Transfers

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Introduction

The transfers classes describe how utilities can be moved between two sides of a market.

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Implicit Parameterization

Systematic utilities are in a feasible set $(U_{xy},V_{xy}) \in \mathcal{F}_{xy}$. We can represent the feasible set $\mathcal{F}_{xy}$ by means of a function $\Psi$, defined as $$\Psi_{xy} (u,v) = \min \left\{ t \in \mathbb{R} : (u - t, v - t) \in \mathcal{F}_{xy} \right\}$$

• ETU $:=$ Exponentially Transferable Utility: $$\Psi(u,v) = \tau \times \log \left( \frac{\exp((u-\alpha)/\tau) + \exp((v-\gamma)/\tau)}{2} \right)$$
• LTU $:=$ Linearly Transferable Utility: $$\Psi (u,v) = \lambda \times u + (1-\lambda) \times v - \Phi_{xy}$$
• NTU $:=$ Non-Transferable Utility: $$\Psi (u,v) = \max( u - \alpha_{xy}, v - \gamma_{xy} )$$
• TU $:=$ Transferable Utility: $$\Psi (u,v) = \frac{u + v - \Phi_{xy}}{2}$$

where $\Phi = \alpha + \gamma$.

Explicit Parameterization

As an alternative to the implicit parameterization, we can define the efficient frontier of $\mathcal{F}_{xy}$ by $u = \mathcal{U}_{xy} (w)$ and $v = \mathcal{V}_{xy} (w)$, which are increasing and decreasing in $w$, respectively, where $u$ and $v$ solve $$\Psi_{xy} (u,v) = 0$$ where $$u - v = w$$ In many cases, $\mathcal{U}$ and $\mathcal{V}$ have simple expressions:

• ETU: $$\mathcal{U}_{xy} (w) = - \tau \log \left[ \frac{1}{2} \left( \exp \left(- \frac{\alpha}{\tau} \right) + \exp \left( - \frac{w + \gamma}{\tau} \right) \right) \right]$$ and $$\mathcal{V}_{xy} (w) = - \tau \log \left[ \frac{1}{2} \left( \exp \left(- \frac{\gamma}{\tau} \right) + \exp \left( \frac{w - \alpha}{\tau} \right) \right) \right]$$
• LTU: $$\mathcal{U}_{xy} (w) = \Phi_{xy} + (1-\lambda) w \ \text{ and } \ \mathcal{V}_{xy} (w) = \Phi_{xy} - (1-\lambda) w$$
• NTU: $$\mathcal{U}_{xy} (w) = \min \{ \alpha_{xy}, w + \gamma_{xy} \} \ \text{ and } \ \mathcal{V}_{xy} (w) = \min \{ \alpha_{xy} - w, \gamma_{xy} \}$$
• TU: $$\mathcal{U}_{xy} (w) = \frac{\Phi_{xy} + w}{2} \ \text{ and } \ \mathcal{V}_{xy} (w) = \frac{\Phi_{xy} - w}{2}$$

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