Consider the setting of section 12.3 .4 but now allow for multiple risky assets \(j=1, \ldots, J\). The speculators' budget constraint and margin constraint are now given by \(W_{t}=W_{t-1}+\) \(\sum_{j=1}^{J}\left(p_{t}^{j}-p_{t-1}^{j}ight) x_{t-1}^{j}+\eta_{t}\) and \(\sum_{j=1}^{J} \tilde{m}_{t}^{j}\left|x_{t}^{j}ight| \leq W_{t}\), respectively. Customers' quantity demanded of asset \(j\) at date 1 is \(D_{1}^{j}\left(\Lambda_{1}^{j}, Z_{1}^{j}ight)\), where \(\Lambda_{1}^{j} \equiv v_{1}^{j}-p_{1}^{j}\). The speculators' date- 1 wealth shock \(\eta_{1}\) has zero mean and is uncorrelated with the demand shocks \(Z_{1}^{j}\).

Assume \(p_{1}^{j} eq v_{1}^{j}\) for at least some \(j\), so that speculators are capital-constrained at date 1 (with \(W_{1}>0\) ).

(a) Derive speculators' demand \(x_{1}^{j}\) for each asset. Hint: Explain why speculators only trade assets with the highest \(\left|\Lambda_{1}^{j}ight| / \tilde{m}_{1}^{j}\).

(b) Denote speculators' date- 1 value function by \(J_{1}\left(W_{1}ight)=\mathrm{E}_{1}\left[W_{2}ight]\). Their date- 1 shadow cost of capital \(\varphi_{1}\), a measure of funding illiquidity, is defined through \(\varphi_{1} W_{1}=J_{1}\left(W_{1}ight)\). Express \(\varphi_{1}\) in terms of \(\Lambda_{1}^{j}\) and \(\widetilde{m}_{1}^{j}, j=1, \ldots, J\).

(c) Express each asset's market illiquidity, \(\left|\Lambda_{1}^{j}ight|\), in terms of \(\varphi_{1}, \widetilde{m}_{1}^{j}\), and parameters of the model.

(d) Now consider an additional initial period \(t=0\) with a margin constraint on speculators \(\sum_{j=1}^{J} \widetilde{m}_{0}^{j}\left|x_{0}^{j}ight| \leq W_{0}\). Assume that speculators have unlimited liability, and in particular \(J_{1}\left(W_{1}ight)=\varphi_{1} W_{1}\) for all values of \(W_{1}\) (including negative ones), with \(\varphi_{1}\) from part (b). This assumption raises the possibility that speculators do not trade to their constraint at date 0 if their wealth is high enough.
Assuming that speculators are unconstrained at \(t=0\), show that \(\varphi_{1} / \mathrm{E}_{0}\left[\varphi_{1}ight]\) functions as an SDF in the economy. Explain intuitively why speculators choose to not trade to their constraint even though they are risk-neutral. Show that their date-0 cost of capital \(\varphi_{0}\), defined through \(\varphi_{0} W_{0}=J_{0}\left(W_{0}ight)=\mathrm{E}_{0}\left[W_{2}ight]\), equals \(\mathrm{E}_{0}\left[\varphi_{1}ight]\).

(e) Now suppose that speculators' wealth \(W_{0}\) is low enough that speculators are constrained at \(t=0\). Derive their date- 0 cost of capital, \(\varphi_{0}\).

Data from section 12.3.4

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Brunnermeier and Pedersen (BP 2009) develop themes in Shleifer and Vishny (1997) to link the literatures on trading costs and margins. Margins affect trading costs if margin- constrained agents are marketmakers. Trading costs affect margins if capital providers set margins in relation to the short-run volatility of prices (as opposed to the long-run volat- ility of fundamentals or the deviation of current price from fundamentals). Together, these effects create the possibility of "liquidity spirals," in which funding problems for arbitrageurs disrupt markets and increase trading costs, allowing larger deviations of prices from fundamentals and in turn worsening funding conditions. Figure 12.4, a reproduction of BP Figure 2, illustrates this possibility. To illustrate the BP mechanism, we consider a model with dates t = 0, 1, 2, a single risky asset in zero net supply whose payoff at time 2 is 02, and a market for collateralized lending and borrowing (repo market) with the interest rate normalized to zero. A = -p is the deviation of the price from the fundamental value = E [2] at date 1. The absolute value of A is BP's measure of market illiquidity.