The utility loss from transaction costs measures the difference in each agent’s value function between the frictionless and frictional equilibria. In Shelley (2023), this is decomposed into three components:
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Direct loss (U_d^{n,eps}): the deadweight cost of paying the tax, equal to lambda * E[integral (phi-dot^{lambda,n})^2 dt]. Always non-negative.
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Return loss (U_r^{n,eps}): the gain or loss from the change in equilibrium returns caused by the tax. Can be positive (agent benefits from changed prices) or negative.
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Portfolio loss (U_p^{n,eps}): the loss from deviating from the frictionless optimal portfolio due to trading costs.
The total utility loss U^{n,eps} = U_d + U_r + U_p can be either positive or negative for individual agents. The aggregate utility loss under belief eps is:
U^eps = E^{P(eps)}[integral e^{-delta t} sum_n Delta_t^n * (gamma_n sigma^2 / 2 * Delta_t^n + sigma * eps_t^n) dt] + lambda * E[integral (phi-dot)^2 dt]
The post-rebate aggregate loss (assuming tax revenue is rebated) simplifies to the integral of Delta_t^n weighted by risk aversion and belief terms. This is negative (tax beneficial) precisely when the sample covariance between agents’ belief-weighted portfolio deviations is positive — i.e., when the tax “punishes false beliefs” (Section 2.4.1 in Shelley).
Key Details
- Homogeneous beliefs: post-rebate aggregate loss is always non-negative ⇒ tax always detrimental
- Heterogeneous beliefs: tax can be beneficial when it dampens speculative trading driven by false beliefs
- Keynes’s intuition confirmed: the transaction tax is beneficial precisely when it curbs speculation, matching Keynes’s 1936 argument
- Belief-independent component: the direct loss is always non-negative and independent of beliefs
- Sign condition: tax beneficial iff (1/N) sum (-Delta_t^n) * sigma * eps_t^n > 0, i.e. the covariance between portfolio sluggishness and scaled beliefs is positive