“chap02” — 2004/9/14 — page 35 — #12Applying the Basic Model2.1 Assumptions and ApplicabilityWriting p = E (mx),wedonot assume1. Markets are complete, or there is a representative investor2. Asset returns or payoffs are normally distributed (no options), or indepen-dent over time3. Two-period investors, quadratic utility, or separable utility4. Investors have no human capital or labor income5. The market has reached equilibrium, or individuals have bought all thesecurities they want toAll of these assumptions come later, in various special cases, but we havenot made them yet. We do assume that the investor can consider a smallmarginal investment or disinvestment.The theory of asset pricing contains lots of assumptions used to deriveanalytically convenient special cases and empirically useful representations.In writing p = E (mx) or pu(ct) = Et[βu(ct+1)xt+1], we have not made mostof these assumptions.We have not assumed complete markets or a representative investor.These equations apply to each individual investor, for each asset to whichhe has access, independently of the presence or absence of other investorsor other assets. Complete markets/representative agent assumptions areused if one wants to use aggregate consumption data in u(ct), or otherspecializations and simplifications of the model.We have not said anything about payoff or return distributions. Inparticular, we have not assumed that returns are normally distributed orthat utility is quadratic. The basic pricing equation should hold for anyasset, stock, bond, option, real investment opportunity, etc., and any mono-tone and concave utility function. In particular, it is often thought that35“chap02” — 2004/9/14 — page 36 — #236 2. Applying the Basic Modelmean-variance analysis and beta pricing models require these kinds oflimiting assumptions or quadratic utility, but that is not the case. A mean-variance efficient return carries all pricing information no matter what thedistribution of payoffs, utility function, etc.This is not a ‘‘two-period model.’’ The fundamental pricing equationholds for any two periods of a multiperiod model, as we have seen. Really,everything involves conditional moments, so we have not assumed i.i.d.returns over time.I have written things down in terms of a time- and state-separable utilityfunction and I have focused on the convenient power utility example.Nothing important lies in either choice. Just interpret u(ct) as the partialderivative of a general utility function with respect to consumption at timet. State- or time-nonseparable utility (habit persistence, durability) compli-cates the relation between the discount factor and real variables, but doesnot change p = E (mx) or any of the basic structure.We do not assume that investors have no nonmarketable human capital,or no outside sources of income. The first-order conditions for purchase ofan asset relative to consumption hold no matter what else is in the budgetconstraint. By contrast, the portfolio approach to asset pricing as in theCAPM and ICAPM relies heavily on the assumption that the investor has nononasset income, and we will study these special cases below. For example,leisure in the utility function just means that marginal utility u(c, l ) maydepend on l as well as c.We do not even really need the assumption (yet) that the market is‘‘in equilibrium,’’ that the investor has bought all of the asset that he wantsto, or even that he can buy the asset at all. We can interpret p = E (mx) asgiving us the value, or willingness to pay for, a small amount of a payoff xt+1that the investor does not yet have. Here is why: If the investor had a littleξ more of the payoff xt+1at time t + 1, his utility u(ct) + βEtu(ct+1) wouldincrease byβEtu(ct+1+ ξ xt+1) − u(ct+1)= βEt u(ct+1)xt+1ξ +12u(ct+1)(xt+1ξ)2+···.If ξ is small, only the first term on the right matters. If the investor has togive up a small amount of money vtξ at time t, that loss lowers his utility byu(ct− vtξ) − u(ct) =−u(ct)vtξ +12u(ct)(vtξ)2+···.Again, for small ξ , only the first term matters. Therefore, in order toreceive the small extra payoff ξ xt+1, the investor is willing to pay the small“chap02” — 2004/9/14 — page 37 — #32.2. General Equilibrium 37amount vtξ, wherevt= Et βu(ct+1)u(ct)xt+1.If this private valuation is higher than the market value pt, and if theinvestor can buy some more of the asset, he will. As he buys more, hisconsumption will change; it will be higher in states where xt+1is higher,driving down u(ct+1) in those states, until the value to the investor hasdeclined to equal the market value. Thus, after an investor has reached hisoptimal portfolio, the market value should obey the basic pricing equationas well, using post-trade or equilibrium consumption. But the formula canalso be applied to generate the marginal private valuation, using pre-tradeconsumption, or to value a potential, not yet traded security.We have calculated the value of a ‘‘small’’ or marginal portfolio changefor the investor. For some investment projects, an investor cannot takea small (‘‘diversified’’) position. For example, a venture capitalist orentrepreneur must usually take all or nothing of a project with payoffstream {xt}. Then the value of a project not already taken, Ejβj[u(ct+j+xt+j) − u(ct+j)], might be substantially different from its marginal counter-part, Ejβju(ct+j)xt+j. Once the project is taken, of course, ct+j+ xt+jbecomes ct+j, so the marginal valuation still applies to the ex post consump-tion stream. Analysts often forget this point and apply marginal (diversified)valuation models such as the CAPM to projects that must be bought in dis-crete chunks. Also, we have abstracted from short sales and bid/ask spreads;this modification changes p = E (mx) from an equality to a set of inequalities.2.2 General EquilibriumAsset returns and consumption: which is the chicken and which is theegg? I present the exogenous return model, the endowment economy model,and the argument that it does not matter for studying p = E (mx).So far, we have not said where the joint statistical properties of thepayoff xt+1and marginal utility mt+1or consumption ct+1come from. Wehave also not said anything about the fundamental exogenous shocks thatdrive the economy. The basic pricing equation p = E (mx) tells us only whatthe price should