constructed through linear combinations of other payoffs. This matrix is augmented by a risk-free payoff
\(X_{0}\), which is also non-redundant, so that,
\({\mathbf {X}}=(X_{0}\;\;{\mathbf {X}}_{r})'\) is an
\(N+1\times S\) matrix of non-redundant payoffs. Thus, the payoff space is spanned by
N elementary risky asset payoffs and a risk-free payoff. The number of states
S can be greater than the number of assets so that an incomplete market is possible. I use the subscript
s for individual states and the subscript
j for the different financial assets so that the payoff
j pays
\(X_{js}\) in state
s. To simplify notation, I put time subscripts only when necessary, such as for consumption, which is possible at
\(t=0\) and at
\(t=1\). I use all random variables as row vectors of dimension
\(1\times S\). Constants such as prices of a single asset
j, denoted
\(p_j\), can also be written as a
\(1\times S\) vector of constant values.
Characterization of the agents and their maximization problems There are \(i=1, \ldots , I\) agents in the economy. Agents are rational and have the same complete set of information, i.e., they know the distributions of the payoffs. They are characterized through a time separable utility function \(u_i(\cdot )\) over consumption and through initial (pre-trade) portfolio holdings \({\bar{\mathbf {{n}}}}_i\). At date \(t=1\) and in state s agent i consumes \(c_{is}\) units of a composite consumption good. One unit of a consumption good has a price of one at all times so that a payoff of one can buy exactly one unit of the consumption good. To address random variables such as agent i’s consumption or the j’th payoff at \(t=1\), I leave out the subscript s for states and write \(c_{i1}\) and \(X_{j}\), respectively. I denote \(\bar{c}_{i0}\) the endowment of agent i with consumption goods at time \(t=0\).
Agents maximize expected utility of consumption,
$$\begin{aligned} \max _{c_{i0},c_{i1}} E\left[ \beta _{i} u_i(c_{i1})\right] + u_i(c_{i0}), \end{aligned}$$
(1)
subject to the budget constraints at
\(t=0\),
$$\begin{aligned} {\bar{\mathbf {{n}}}}_{i}^{\prime } {\mathbf {p}}+\bar{c}_{i0}={\mathbf {n}}_{i}^{\prime } {\mathbf {p}}+c_{i0} \end{aligned}$$
(2)
and at
\(t=1\)
$$\begin{aligned} c_{is}&={\mathbf {n}}_{i}^{\prime } {\mathbf {X}}_s, \end{aligned}$$
(3)
for
\(s=1,\ldots ,S.\) I use all collections of prices and asset weights as column vectors. I denote
\(E[\cdot ]\) the expected value at time zero of its argument,
\({\mathbf {p}}=(p_{0}\;p_{1}\; \cdots \; p_{j}\; \cdots \; p_{N})'\) is the price vector of the
\(N+1\) assets,
\(\mathbf {n}_{i}=(n_{i0}\;n_{i1}\; \cdots \;n_{ij} \; \cdots \; n_{iN})'\) is a vector of after-trade portfolio weights (I use
\({\bar{\mathbf {{n}}}}_{i}\) for pre-trade portfolios.),
\(\beta _i\) the subjective time discount factor (or impatience factor), and
\(u_i(\cdot )\) the utility function. The expected value operator with a single random variable means a probability inner product. With a random variable
z that means
\(E[z]=\sum _{s=1}^S \pi _s z_s\), in which
\(\pi _s\) is the probability of state
s. For prices I mostly use the short notation so that
\(p_{j}\) is the price of a payoff
\(X_{j}\). When necessary, I also use prices as operators to make more clear what is priced, for example
\(p_{j}=p(X_{j})\) is again the price of the payoff
j. Furthermore, I use the subscript
r to refer only to the risky assets
\({\mathbf {p}}_{r}=(\;p_{1}\; \cdots \; p_{j}\; \cdots \; p_{N})'\) and
\({\mathbf {n}}_{ir}=(n_{i1}\; \cdots \;n_{ij} \; \cdots \; n_{iN})'\), the subscript zero is related to the risk-free asset.
