The optimal investment decision problem for an individual investor under idiosyncratic risk resembles the one under aggregate risk with central bank intervention in (14)39 max s,c1,c2,ˆs,z,l E[U(c′ 1, c2)] = Z ∞ −∞ (ζiln (c1+ρz) +βlnc2)f(γ)dγ s.t. (24) p1c1+p2c2≤w−s+Rs+ (R−q)ˆs−Rz− R−q l p1c1+qˆs≤w−s+ql 0≤z≤s; 0≤l≤s; l+z≤s
and is solved analogically. The first-order conditions are the same as in section 3.2 with ζi replacingγ.40 Again, market clearing int = 1requires thatSˆ =
R
i∈Iˆsdi= 0. Hence, the asset priceqhas to adjust to equalise excess demand
and supply of assets by individual investors. The crucial difference to the case with only aggregate risk is that the market clearing condition doesnotimply thatˆs= 0and no assets are traded.
The demand for shares int= 1by investori,ˆsi, is determined by
ˆ si=
β(w−s)−qζis
q(β+ζi)
. (25)
Note that that the Cobb-Douglas utility function (2) implies thatsˆiis a convex
function ofζifor a given asset priceqas∂2ˆsi/(∂ζi)2>0.41
Assume that each investor has an ex-ante probability of one half of belonging to groupAwho receive a shockζAand to groupBwith shockζB, respectively,
andζA ≥ζB without loss of generality. The conditionE[ζi] = 1and the pos-
itive support ofζi imply ζA ∈ [1; 2),(ζA+ζB)/2 = 1and the absence of an
aggregate shock. As usual, market clearing requires
Z i∈A β(w−s)−qζAs q(β+ζA) di+ Z i∈B β(w−s)−qζBs q(β+ζB) di= 0.
The pricing kernel forqbecomes
q= min β(1 +β) (W−S) [ζA(2−ζA) +β]S ;R for β(W −S) S ≥p1ρ (26)
with W andS defined as before. Without idiosyncratic shocks, i.e. ζi = 1, 39In order to explicitly exclude short sales, the constraintsˆ≥ −shad to be added to (24). Foot-
note 41 shows that this is redundant given the specification of the model.
40That means equations (9a) to (9d), (9g), (15a), (15b) and (16). 41Furthermore,ˆs
1 1.2 1.4 1.6 1.8 2 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 q = 1 Value of ζ A 1 1.2 1.4 1.6 1.8 2 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 q = R Value of ζ A sA sB Σ si sA sB Σ si
Figure 7: Convexity ofsˆi:sˆA,sˆB andΣbsi = ˆsA+ ˆsBas a function ofζAforq= 1
andq= 2.
equation (26) simplifies toq=β(W−S)/S, the same asset price as forγ = 1
in section 2.3.1. The conditionβ(W −S)/S≥p1ρexcludes liquidationsz >0
forζi= 1.42
Note that q in (26) increases in the heterogeneity ofA and B, i.e. in the ab- solute value|ζi−1|. The reason is the convexity ofˆsiin (25) mentioned above.
The convexity implies that the additional demand of the agents with the low liquidity shockζB is always sufficiently large such that agents with the high
shockζAdo not need to liquidate their asset. Figure 7 illustrates the convexity
ofsˆi forR = 1/β = 1.1 andS = β/(1 +β)W, the investment in the case of
certainty. The solid line represents the excess demand for the asset which is 0 forζA = 1, given an asset price ofq = 1in the left panel. For ζA > 1(and
thusζB = 2−ζA<1),sˆBrises faster thanˆsAfalls, the excess demand becomes
positive andq >1forζA>1. ForζA≈1.413, the asset price increases toq=R,
since the excess demand is 0 at this combination ofqandζAin the right panel.
ForζA >1.413, investors hit by the low shockζBtransfer money intot= 2as
their CIA becomes unbinding.
To summarise the effects, the structure of the model, in particular the Cobb- Douglas utility function (2) that causes the convexity ofˆsiand the dissolution
of risk int= 1, imply that idiosyncratic shocks alleviate the CIA given a fixed initial investmentS. In general, however, idiosyncratic shocks can have a neg-
ative impact on asset prices if the absorption capacity of the market is limited. This happens in reality and in other models for example if investors are risk-
42The more general form of (26) isq= min
h max β(1+β)(W−S) [ζA(2−ζA)+β]S;p1ρ ;R i .
averse and future returns are risky (see, e.g., Huang and Wang, 2006).A further feature of reality is the presence of brokers and market-makers on financial markets rather than a Walrasian auctioneer. As they smooth price fluctuations by providing liquidity to financial markets, they earn income in the form of bid-ask spreads. Models that analyse the microstructure of financial markets explain the behaviour of these market participants and the implications for transaction prices. The following section presents an extension to the stan- dard model of this section that includes transaction costs in the form of bid-ask spreads, and section 4.3 discusses different mechanisms how small shocks can have large impacts.