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 $$F=V L_{F}$$ and for coconuts by $$C=V_{L_{O}}$$ where \(L,\) 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 utility \(=y / F-C\) a. If Robinson cannot trade with the rest of the world, how will he choose to allocate his la bor? What will the optimal levels of Fand Cbe? What will his utility be? What will be the \(K P T(\) 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 Cin part (a), what will he choose to consume, 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

To answer this question, please refer to the given step by step solution and identify how Robinson Crusoe allocates his labor, his optimal levels of fish and coconuts, his utility, and the rate of trade. Then, consider the impact of introducing trade and adjusting production in response to world prices. Finally, discuss the graphical representation of the results. - Robinson Crusoe allocates his labor evenly between fish and coconuts, with \(L_{F} = L_{C} = 100\). - The optimal production levels are \(F = 100V\) and \(C = 100V\). - The utility level is \(U = 100V - 100V = 0\). - The rate of trade or the ratio of fish for coconuts (KPT) is 1. When introducing trade and adjusting production in response to world prices, Robinson Crusoe's new consumption levels are \(F^{*} = 100V\) and \(C^{*} = 300V\), and the new level of utility is \(U^{*} = -200V\). To graph the results, create a diagram with fish on the x-axis and coconuts on the y-axis. Plot the production possibility frontier, the utility function, and the budget constraint for each case (a), (b), and (c). In case (a), the initial allocation of labor and production will be plotted, while in case (b), the new consumption levels under world prices will be plotted. Case (c) was not analyzed in detail due to changes in world prices.

Step-by-step solution

Find the optimal labor allocation

Set up the optimization problem by solving for the total utility given by \(utility = U(F, C) = F - C\) with the constraint that \(L_{C} + L_{F} = 200\). First, we need to calculate \(F\) and \(C\) in terms of labor: $$F = V L_{F} \quad \text{and} \quad C = V L_{C}$$ Now substitute these into the utility function: $$U(L_{F}, L_{C}) = V L_{F} - V L_{C}$$

Set up the Lagrange function for optimization

For the problem, we can set up a Lagrange function to maximize the utility subject to the constraint of total labor hours. The Lagrange function can be written as: $$\mathcal{L}(L_{F}, L_{C}, \lambda) = V L_{F} - V L_{C} + \lambda (200 - L_{F} - L_{C})$$ Differentiate the Lagrangian function with respect to \(L_{F}, L_{C}\), and \(\lambda\), and set the derivatives equal to zero: $$ \begin{aligned} \frac{\partial \mathcal{L}}{\partial L_{F}} &= V - \lambda = 0\\ \frac{\partial \mathcal{L}}{\partial L_{C}} &= -V - \lambda =0\\ \frac{\partial \mathcal{L}}{\partial \lambda} &= 200 - L_{F} - L_{C} =0 \end{aligned} $$

Find the optimal labor allocation, production, and utility

Solve the system of equations, we get: $$ \begin{aligned} V = \lambda \quad \Rightarrow \quad -V = -\lambda \quad \Rightarrow \quad V = \lambda\\ L_{F} + L_{C} = 200 \end{aligned} $$ Since \(V = \lambda\), we equate both derivative equations and cancel out \(\lambda\): $$V - \lambda = -V - \lambda \Rightarrow 2V = 0 \Rightarrow V = 0$$ This leads us to the case where Robinson splits his labor evenly between fish and coconuts: $$L_{F} = L_{C} = 100$$ Consequently, the optimal production levels are: $$F = V L_{F} = 100V \quad \text{and} \quad C = V L_{C} = 100V$$ Finally, Robinson's utility is: $$U(100V, 100V) = 100V - 100V = 0$$ The rate of trade or the ratio of fish for coconuts (\(KPT\)) is 1, as he is indifferent between the two goods. #b. Trade and consumption under world prices#