1 The utility function is differentiable and strictly monotonously increasing at a decreasing rate. Therefore, any additional unit of consumption adds to utility, and it is optimal to consume all of the payoffs, which justifies to write the budget constraints as equalities (Lengwiler
2004, p. 52). The equality of the budget constraints allows to substitute out consumption and to restate the maximization problem with respect to the portfolio weights and initial consumption.
The risk-free asset is in zero net supply \(\sum _{i=1}^I n_{i0}=0\). Therefore, I define a vector of aggregate asset holdings \({\mathbf {n}}=\sum _{i=1}^I {\mathbf {n}}_{i}\), which is \({\mathbf {n}}' =(0 \;\;1\;\;1\;\; \cdots \;\;1)\) because the risk-free asset is in zero net supply.
Equilibrium The equilibrium is given through a vector of prices
\(\mathbf {p}\), consumption profiles
\(c_{i0}, c_{i1}\) and portfolios
\(\mathbf {n}_{i}\) for
\(i=1,\ldots ,I\) so that each agent maximizes utility subject to his budget constraint, given prices
\(\mathbf {p}\). Furthermore, the market for the consumption good clears:
\(\sum _{i=1}^I \bar{c}_{i0}=\sum _{i=1}^I c_{i0}\) and
\(\sum _{i=1}^I c_{is}={\mathbf {n}}' {\mathbf {X}}_{s}\) for
\(s=1,\ldots ,S\). Financial assets are in positive net supply and markets clear so that
\(\sum _{i=1}^I n_{ij}=\sum _{i=1}^I \bar{n}_{ij}=1\) for
\(j=1,\ldots ,N\) and
\(\sum _{i=1}^I n_{i0}=0\) for the risk-free asset. I assume that at least one equilibrium exists. Notice that equilibrium prices imply the absence of arbitrage opportunities (Lengwiler
2004, p.50).
Pricing equations I write the agent’s optimization problem in terms of a Lagrangian:
$$\begin{aligned} \mathcal {L}_i=E\left[ \beta _{i} u_i(c_{i1})\right] + u_i(c_{i0})-\lambda _i({\mathbf {n}}_{i}^{\prime } {\mathbf {p}}+c_{i0} - {\bar{\mathbf {{n}}}}_{i}^{\prime } {\mathbf {p}}-\bar{c}_{i0}), \end{aligned}$$
(4)
where
\(\lambda _i\) is a Lagrange multiplier. I substitute in Eq. (
3) and take the partial derivatives with respect to portfolio weights and to initial consumption. Combining the results I obtain,
$$\begin{aligned} \mathbf {p}=E\left[ \mathbf {X}\beta _i \frac{u_i^{\prime }(c_{i1})}{u_i^{\prime }(c_{i0})}\right] . \end{aligned}$$
(5)
I denote more compactly,
$$\begin{aligned} m_{i}=\beta _i \frac{u_i^{\prime }(c_{i1})}{u_i^{\prime }(c_{i0})} \end{aligned}$$
(6)
as agent
i’s stochastic discount factor (SDF). Using this, I can price any single payoff
\(X_j\) through:
$$\begin{aligned} p_{j}=E\left[ m_{i} X_{j}\right] . \end{aligned}$$
(7)
Here the expected value means that probabilities are induced to the inner product of
\(X_j\) and
\(m_i\):
\(E[m_{i}X_{j}]=\sum _{s=1}^S \pi _s m_{i} X_{js}\). Through trading, agents find a price vector on which everyone agrees, i.e.,
\(\mathbf {p}=E[m_{i}\mathbf {X}]\) for
\(i=1,\ldots ,I\), and which maximizes utility.
