The Set of Pareto-Efficient Allocations under General Preferences

An individual's well-being is described by a utility function u(c, L), where 'c' represents consumption and 'L' represents leisure. The individual earns a constant wage 'w' for each hour worked, 'h'. Total time available is 'T'. Therefore, consumption is a function of hours worked (c = w*h) and so is leisure (L = T - h). The individual chooses the number of hours to work, 'h', to maximize their well-being. What is the first-order condition that defines the optimal choice of 'h'?

Optimizing Social Benefit with Economic Trade-offs

Deriving the First-Order Condition for a Producer-Consumer

An agent's utility depends on money (m) and the quantity of a good (Q), represented by the function u(m, Q). The agent's money is determined by m = m_0 - τ, where τ is a transfer. This transfer is linked to Q by the relationship τ = k - C(Q), where k is a constant and C(Q) is a cost function. The agent chooses Q to maximize utility. After substituting the constraints, the problem is to maximize u(m_0 - k + C(Q), Q). Arrange the following steps in the correct logical order to derive the first-order condition for the optimal Q.

Consider an individual who chooses a quantity Q to maximize their utility, which is represented by the function u(m, Q). The individual's money, m, is not fixed but depends on the choice of Q according to the relationship m = 100 - 5Q. The first-order condition for finding the optimal Q is found by setting the partial derivative of the utility function with respect to Q equal to zero (i.e., ∂u/∂Q = 0).

The Role of the Chain Rule in Constrained Optimization

An individual's utility is given by u(c, h), where c is consumption and h is the number of hours worked. The individual chooses h to maximize utility. Consumption is also a function of h, according to the relationship c = c(h). The first-order condition for this problem is found by setting the total derivative with respect to h to zero: (∂u/∂c) * (dc/dh) + (∂u/∂h) = 0. Match each mathematical term from this equation to its correct interpretation.

An agent chooses the quantity of a good, y, to maximize their utility, which is given by the function u(x, y) = x * y. The quantity of another good, x, is determined by the agent's choice of y through the relationship x = 120 - 3y. The first-order condition that defines the optimal choice of y can be expressed as ____ - 6y = 0.

A city planner is choosing the optimal size of a new public park, A, to maximize the city's welfare, represented by the function W(M, A). M represents the remaining public funds available for other projects, and A is the park area in acres. The available funds M are dependent on the park's size due to construction and maintenance costs, according to the relationship M = B - C(A), where B is the total initial budget and C(A) is the total cost function for the park. Which expression correctly represents the first-order condition that must be satisfied to find the welfare-maximizing park area, A?

Identifying an Error in a Constrained Optimization Derivation

An individual's well-being is described by a utility function u(c, L), where 'c' represents consumption and 'L' represents leisure. The individual earns a constant wage 'w' for each hour worked, 'h'. Total time available is 'T'. Therefore, consumption is a function of hours worked (c = w*h) and so is leisure (L = T - h). The individual chooses the number of hours to work, 'h', to maximize their well-being. What is the first-order condition that defines the optimal choice of 'h'?

Optimizing Social Benefit with Economic Trade-offs

Deriving a First-Order Condition with Dependent Variables

A planner seeks to maximize a society's welfare, represented by the function W(C, G), where C is total private consumption and G is the level of a public good. The society has a fixed total income Y. The cost of providing the public good is given by the function c(G), which means the total amount available for private consumption is C = Y - c(G). A student attempts to find the optimal level of G by setting the partial derivative of the welfare function with respect to G equal to zero: ∂W/∂G = 0. Why is this approach incorrect for finding the optimal G?

A consumer's satisfaction is represented by a function U(x, y), where x and y are quantities of two goods. The consumer's choice is limited by a budget, which creates a direct trade-off between the goods that can be expressed as y = f(x). To find the optimal amount of good x, this trade-off relationship is substituted into the satisfaction function, resulting in a new function of a single variable: V(x) = U(x, f(x)).

Statement: The first-order condition for finding the optimal x that maximizes satisfaction is ∂U/∂x = 0.

A decision-maker wants to choose the optimal level of an activity, z, to maximize an objective F(x, y). The variables x and y are not chosen directly but are both determined by the level of z according to the relationships x = g(z) and y = h(z). Arrange the following steps in the correct logical order to find the first-order condition that defines the optimal z.

Interpreting the First-Order Condition with Dependent Inputs

A decision-maker's objective is represented by the function F(x, y). The values of x and y are determined by a single choice variable, z, through the functions x = g(z) and y = h(z). The first-order condition for the optimal choice of z is found by setting the total derivative of F with respect to z equal to zero. This total derivative is given by: dF/dz = (∂F/∂x) * (dg/dz) + (∂F/∂y) * (dh/dz). Match each mathematical term from this derivative with its correct interpretation.

An agent's objective is to maximize a function V(a, b). The values of a and b are not chosen independently but are both determined by a single decision variable, q, according to the relationships a = f(q) and b = g(q). To find the optimal q, the first-order condition is found by setting the total derivative of V with respect to q equal to zero. This total derivative is expressed using the chain rule as: dV/dq = (∂V/∂a) * (df/dq) + ________.

Critiquing a Profit Maximization Strategy