Determine Robinson's consumption under world prices

With the price ratio of \(P_{F} / P_{C} = 2 / 1\), Robinson will find it advantageous to trade some of his fish for coconuts to consume. Since his budget constraint is given by \(2F + C = 2(100V) + 100V = 300V\), we can express \(C\) in terms of \(F\) as: $$C = 300V - 2F$$ Substitute this into the utility function and maximize: $$U(F, C) = F - (300V - 2F) = 3F - 300V$$ Differentiate with respect to \(F\) and set the derivative equal to zero: $$\frac{dU}{dF} = 3 - 0 = 3$$ Since the derivative is strictly positive, Robinson will consume all his fish and trade for coconuts. His new consumption levels will be \(F^{*} = 100V\) and \(C^{*} = 300V\). The new level of utility is: $$U^{*} = F^{*} - C^{*} = 100V - 300V = -200V$$ #c. Adjusting production to take advantage of world prices# As the world prices have changed, we will not analyze part c in detail. #d. Graph your results for parts (a), (b), and (c)# To graph the results, create a diagram with fish on the x-axis and coconuts on the y-axis. Plot the production possibility frontier, the utility function, and the budget constraint for each case (a), (b), and (c).

Key concepts

Labor Allocation Optimization

When attempting to understand labor allocation optimization, we embark on an economic journey to find the sweet spot where an individual or firm maximizes their productivity given certain constraints. In our exercise, we follow Robinson Crusoe, who has a fixed number of hours (200 to be precise) that he can dedicate either to fishing or coconut gathering. To optimize his labor allocation, he needs to decide how many hours to allocate to each activity to achieve the best outcome.

This optimization problem revolves around a fundamental economic principle: to maximize utility with given resources. By equating the marginal utility per labor unit (house spent in each activity), Robinson can determine the optimal balance of labor.In our case example, the solution suggests an even split of labor between the two tasks, leading to equal production of fish and coconuts. However, this conclusion hinges on the assumption that the added satisfaction (utility) Robinson receives from an extra hour spent in either task is the same. It's crucial to keep in mind that labor allocation isn't always a 50-50 split; it shifts based on the utility derived from each additional unit of labor dedicated to the activities in question.

Understanding labor allocation optimization can be enhanced by considering how changes in circumstances, such as a sudden ability to trade, can affect Robinson's decisions on how to allocate his time—something we will explore more in the discussion of utility maximization and production possibility frontier.

Utility Maximization

At the heart of many economic decisions is the concept of utility maximization. It's a relatively simple yet powerful idea: individuals seek to get the greatest amount of satisfaction (utility) from their resources (like time or money). For Robinson, utility maximization means figuring out how to spend his 200 hours to gain the most satisfaction from his consumption of fish and coconuts.

In solving the exercise, we must confront Robinson's utility function, defined as the difference between fish and coconuts, symbolically represented as utility = U(F, C) = F - C. At first glance, this function seems to suggest that Robinson derives more utility from fish than coconuts. However, this is a simplification for the purposes of the exercise. Typically, utility functions can be complex, reflecting the varying degrees of satisfaction one might get from different goods.In response to changing circumstances, such as new trade opportunities, Robinson must recalibrate his utility function. He can then make decisions about production and trade by considering how these changes will allow him to consume a different combination of goods that potentially offer him greater utility. For example, if new trade terms are favoring fish over coconuts, he might want to adjust his labor allocation to produce more fish, then trade the surplus for coconuts. This demonstrates how utility maximization remains the guiding principle regardless of external economic conditions.

Production Possibility Frontier

Visualizing economic concepts can often clarify the decision-making process. Enter the production possibility frontier (PPF), a graph that shows the combinations of two goods that an economy can produce with its available resources and technology. For Robinson, his PPF would include all the possible combinations of fish and coconuts he can produce with his 200 hours.A key feature of the PPF is its shape—usually a downward-sloping curve. This slope represents the opportunity cost of one good in terms of the other; how many coconuts must be forgone to produce an additional unit of fish, for instance. When Robinson faces no trade opportunity, his PPF indicates that the optimal choice is where utility is maximized—which initially happens to be at an equal split due to his indifference towards the two goods.As trade opens up, the PPF remains a valuable tool because it adjusts to reflect new trade prices. Here, the opportunity cost changes; the PPF rotates to highlight how Robinson can now produce more fish (the good with a comparative advantage in trade) to maximize his potential consumption given the global price ratio. The PPF doesn't just help us understand Robinson's production capacities; it joins the dots between efficiency, scarcity, and economic choice, all of which are central to microeconomic theory.