proposition. Retained-finance loss under the competitive allocation [ftip-00OC]
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    In Definition [ftip-00OB], write total profit at a common choice \(a\) as \(\Pi (a)=N\Pi _0+NL((s-\ell )a-ka^2/2)\). Then

    \[\Delta \Pi =\Pi (\alpha ^{\rm CO})-\Pi (\alpha ^{\rm NE}) =\frac {NLk}{2}(\alpha ^{\rm NE}-\alpha ^{\rm CO})^2>0.\]

    Proof. 

      Complete the square around \(a=(s-\ell )/k\). The difference of the choices is \(\ell (1-1/N)/k>0\).

      Suppose both total profits are nonnegative, a fixed fraction \(\rho \in [0,1]\) funds the next training round, outside finance is unavailable, and its compute price is a fixed \(p>0\). The difference between the monetary training allocations is \(\rho \Delta \Pi \), and the difference between their financially purchasable compute quantities is \(\rho \Delta \Pi / p\). This follows directly from the stipulated rule \(C(a)=\rho \Pi (a)/p\); other physical constraints can prevent those quantities from being realized.

      The result compares two allocations. It does not bound all autonomous policies, which may coordinate, borrow or maintain demand when permitted. For \(\rho =0\) the financing difference vanishes. Changing institutions or adding productive investment changes the model rather than contradicting the displayed algebra.