In models with general, non-quasi-linear preferences, the Pareto-efficient quantity (Q) is not independent of income distribution. This dependency arises because the partial derivatives in the first-order condition are influenced by a party's income level (mf). Since the transfer payment (τ) and, consequently, mf are linked to the other party's target payoff (y₀), the choice of y₀ affects the optimal Q. This results in a set of Pareto-efficient allocations, where each combination of quantity (Q) and transfer (τ) corresponds to a different value chosen for y₀.

The Set of Pareto-Efficient Allocations under General Preferences

To derive the first-order condition for a constrained choice problem with general preferences, differentiation is performed after using the substitution method. Once the constraint is substituted into the objective function, such as the utility function `u(m_f^0 - τ, Q)`, both arguments of the function become dependent on the production quantity, Q. As a result, the chain rule is required when differentiating the objective function with respect to Q. The final step is to calculate this derivative, which yields the first-order condition for the optimal quantity. [4]

Google

The constrained choice problem for determining a Pareto-efficient allocation under general, non-quasi-linear preferences is formally stated as maximizing one party's utility, such as the fishermen's, represented by a general function $u(m_f^0 - \tau, Q)$. This maximization of production quantity (Q) and monetary transfer ($\tau$) is subject to the constraint that the other party's payoff, in this case the plantation owner's, remains at a constant level, $y_0$. The constraint is defined by the equation $\tau + P^W Q - C_p(Q) = y_0$.

Mathematical Statement of the Constrained Choice Problem with General Preferences

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

Learn Before

Related

Learn After