Question

Suppose that Robinson Crusoe produces and consumes fish \((F)\) and coconuts \((C) .\) Assume that, during a certain period, he has decided to work 200 hours and is indifferent as to whether he spends this time fishing or gathering coconuts. Robinson's production for fish is given by \\[ F=\sqrt{l_{F}} \\] and for coconuts by \\[ C=\sqrt{l_{C}} \\] where \(l_{F}\) and \(l_{C}\) are the number of hours spent fishing or gathering coconuts. Consequently, \\[ l_{C}+l_{F}=200 \\] Robinson Crusoe's utility for fish and coconuts is given by \\[ \text { utility }=\sqrt{\boldsymbol{F} \cdot \boldsymbol{C}} \\] a. If Robinson cannot trade with the rest of the world, how will he choose to allocate his labor? What will the optimal levels of \(F\) and \(C\) be? What will his utility be? What will be the \(R P T\) (of fish for coconuts)? b. Suppose now that trade is opened and Robinson can trade fish and coconuts at a price ratio of \(p_{F} / p_{C}=2 / 1 .\) If Robinson continues to produce the quantities of \(F\) and \(C\) from part \((a)\) what will he choose to consume once given the opportunity to trade? What will his new level of utility be? c. How would your answer to part (b) change if Robinson adjusts his production to take advantage of the world prices? d. Graph your results for parts (a), (b), and (c).

Short answer

In closed economy: work 100 hours on both; produce 10 fish and 10 coconuts, utility = 10. With trade and no change: swap for 20 coconuts, utility remains same. Adjusting for trade: allocate labor for price, produce \(F\approx7.07\), \(C\approx12.25\), utility \(\approx9.33\).

Step-by-step solution

Setting up the Equations

For both fish and coconuts, the production functions are given by \( F = \sqrt{l_F} \) and \( C = \sqrt{l_C} \). In a closed economy, Robinson seeks to maximize his utility \( \sqrt{F \cdot C} \) subject to the constraint \( l_F + l_C = 200 \).

Expressing Utility in Terms of Labor

Substitute the expressions for \( F \) and \( C \) into the utility function: \( \text{utility} = \sqrt{(\sqrt{l_F}) \cdot (\sqrt{l_C})} = \sqrt{l_F \cdot l_C} \). This is the function to maximize.

Solving for Labor Allocation Without Trade

To maximize \( \sqrt{l_F \cdot l_C} \), use the constraint \( l_F + l_C = 200 \). Assume \( l_F = l_C \) because the marginal utility per hour of labor should be equal (since Robinson is indifferent). So, \( l_F = l_C = 100 \).

Finding Optimal Levels of Fish and Coconuts

With \( l_F = 100 \) and \( l_C = 100 \), calculate the optimal \( F \) and \( C \) respectively: \( F = \sqrt{100} = 10 \) and \( C = \sqrt{100} = 10 \).

Utility and Rate of Product Transformation

Plug \( F = 10 \) and \( C = 10 \) into the utility function: \( \text{utility} = \sqrt{10 \cdot 10} = 10 \). The RPT (Rate of Product Transformation) is the ratio of marginal products and equals 1, since \( l_F = l_C \).

Effect of Opening Trade (b)

With a price ratio \( p_F/p_C = 2/1 \) and initial production levels \( F = 10 \) and \( C = 10 \), Robinson would trade \( F \) for \( C \) until \( 2F = C \), consuming 20 coconuts and no fish if he swaps all.

New Utility Level with Trade

Robinson's new utility when consuming 0 fish and 20 coconuts is \( \text{utility} = \sqrt{0 \cdot 20} = 0 \) if he consumes only C or maintains 10 units of something.

Adjusting Production for Trade (c)

To maximize utility using the price ratio \( 2F = C \), shift labor such that \( l_F = 50 \) and \( l_C = 150 \). Calculate new values: \( F = \sqrt{50} \approx 7.07 \) and \( C = \sqrt{150} \approx 12.25 \).

New Utility with Adjusted Production

Plug new production levels into the utility function: \( \text{utility} = \sqrt{7.07 \times 12.25} \approx 9.33 \). Maximum achievable on the constraint line.

Key concepts

Utility Maximization

Utility maximization is about getting the most satisfaction out of what you have. Robinson Crusoe needs to decide how to distribute his 200 hours of labor to get the maximum joy or utility from catching fish and gathering coconuts. The utility function given is \( \sqrt{F \cdot C} \), where \( F \) is the quantity of fish and \( C \) is the quantity of coconuts. This kind of utility function suggests that the satisfaction depends on both products, combining them multiplicatively.

To maximize utility, he must find the allocation of labor that makes the product \( F \cdot C \) as large as possible, given the constraint that \( l_F + l_C = 200 \). By making the marginal gain in utility equal from both fish and coconuts, Robinson learns that he’ll get the best outcome by dividing his time equally. This balance ensures that each additional hour spent either gathering coconuts or fishing results in the same increase in satisfaction.

Rate of Product Transformation

The rate of product transformation (RPT) indicates how the production of one good can be changed into the production of another while keeping the total output constant. In the context of Robinson's labor allocation, it represents the trade-off between producing fish and coconuts.

In a closed economy, the RPT is captured mathematically as the ratio of the slopes of the production functions for fish and coconuts. For Robinson, the important note is that when \( l_F = l_C \), this ratio is 1. This means that reshaping one hour from fishing to coconut gathering (or vice versa) won't change his ultimate utility because the productivity, measured through the slope, remains consistent across the two activities.

This changes with trade opportunities, where prices affect how products transform. Thus, when he can trade, RPT aligns with price ratios offered in the market, altering how Robinson should allocate his labor for optimal utility.

Labor Allocation

Labor allocation is about deciding how best to divide work hours between different productive activities. For Robinson Crusoe, this means deciding how many hours he spends fishing versus gathering coconuts. With 200 hours available, the challenge is to spread this time optimally to gain the most utility, given by the interplay of fish and coconut gathered.

In his closed economy, Robinson realizes that due to the symmetrical nature of his production functions \( F = \sqrt{l_F} \) and \( C = \sqrt{l_C} \), distributing his labor evenly means half for fishing and half for coconuts. This results in equal amounts produced and maximizes his utility, given the constraints.

• \( l_F = 100 \)
• \( l_C = 100 \)

This balance changes if he's allowed to trade since external price ratios can suggest different optimal labor allocations. Adjustments must be made to respond to new market opportunities, potentially skewing labor towards the higher-priced commodity.

Closed Economy Model

A closed economy model is one where no trade occurs with external economies. Robinson's isolated scenario on his island is a classic example of this: he produces and consumes goods entirely on his own, without any input or output exchange with others.

In such a setting, he must optimize his production to match his consumption, using only his labor. This close-looped system requires a continued assessment of internal trade-offs without the benefit of outside market prices affecting decisions.

This model simplifies understanding how resource allocation works without distractions from external factors. It also highlights the importance of self-sufficiency and balanced production, as Robinson cannot rely on importing or exporting goods to adjust for production shortfalls or excesses.