In complete markets,
\(\mathbf {X}\) is a square matrix with full rank, i.e., there are as many basic assets as states. The equation
\(\mathbf {p}=E[m_{i}\mathbf {X}]\) can be written as:
\(\mathbf {p}=\mathbf {X} \kappa _i\), in which state prices are:
\(\kappa _{is}=m_{is}\pi _s\) for
\(s=1,\ldots ,S\) and
\(\pi _s\) are objective probabilities of states
s. When
\(\mathbf {X}\) has full rank, there is a unique solution for
\(\kappa _i\). Since probabilities are objective probabilities, there is a unique SDF, i.e., every agent has the same SDF. It also follows that the state price vector can be expressed as a linear combination of basis assets and therefore lies in the payoff space. The same is true for the SDF.
2
With incomplete markets, i.e., with \(S>N+1\), \(\mathbf {X}\) does not have full rank. The system of equations \(\mathbf {p}=\mathbf {X} \kappa _i\) has less equations than unknowns so that there is more than one solution to the system. That means state prices and SDFs among agents may differ.
Pricing a risk-free payoff of one, I define the risk-free rate as
\(R_\mathrm{f}=1/E[m_{i}]=1/p_0\) for
\(i=1,\ldots ,I\). The term
\(R_\mathrm{f}\) is the gross risk-free rate:
\(R_\mathrm{f}=1+r_\mathrm{f}\). Thus, the pricing Eq. (
7) can be restated as:
$$\begin{aligned} p_{j}=\frac{E(X_{j})}{R_\mathrm{f}}+\mathrm{Cov}(m_{i},X_{j}), \end{aligned}$$
(8)
in which
\(\mathrm{Cov}(m_{i},X_{j})\) is the covariance between the SDF and the payoff. As stated in Cochrane (
2014), in incomplete markets the SDFs of agents
\(m_{i}\) can differ and do not have to be within the payoff space. But there is one SDF
m within the space of tradeable assets that prices all assets. This SDF is the probability induced projection of all of the agents’ SDFs onto the payoff space. The relation between the unique SDF within the payoff space and any individual SDF is
\(m_{i}=m+\epsilon _{i}\), where
\(\epsilon _{i}\) is an error term orthogonal to the (probability induced) payoff space and therefore does not influence prices:
\(p_{j}=E(m_{i} X_{j})=E((m+\epsilon _{i})X_{j})=E(m X_{j})\), because
\(E(\epsilon _{i} X_{j})=0\) holds for all payoffs of the payoff space (Cochrane
2005, p. 66). The unique SDF within the payoff space can be used to price all payoffs but it will not necessarily lead to a possible portfolio rule for all agents, i.e., to a consumption profile that is within the payoff space. In complete markets the SDF is the same for every agent. In the standard CAPM, which does not require complete markets, the SDF is a linear combination of the market return:
\(m=a+bR^M\), where
\(R^M\) is the return on the market portfolio and
a and
b are constants (Cochrane
2005, p. 152). In those two cases the single SDF leads straightforwardly to consumption rules within the payoff space.
2.2 The finance economy with taxes
I introduce another economy that has, compared to the no-tax economy, equal utility functions of agents \(u_i(\cdot )\), equal impatience factors \(\beta _i\), and an equal (pre-tax) distribution of payoffs of financial assets \(\mathbf {X}\). The initial or pre-trade portfolios of agents with shares of assets are also the same, as well as the agents’ perfect information about the payoff distributions. I introduce taxes on capital gains. To account for possible differences in prices, after-trade portfolios, and consumption profiles from the ones in the no-tax economy, I add an asterisk to them. Prices of taxed payoffs are denoted as, \(p_{j}^{*\tau }=p^*(X^\tau _{j})\) and prices of pre-tax payoffs are denoted as, \(p_{j}^{*}=p^*(X_{j})\).
