# Asset rate (elasticities ?) of substitution

I'm kind of a newbie in the finance research area. However, I'm working on cross-asset spillovers (transmission of shocks between assets) and my guess is that it comes from investors behaviors. Particularly, following a shock on asset $$i$$, investors will reduce their exposure to $$i$$ and also move their exposure to the other assets. For $$n=2$$ assets, there is a perfect elasticity, however for $$n>2$$ I didn't find any paper discussing the determinants of this elasticity. So I've made a model that works quite well, however I didn't add constraints for simplicity and now I can't add them (maybe I'm bad at it though).

So let's assume an investor with mean-variance preferences $$U=E(R)-\frac{\gamma}{2}Var(R)$$ with

$$E(R)=\sum_{i=1}^{N}w_ir_i$$

$$Var(R)=\sum_{i=1}^{N} w_i^2\sigma^2_i +2\sum_{i,j=1}^{N}\sum_{i\ne j}w_iw_j\sigma_{ij}$$

So you can have the FOC and isolate $$w_i$$ in function of the others $$w_j\ \forall j \neq i$$, and differentiate $$w_i$$ w.r.t $$w_j$$, and I get something really simple yet really useful :

$$\frac{\partial w_i}{\partial w_j} = -\frac{\sigma_j}{\sigma_i}\rho_{ij}$$

So the elasticity of substitution sign depends on the correlation signs, which gives many cool interpretations for the investors-driven financial spillover.

However, there is not the constraint that $$\sum_{i=1}^N=1$$, and when I try to add I can't get rid of the $$\lambda$$ of the constraint. Notably, I get the following FOCs :

$$\frac{\partial \mathcal L}{\partial w_i}=0 \rightarrow r_i - 2 \sum_{i,j=1}^{n} \sigma_{ij}w_j = 0$$

$$\frac{\partial \mathcal L}{\partial \lambda}=0 \rightarrow 1- \sum^n_{i=1}w_i=0$$

which is a result that is also found in some (old) research papers (notably Aivazian & al., 1983) but they have different research question than mine.

Does anyone has a guess about how to do such thing ? Maybe use matrix notation (yet, again, I'm bad at it) ?

Your language needs to be more clear. What do you mean by a shock to asset 2? A decrease in price of asset 2, should not change your allocation and have no spillovers (since the expected return of that asset and the variance covariance matrix did not change). Now changing the return of one asset will indeed change the weights investor $$j$$ invests on that asset, but without a resource contra int (or in other words a general equilibrium model).
• Thank you for the answer ! A shock to asset 2 is anything that will make the investors change its weights, so a decrease in the expected price, an increase in the volatility, or a change in the covariance matrix. I'm not sure to understand clearly what you mean by "Now changing the return of one asset will indeed change the weights investor $j$ invests on that asset, but without a resource constraint". Could you provide some paper that explains this if you have one please ? Commented Jun 14 at 9:09