Moreover, in many non-profit institutions, funds are substantially fungible, and even specifically targeted gifts can be used for general purposes. To understand the role of general gifts to a non-profit institution, we must return to the previous model and explicitly incorporate an altruistic donor. Furthermore, we now assume that V(Z) is not linear, but an increasing, strictly concave function.

The timing of the model must be adjusted to include a donor. In period zero, the entrepreneur decides on the not-for-profit or the for-profit status. In period one, a donor decides on a level of general donations, denoted by D. The donor correctly anticipates the effect of his donation on the future price and the non-contractible quality level. In period two, the entrepreneur sells the good to the consumer at a price P and a contractible quality level Qx. As in the previous section, we assume that there is only one possible level of contractible quality. In period three, the entrepreneur chooses his effort level E, which in turn determines non-contractible quality.

We assume that a donor who wishes to improve Q, the overall quality of services provided by the firm, can only do so through general donations and cannot in any sense contract to directly induce the firm to deliver a higher quality product. The donor chooses the level of general donations, denoted by D, to maximize (1-t)(I-D)+F(Q), where I is the donor’s taxable income, t is the tax rate and F(Q) is an increasing, twice differentiable concave function. The function F(Q) is meant to capture the idea that the donor just wants to see good health, good universities or good theater. We assume that there is no competition, so a single entrepreneur is maximizing the utility function specified previously. If there is an interior solution for D, the donor sets its level so that dQ/dD*F'(Q)=1-t. To ensure that this first order condition is a maximum, we assume that second order conditions hold.
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In a for-profit firm, effort is set so that 1=K'(E). Increases in income do not change this first order condition, and donations have no effect on quality. This conclusion is too strong if the entrepreneur has diminishing marginal utility of income because of satiation. However, satiation with consumption as a whole is likely to set in much slower than satiation with perquisites, and hence for the comparison of non-profit and for-profit firms, we can assume constant marginal utility of income.

In a non-profit firm, in contrast, donations influence the marginal utility of perquisites and thereby affect quality. To solve the model, we proceed recursively and first solve for effort. The first order condition for effort is K’(E)*V’(Y)=1, where we use Y to denote net income: Y = P+D+K(E)-C(QX). We can then use the equilibrium relationship P=QrmE to find the relationship between E and D that incorporates the idea that donations affect the price. We assume that K’(E)+[V’(Y)*K”(E)]/[V”(Y)*K’(E)] > m holds everywhere, which means that an increase in effort does not depress prices and profits so much that the marginal benefit from more effort itself increases. This condition must hold locally for the equilibrium to be stable. This condition ensures a unique equilibrium and that dE/dD