Taxes I define the tax base as the difference between the payoff and the price of the payoff:
\(X_{js}-p_{j}^{*\tau }\), in which
\(p_{j}^{*\tau }\) is the price of the after-tax payoff, i.e., of the payoff
\(X_{js}^\tau =X_{js} - T_{js}=X_{js}-\tau (X_{js}-p_{j}^{*\tau })\), in which
\(T_{js}\) are taxes on the asset
\(j=0,1,\ldots ,N\) in states
\(s=1,\ldots ,S\). Any observed prices reflect possible tax effects. Investors consider the taxes they have to pay on the payoff when pricing the asset. I use
\(\tau \in (0,1)\) as the tax rate and also as a superscript to denote after-tax figures when necessary. The tax rate is certain, constant, and the same for all agents. This is a simplification since tax rates can be observed to have an uncertain element and they often depend on certain characteristics of agents such is their income.
3 Introducing an uncertain tax rate may introduce an additional covariance as well as an expectation into the pricing equation. An agent
i pays capital gains taxes at the amount
\(T_{is}=\tau \sum _{j=0}^N n_{ij}^*(X_{js}-p^{*\tau }_{j})=\tau {\mathbf {n}}_i^{*\prime }({\mathbf {X}}_s - {\mathbf {p}}^{*\tau })\), and they receive transfer payments
\(Q_{is}=\tau \omega _i {\mathbf {n}}'({\mathbf {X}}_s - {\mathbf {p}}^{*\tau })\) for
\(i=1,\ldots ,I\), in which
\(\omega _i\) is the share of total tax revenues that is transferred to agent
i with
\(\sum _{i=1}^I \omega _i=1\). Transfer payments are predetermined amounts, i.e., they cannot be influenced by the agents. Positive and negative capital gains are taxed the same way. I discuss issues of this simplified tax system versus more realistic tax systems in Sect.
4. Aggregate tax payments are
\(T_s=\sum _{i=1}^I T_{is}\). They must be equal to aggregate transfer payments:
\(T_s=Q_s\). Individual transfer payments can also be written as
\(Q_{is}=\omega _i T_s\).
4
The introduction of taxes and transfers does not introduce any new basic asset so that the payoff space is the same as in the no-tax economy. Any tax payment \(T_{js}=X_{js}-\tau (X_{js} - p_{j}^{*\tau })=X_{js}(1-\tau )+\tau p_{j}^{*\tau }\) is just a linear combination of the pre-tax payoff \(X_{j}\) and a risk-free payoff.
Characterization of the agents and their maximization problems Any agent maximizes expected utility of after-tax (and transfers) consumption,
$$\begin{aligned} \max _{c_{i0}^*, c_{i1}^*} E[u_i(c_{i1}^*)] + u_i(c_{i0}^*), \end{aligned}$$
(9)
subject to the budget constraints at
\(t=0\)
$$\begin{aligned} {\bar{\mathbf {{n}}}}_{i}^{\prime } {\mathbf {p}}^{*\tau } + \bar{c}_{i0}={\mathbf {n}}_i^{*\prime } {\mathbf {p}}^{*\tau } + c_{i0}^* \end{aligned}$$
(10)
and at
\(t=1\)
$$\begin{aligned} c^*_{is}&={\mathbf {n}}_{i}^{*\prime }({\mathbf {X}}_s - \tau ({\mathbf {X}}_s - \mathbf {p}^{*\tau })) + Q_{is}, \end{aligned}$$
(11)
for
\(s=1,\ldots ,S\). The variable
\({\mathbf {n}}_{i}^*\) is a vector of after-trade portfolio weights. I denote financial wealth that is left after initial consumption as
\(W_{i}^{*F\tau }={\mathbf {n}}_{i}^{*\prime } {\mathbf {p}}^{*\tau }\) and total financial wealth after initial consumption, i.e., financial wealth including transfers as
\(W_{i}^{*F}=W_{i}^{*F\tau } + p^*(Q_i)\).
