Asset PricingThe objective of this section of the course is to introduce the asset pricingformula developed by Lucas [1978]. We will study the pricing of assets thatis consistent with the neoclassical growth model. More generally, this is thepricing methodology that is implied by the ”microfoundations” approach tomacroeconomics - which we endorse!.Lucas works out his formula using an endowment economy inhabited by oneagent. The reason for doing so is that in such an environment the allocationproblem is trivial; therefore only the prices that support a no-trade generalequilibrium need to be sorted out.In the second part of this section, we study the application of the Lucaspricing formula performed by Mehra and Prescott [1985]. The authors utilizedthe tools developed by Lucas [1978] to determine the asset prices that wouldprevail in an economy whose endowment process mimicked the consumptionpattern of the United States economy during the last century. They then com-pared the theoretical results with real data. Their findings were striking andhave produced an extensive literature.1 Lucas Asset Pricing FormulaThe ModelThe representative agent in Lucas’ economy solves:max{ct(zt)}∞t=0, ∀zt(XtXztπzt· uctzt )s.t.XtXztptzt· ctzt=XtXztptzt· $tztctzt= $tzt∀t, ∀zt(market clearing)The last condition is the feasibility condition. Notice that it implies thatthe allocation problem is trivial, and only the prices pt(zt) supporting thisallocation as a (competitive) equilibrium must be sought. (Note: Lucas’ paperuses continuous probability.)Therefore, we have two tasks:1stTask: Find an expression for pt(zt) in terms of the primitives.12ndTask: Apply the resulting formula pt(zt) to price arbitrary assets.1stTaskFirst order conditions from the consumer’s problem:ctzt: βt· πzt· u0ctzt = λ · ptztwhere ct(zt) = $t(zt) will need to hold, and λ will be endogenous. We can getrid of this shadow value of income by normalizing p0= 1:c0: u0($0) = λ · p0≡ λThenptzt= βt· πzt·u0[$t(zt)]u0($0)(LPF)The Lucas Pricing Formula (LPF) shows that pt(zt) is the price of a claimon consumption goods at t that yields 1 unit if the realized state is zt, and 0units otherwise.We can distinguish three separate components in the price of this claim:1. Time: pt(zt) is decreasing in t (since β < 1).2. Likelihood: pt(zt) is increasing in the probability of occurrence of zt.3. Marginal rate of substitution: pt(zt) is increasing in the marginal rate ofsubstitution between goods at (t, zt) and t = 0 (don’t forget that pt(zt)is in fact a relative price).For the case of a concave felicity index u (·) (which represents risk aversebehavior), the third effect will be high if the endowment of goods is scarce at(t, zt) relative to t = 0.2ndTaskAny asset is in essence nothing but a sum of contingent claims. Thereforepricing an asset consists of summing up the prices of these rights to collectgoods. You may already (correctly) suspect that the key is to properly identifythe claims to which the asset entitles its owner. This involves specifying thetime and state of nature in which these rights get activated, and the quantities.We must find the price at (t, zt) of an asset that pays 1 unit at t + 1 forevery p oss ible realization zt+1such that zt+1= (zt+1, zt) for zt+1∈ Z.2The date-0 price of such an asset is given byqrf0zt=Xz0∈Zpt+1z0, zt| {z }price of claim· 1|{z}quantityThe date-t price is computed byqrftzt=qrf0(zt)pt(zt)qrftzt=Pz0∈Zpt+1(z0, zt) · 1pt(zt)Using (LPF) to replace for pt(zt), pt+1(zt+1, zt):qrftzt=βt+1·Pz0∈Zπ (z0, zt) ·u0[$t+1(z0, zt)]u0[$0]βt· π (zt) ·u0[$t( zt)]u0[$0]qrftzt= β ·Xz0∈Zπ (z0, zt)π (zt)·u0[$t+1(z0, zt)]u0[$t( zt)]Notice that three components identify before now have the following char-acteristics:1. Time: Only one period discounting must be considered between t and t+1.2. Likelihood:π (z0, zt)π (zt)is the conditional probability of the state z0occurringat t + 1, given that ztis the history of realizations up to t.3. Marginal rate of substitution: The relevant rate is now between goods at(t, zt) andt + 1, zt+1for each possible zt+1of the form (zt+1, zt) withzt+1∈ Z.For more intuition, you could also think that qrft(zt) is the price that wouldobtain if the economy, instead of starting at t = 0, was ”rescheduled” to beginat date t (with the stochastic process {zt}∞t=0assumed to start at zt).Next we price a stock that pays out dividends according to the process dt(zt)(a tree yielding dt(zt) units of fruit at date-state (t, zt)). The date-t price of3this portfolio of contingent claims is given byqtreetzt=∞Ps=t+1Pzsps(zs) · ds(zs)pt(zt)qtreetzt=∞Xs=t+1Xzsβs−t·π (zs)π (zt)·u0[$s(zs)]u0[$t(zt)]· ds(zs)qtreetzt= Et"∞Xs=t+1βs−t·u0[$s]u0[$t(zt)]· ds#Notice that the price includes the three components enumerated ab ove , mul-tiplied by the quantity of goods to which the asset entitles in each date-state.This quantity is the dividend process dt(zt).We can also write the price of the tree in a recursive way. In the deterministiccase, this would mean thatpt=pt+1+ dt+1Rt+1where Rt+1is the (gross) interest rate between periods t and t + 1. This isrecursive b ec ause the formula for ptinvolves pt+1.The uncertainty analogue to this expression isqtreetzt=Xzsβ ·π (z0, zt)π (zt)·u0[$t+1(z0, zt)]u0[$t(zt)]·dt+1z0, zt+ qtreet+1z0, zt qtreet= β · Etu0[$t+1]u0[$t]·dt+1+ qtreet+1 You can check that this corresponds to the previous formula by iterativelysubstituting for qtreet+1(z0, zt). More importantly, notice that the price includesthe usual three components: what about quantities? This expression reads likethe price of a one period tree that e ntitles to the dividend dt+1(z0, zt), plus theamount of ”money” needed to purchase the one-period tree again next period.If you think about how this price fits into the endowment economy, then theamount qtreet+1(z0, zt) will have to be such that at date-state [t + 1, (z0, zt)] , theconsumer is marginally indifferent between purchasing the tree again, or usingthe proceeds to buy consumption goods. Given an equilibrium price qtreet+1(z0, zt)for each date-state, the equilibrium price qtreet(zt) will be determined recur-sively.Finally, the jargon has it thatβ ·Xzsπ (z0, zt)π (zt)·u0[$t+1(z0, zt)]u0[$t(zt)]≡ β