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 $$\boldsymbol{C}=\sqrt{\boldsymbol{L}_{c}}$$ where \(L_{F}\) and \(L_{c}\) are the number of hours spent fishing or gathering coconuts. Consequently, $$\boldsymbol{L}_{c}+\boldsymbol{L}_{F}=200$$ 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 \(C\) in 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

Answer: The optimal allocation of labor in autarky is 100 hours for fishing and 100 hours for gathering coconuts. The optimal production levels in autarky are 10 fish and 10 coconuts. The utility is 10. With the opportunity to trade, Robinson consumes the same levels of fish and coconuts (10 each), and his utility remains the same (10). When adjusting production to take advantage of world prices, the new production levels are 20√(40) fish and √(40) coconuts.

Step-by-step solution

Analyze production constraints

Given the limited time of 200 hours, we need to consider the relation of time spent on each activity, which is: $$L_F + L_C = 200$$

Determine production functions for fish and coconuts

Production functions are given as follows: $$F = \sqrt{L_F}$$ and $$C = \sqrt{L_C}$$

Plug in utility function and set up the Lagrange function

The utility function is: $$U = \sqrt{F \cdot C}$$ Define the Lagrange function as: $$L = \sqrt{F \cdot C} + \lambda(200 - L_F - L_C)$$

Solve the first-order conditions and find the optimal levels of labor allocation

Compute the partial derivatives for the Lagrange function with respect to F, C, and λ and set them equal to zero: $$\frac{\partial L}{\partial F} = \frac{C}{2 \sqrt{F \cdot C}} - \lambda \frac{1}{2 \sqrt{L_F}} = 0$$ $$\frac{\partial L}{\partial C} = \frac{F}{2 \sqrt{F \cdot C}} - \lambda \frac{1}{2 \sqrt{L_C}} = 0$$ $$\frac{\partial L}{\partial \lambda} = 200 - L_F - L_C = 0$$ Solving the first-order conditions yields:$$L_F = L_C$$ Plug the result into the constraint equation: $$200 = 2 L_F \Rightarrow L_F = 100$$ $$200 = 2 L_C \Rightarrow L_C = 100$$

Determine the optimal levels of fish and coconuts production

Now, plug the optimal levels of labor allocation into the production functions to find the optimal levels of F and C: $$F = \sqrt{100} = 10$$ $$C = \sqrt{100} = 10$$

Compute the utility and rate of product transformation

Use the optimal levels of fish and coconuts production in the utility function: $$U = \sqrt{10 \cdot 10} = 10$$ Calculate the marginal productivities of fish and coconuts: $$MP_F = \frac{dF}{dL_F} = \frac{1}{2\sqrt{L_F}} = \frac{1}{20}$$ $$MP_C = \frac{dC}{dL_C} = \frac{1}{2\sqrt{L_C}} = \frac{1}{20}$$ Compute the rate of product transformation (RPT): $$RPT = \frac{MP_F}{MP_C} = 1$$ Part b: Consumption and utility given the opportunity to trade.

Determine trade prices

The given prices of fish and coconuts are: $$P_F/P_C = 2/1$$

Determine consumption choices

Trade allows Robinson to consume at a new tangent to his original utility function: $$\frac{2}{1} = \frac{MP_F}{MP_C}$$ As the world prices are aligned with his RPT, Robinson will just consume his production: F = 10 and C = 10.

Calculate new utility with trade

Compute the utility for these consumption levels: $$U = \sqrt{10 \cdot 10} = 10$$ The utility remains the same at 10. Part c: Adjusting production to take advantage of world prices.

Find new production levels

Following world prices, Robinson will trade off 1 coconut for 2 fish (or 1 fish for 0.5 coke). He will consume less C and get more F through trade by allocating more labor to fish production so: $$ L_F^* > L_C^* $$ As Robinson is indifferent between fish and coconuts, the optimal production will be at the relative price of the world prices $$ F^* = 2 C^* $$ Let's use production function and labour constraint to find new production function $$L_C^* + L_F^* = 200$$ $$C^* = \sqrt{L_C^*} \Rightarrow L_C^* = C^{*2} $$ $$F^* = \sqrt{L_F^*} \Rightarrow L_F^* = F^{*2} $$ Substitute for L_C and L_F in the labour constraint $$C^{*2} + F^{*2} = 200$$ But $$F^* = 2 C^* ,$$ so substitute again: $$(C^{*})^2 + (2C^{*})^2 = 200 \rightarrow 5(C^{*})^2 = 200$$ Now, we find new \(C^*\) and \(F^*\): $$C^* = \sqrt{40} \Rightarrow F^* = 2\sqrt{40}$$ \(new \text{ production} = (2\sqrt{40}, \sqrt{40})\) Part d: Graph your results. Create a graph with coconuts on the x-axis and fish on the y-axis. Plot the production function, the optimal production levels in autarky, the new production with higher labor for fish (2F=C), and the new production with the adjustment to world prices.

