Appendix A: Demand, Marginal Valuation and the Cumulative Bid Function | RDP 2024-03: Demand in the Repo Market: Indirect Perspectives from Open Market Operations from 2006 to 2020

RDP 2024-03: Demand in the Repo Market: Indirect Perspectives from Open Market Operations from 2006 to 2020 Appendix A: Demand, Marginal Valuation and the Cumulative Bid Function

Chris Becker, Anny Francis, Calebe de Roure and Brendan Wilson

May 2024

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The Reserve Bank auction is a multi-unit discriminatory price (pay-your-bid) auction. Before each auction commences, the Reserve Bank publishes an intended size of operation in millions of dollars, denoted by K, and preferred repo terms in days. Suppose that $N \geq 0$ bidders, each denoted by $i \in {1, 2, ..., N},$ participate in the auction. We restrict our attention to repo approaches and exclude approaches for outright sales of securities.

We assume that each participant has their own downward-sloping demand schedule for repo funding which summarises their marginal valuation of successive units. This demand schedule varies by the term of repo funding on a given day and over time. The term dimension allows participants to have different demand for different terms and the time dimension allows demand for each term to vary over time. On each day t, bidder i 's private valuation for each unit of available repo funding at a term of d days is given by the vector $x_{i d t} = (x_{i d t}^{1}, x_{i d t}^{2}, ... x_{i d t}^{k}),$ where $x_{i d t}^{k} \in ℝ$ is i 's marginal valuation of obtaining the k th unit. We assume, by construction, that i 's marginal valuation is decreasing in the number of units obtained: $x_{i d t}^{1} \geq x_{i d t}^{2} \geq \dots \geq x_{i d t}^{K} .$ Each unit is equal to $1 million of exchange settlement balances; the smallest allowed increment in bid size.^[15] Since x_idt maps each unit to a marginal valuation, it represents i 's inverse demand function (Krishna 2002, p 180). By inverting x_idt we obtain i 's demand function for term d on day t:

q_{i d t} : ℝ \to {0, 1, 2, 3, ..., K}

Thus, for a given repo rate r we have:

q_{i d t} (r) \equiv \max {k : r \leq x_{i d t}^{k}}

An aggregated demand function for each term and day is obtained by adding the demand functions for each term for all bidders:

Q_{d t} (r) \equiv \sum_{i = 1}^{N} q_{i d t} (r)

Since we do not observe the true private valuations of bidders, we cannot construct the demand function. However, we do observe the bids submitted by participants and we can use these bids to construct the cumulative bid function which serves as a close approximation.

On each day, bidder i submits a vector of bids $b_{i d t} = (b_{i d t}^{1}, b_{i d t}^{2}, ..., b_{i d t}^{K}), b_{i d t}^{k} \in ℝ$ for each term satisfying $b_{i d t}^{1} \geq b_{i d t}^{2} \geq ... \geq b_{i d t}^{K},$ indicating how much is bid for each unit. Since b_idt maps each unit to a bid, it represents the inverse bid function. By inverting b_idt we obtain i 's bid function for term d on day t:

{\hat{q}}_{i d t} (r) \equiv \max {k : r \leq b_{i d t}^{k}}

By aggregating over N bidders, we can construct the cumulative bid function:

{\hat{Q}}_{d t} (r) = \sum_{i = 1}^{N} {\hat{q}}_{i d t} (r)

In equilibrium, it must be that ${\hat{q}}_{i d t} (r) \leq q_{i d t} (r)$ for all repo rates $r \in ℝ$ . To see this, suppose that there exists some repo rate such that ${\hat{q}}_{i d t} (r) > q_{i d t} (r)$ . This is the case if, and only if, for some unit $u, b_{i d t} (u) > x_{i d t} (u)$ , meaning that for some unit a bid is made that exceeds the marginal valuation of that unit. Assume that there is some positive probability that this will be the winning bid for that unit. This is true if there exists a bidder j such that the support of j 's valuation distribution intersects with that of bidder $i, i \neq j .$ Then, bidder i would be better off by submitting a lower bid (than her true marginal valuation). This is because paying a bid that is greater than the marginal valuation results in a negative pay-off.

This means that the cumulative bid function lies on or below the demand function. The cumulative bid function is our best proxy for demand but not necessarily the same as the private marginal valuation held by auction participants. Our working assumption is that this approximation is very close to the true demand curve.

Footnote

Although the minimum bid is $20 million, for simplicity this can be thought of as 20 bids for $1 million. [15]