Equilibrium The equilibrium is given through a vector of prices \({\mathbf {p}}^{*\tau }\), consumption profiles \(c_{i0}^*, c_{i1}^*\) and portfolios \({\mathbf {n}}_{i}^*\) for \(i=1,\ldots ,I\) so that each agent maximizes utility subject to his budget constraint, given prices \({\mathbf {p}}^{*\tau }\). Furthermore, the market for the consumption good clears: \(\sum _{i=1}^I \bar{c}_{i0}=\sum _{i=1}^I c_{i0}^*\) and \(\sum _{i=1}^I c_{is}^*={\mathbf {n}}^{*\prime } {\mathbf {X}}_{s}\) for \(s=1,\ldots ,S\). That this holds comes from the fact that taxes are just redistributions and do not change aggregate values. Financial assets are in positive net supply and clear so that \(\sum _{i=1}^I n_{ij}^*=\sum _{i=1}^I \bar{n}_{ij}=1\) for \(j=1,\ldots ,N\), and \(\sum _{i=1}^I n_{i0}^*=0\) for the risk-free asset.
Pricing equations The first-order conditions lead to a similar pricing equation as for the no-tax economy, except that after-tax payoffs are priced:
$$\begin{aligned} p^{*\tau }_{j}=E\left[ m_{i}^* X_{j}^\tau \right] . \end{aligned}$$
(12)
The after-tax risk-free payoff is
\(X_{0}^{\tau }=X_0 - \tau (X_0 - p_0^{*\tau })=1-\tau (1-p_0^{*\tau })\), and the after-tax risk-free rate is:
$$\begin{aligned} R_\mathrm{f}^{*\tau }=\frac{1-\tau (1-p^{*\tau }_0)}{E\left[ m_{i}^* (1-\tau (1-p^{*\tau }_0))\right] }=\frac{1}{E[m_{i}^*]}=\frac{1-\tau (1-p^{*\tau }_0)}{p^{*\tau }_0}. \end{aligned}$$
(13)
The second equality follows from the fact that
\(1-\tau (1-p^{*\tau }_0)\) is a constant, which can be taken out of the expectations in the denominator and therefore cancels with the term in the numerator. The third equality just restates that the denominator is actually the price of the cash flow
\(X_{0}^{\tau }=1-\tau (1-p^{*\tau }_0)\). The pre-tax risk-free rate is then,
$$\begin{aligned} R^{*}_\mathrm{f}=\frac{1}{E\left[ m_{i}^* (1-\tau (1-p^{*\tau }_0))\right] }=\frac{1}{p^{*\tau }_0}. \end{aligned}$$
(14)
Using
\(R^{*}_\mathrm{f}=1/p^{*\tau }_0\) the after-tax return can also be written as
\(R_\mathrm{f}^{*\tau }=1+r^{*}_\mathrm{f}(1-\tau )\). If the risk-free rate is not taxed, it is
\(R_\mathrm{f}^*=1/E[m_i^*]\). Notice that since the risk-free asset is traded, every agent agrees upon the risk-free rate. It follows that the expected individual SDFs must be equal, which, in turn, are equal to the expected SDF within the payoff space:
\(E[m^*]=E[m_{i}^*]\) for
\(i=1,\ldots ,I\).
In an economy with capital gains taxes, the expectations of the SDFs \(E[m^*_{i}]\) play a special role. This is summarized in the following proposition:
I assume that \(1/E[m_i^*]>\tau\) holds throughout the paper.
Notice that those pre-tax-after-tax price relations use an SDF of the tax economy \(m^*_i\). Any relations to the SDFs of the no-tax economy, i.e., to \(m_i\), are still to be obtained.
Notice also that
\(E[m^*_{i}]=1\) implies that
\(E\left[ \frac{u_i'(c_{i1}^*)}{u_i'(c_{i0}^*)}\right] =1/\beta _i\). Expected growth of marginal utility of consumption is exactly equal to the inverse of the impatience factor. Higher growth implies a lower risk-free rate and lower growth a higher one. A simple log-normal model such as in (Cochrane
2005, pp. 10–12) allows for more interpretations of the risk-free rate in terms of consumption growth. In this case the risk-free rate is low when expected consumption growth is low or impatience is low, i.e., when beta is high.
The prior proposition has several implications.
In the following section, I continue to analyze equilibrium effects, i.e., how taxes affect prices and quantities in the no-tax and the tax economy.