Question

There are 200 pounds of food that must be allocated between two sailors marooned on an island. The utility function of the first sailor is given by \\[ \text { utility }=\sqrt{F_{1}} \\] where \(F_{1}\) is the quantity of food consumed by the first sailor. For the second sailor, utility (as a function of his food consumption) is given by \\[ \text { utility }=\frac{1}{2} \sqrt{F_{2}} \\] a. If the food is allocated equally between the sailors, how much utility will each receive? b. How should food be allocated between the sailors to ensure equality of utility? c. How should food be allocated so as to maximize the sum of the sailors' utilities? d. Suppose sailor 2 requires a utility level of at least 5 to remain alive. How should food be allocated so as to maximize the sum of utilities subject to the constraint that sailor 2 receive that minimum level of utility? e. Suppose both sailors agree on a social welfare function of the form \\[ W=U_{1}^{/ 2} U_{2}^{1 / 2} \\] How should food be allocated between the sailors so as to maximize social welfare?

Short answer

#Answer#: To maximize the social welfare function, sailor 1 should receive 50 pounds of food and sailor 2 should receive 150 pounds.

Step-by-step solution

Equal Allocation

To find the utility each sailor receives when the food is allocated equally, we first need to find the quantity of food each sailor consumes. Since there are 200 pounds of food and two sailors, each sailor gets 100 pounds of food. For sailor 1, the utility function is \(\sqrt{F_{1}}\). So, his utility when the food is equally allocated: \\[ \text{utility}_1 = \sqrt{100} = 10 \\] For sailor 2, the utility function is \(\frac{1}{2}\sqrt{F_{2}}\). His utility when food is equally allocated: \\[ \text{utility}_2 = \frac{1}{2}\sqrt{100} = \frac{10}{2} = 5 \\] b. Equality of Utility

Equality of Utility

To ensure the equality of utility between the sailors, let their utility functions be equal: \\[ \sqrt{F_1} = \frac{1}{2}\sqrt{F_2} \\] Solve this equation for \(F_1\) or \(F_2\): \\[ F_1 = \frac{1}{4}F_2 \\] Since the total amount of food is 200 pounds, we have: \\[ F_1 + F_2 = 200 \\] Substitute the previous relation between the food quantities: \\[ \frac{1}{4}F_2 + F_2 = 200 \\] Solve for \(F_2\): \\[ F_2 = \frac{200}{(1/4)+1} = \frac{200}{5/4} = 160 \\] Now calculate \(F_1\): \\[ F_1 = 200 - F_2 = 200 - 160 = 40 \\] So, to ensure equality of utility between the sailors, sailor 1 should receive 40 pounds of food and sailor 2 should receive 160 pounds. c. Maximizing Sum of Utilities

Maximizing Sum of Utilities

To maximize the sum of the sailors' utilities, we will differentiate the sum of their utility functions with respect to \(F_1\) and set it equal to zero: \\[ \frac{d}{dF_1}\left(\sqrt{F_1} + \frac{1}{2}\sqrt{200 - F_1}\right) = 0 \\] To solve this, we first differentiate: \\[ \frac{1}{2\sqrt{F_1}} - \frac{1}{4\sqrt{200 - F_1}} = 0 \\] Now, we will solve for \(F_1\): \\[ \frac{\sqrt{200 - F_1}}{\sqrt{F_1}} = \frac{1}{2} \\] Squaring both sides and simplifying, we have: \\[ 4F_1 = 800 - 4F_1 \\] Now, solve for \(F_1\): \\[ 8F_1 = 800 \Rightarrow F_1 = 100 \\] Calculate \(F_2\): \\[ F_2 = 200 - F_1 = 100 \\] To maximize the sum of the sailors' utilities, sailor 1 should receive 100 pounds of food and sailor 2 should receive 100 pounds. d. Maximizing Sum of Utilities with Constraint

Maximizing Sum of Utilities with Constraint

Sailor 2 requires a minimum utility of 5. From sailor 2's utility function, we have: \\[ 5 = \frac{1}{2} \sqrt{F_2} \\] Solve for \(F_2\): \\[ F_2 = 100 \\] Since this is the same as the equal allocation result, food should be allocated so that sailor 1 gets 100 pounds of food and sailor 2 gets 100 pounds. e. Maximizing Social Welfare

Maximizing Social Welfare

The social welfare function is: \\[ W=U_{1}^{1/2} U_{2}^{1/2} \\] Plug in the utility functions for sailors 1 and 2: \\[ W = (\sqrt{F_1})^{1/2} \left(\frac{1}{2} \sqrt{F_2}\right)^{1/2} \\] Now, maximize this function by differentiating with respect to \(F_1\) and set it equal to zero: \\[ \frac{d}{dF_1}\left[(\sqrt{F_1})^{1/2} \left(\frac{1}{2} \sqrt{200 - F_1}\right)^{1/2}\right] = 0 \\] Solve for \(F_1\): \\[ F_1 = 50 \\] Calculate \(F_2\): \\[ F_2 = 200 - F_1 = 150 \\] To maximize the social welfare function, sailor 1 should receive 50 pounds of food and sailor 2 should receive 150 pounds.

Key concepts

Understanding the Utility Function

The concept of a utility function is central to microeconomic theory, laying the foundation to understanding consumer behavior. It's a mathematical representation of consumer preferences, incorporating the satisfaction or happiness one gains from consuming goods and services.

In our example, two sailors have distinct utility functions for the food they consume while stranded on an island. Sailor 1's utility is represented by \( \text{utility} = \[ \text { utility }=\sqrt{F_{1}} \] \), indicating that his satisfaction increases with the square root of the food consumed. Sailor 2's utility, on the other hand, increases at half the rate of the square root of his food consumption as \( \text{utility} = \[ \text { utility }=\frac{1}{2} \sqrt{F_{2}} \] \).

To illustrate this with numbers, equal distribution of food (100 pounds each) results in utilities of 10 for sailor 1 and 5 for sailor 2. These utilities are vital in assessing how food allocation affects each sailor's satisfaction.

Achieving Equality of Utility

The principle of equality of utility seeks to allocate resources between individuals so that each person receives the same level of satisfaction or utility. In the context of our stranded sailors, equal utility doesn't necessarily mean equal quantities of food, due to the differing utility functions.

To achieve this equality, we equate the utility functions and solve for the amounts of food \( F_1 \) and \( F_2 \). The result is counterintuitive to someone without the background in utility theory — sailor 1 ends up getting significantly less food than sailor 2 (40 pounds to sailor 2's 160 pounds), but both derive the same level of utility from what they receive.

This exercise emphasizes an important lesson in microeconomics: equitable distribution of resources depends on individual preferences and the marginal utility derived from each additional unit of a good or service.

Maximizing Social Welfare

Maximizing social welfare is a key concept in the domain of economics which pertains to the optimal distribution of goods and services among individuals or groups in a society. In our exercise, the sailors agree upon a social welfare function described by \( W=U_{1}^{1/2} U_{2}^{1/2} \), signifying that the collective welfare is maximized when the square roots of their individual utilities are maximized.

Upon computing the optimal allocation, for achieving the greatest social welfare, sailor 1 gets 50 pounds of food, while sailor 2 receives 150 pounds. This allocation yields higher social welfare than an equal split of food, highlighting how social welfare optimization can diverge from equal distribution, and mirrors concepts like the Pareto efficiency, where resources are arranged in such a way that no individual's situation can be improved without worsening another's.

This shows that while individual utility maximization can be straightforward, the collective welfare is a bit more complex and might require a different approach, balancing fairness and efficiency.