Key concepts

Robinson Crusoe economy

The Robinson Crusoe economy is a simple framework used in microeconomics to study how individuals make decisions in conditions of scarcity. This theoretical construct imagines a solitary individual, like Robinson Crusoe, stranded on a deserted island with limited resources, who must decide how to allocate time and effort to produce goods for consumption. This scenario sets up a foundational understanding of how time and resource constraints can influence production decisions, much like how individuals and firms decide in the real economy.

In our exercise, Robinson Crusoe must decide how to allocate his 200 hours between fishing (F) and gathering coconuts (C), which represents the labor allocation decision faced by individuals and firms when dividing time and resources between competing activities. His goal is to maximize utility within the constraints of his environment.

Production functions

Production functions show the relationship between the input resources and the output of goods. They provide a mathematical way of showing how much output we can expect to obtain from certain amounts of inputs, under given technical knowledge. In our case, Robinson's production functions for fish and coconuts are given by square root functions: \( F = \sqrt{L_F} \) and \( C = \sqrt{L_C} \), where \( L_F \) and \( L_C \) are the number of hours dedicated to fishing and gathering coconuts respectively.

These functions depict increasing outputs with additional labor input, but at a diminishing rate; this reflects the idea that, for each additional hour spent fishing or gathering coconuts, the increase in the quantity of the resource gained is smaller than the increase from the previous hour. It's a concept similar to diminishing marginal returns in more complex production scenarios.

Utility maximization

Utility maximization is the process where individuals choose among available options to achieve the highest possible satisfaction, often referred to as utility. In microeconomics, we model this behavior through utility functions, which represent an individual's preferences. For Robinson, utility depends on the consumption of fish and coconuts, with the function defined as \( U = \sqrt{F \cdot C} \).

Robinson will choose to allocate his time between fishing and coconut gathering in a way that maximizes his utility. He must consider the production functions for both goods and the time constraint of 200 hours to determine the allocation that provides the highest utility.

Lagrange multiplier method

The Lagrange multiplier method is an optimization technique used to find the maximum or minimum of a function subject to constraints. By introducing a Lagrange multiplier (\( \lambda \)), we can incorporate the constraint into the objective function, effectively transforming a constrained optimization problem into an unconstrained problem.

In our exercise, the method helps Robinson to consider how to best use his 200 hours to balance production of fish and coconuts so that he can maximize his utility. We can see this in the Lagrange function \( L = U + \lambda (200 - L_F - L_C) \), which considers the utility function and the constraint on the total hours of work.

Rate of product transformation (RPT)

The Rate of Product Transformation (RPT) represents the trade-off between the production of different goods. It's similar to the concept of marginal rate of substitution (MRS) but applies to production rather than consumption.

It tells us how many units of one good must be given up to gain an additional unit of another good when resources are shifted from one use to another. In this case, the RPT of fish for coconuts tells us how many coconuts Robinson must forgo to catch an additional fish. Mathematically, this is expressed as the ratio of the marginal products of the two goods, which we calculated to be equal to 1, suggesting an even trade-off between fish and coconuts in Robinson's production possibilities.

Trade and consumption

Trade opens up the possibility for individuals or countries to consume beyond their production possibilities by specializing in actions where they have a relative advantage. In our Robinson Crusoe economy, when trade is not possible, Robinson consumes what he produces. However, once trade is introduced and Robinson can trade fish for coconuts at world prices, he has the opportunity to reallocate his resources to specialize in producing the good that will yield him the highest utility when traded.

These decisions are based on the trade-off between the goods, the rate of product transformation, and the world price ratio. Ideally, Robinson will produce more of the good which he can trade at a value higher than it costs him to produce (in terms of forgone production). This interplay between production and trade is a fundamental economic principle highlighting how individuals, firms, and countries can benefit from specialization and trade.

World prices adjustment

Adjusting to world prices means producers and consumers respond to international market prices when deciding what to produce and consume. For Robinson, when the price of fish is higher relative to coconuts, represented by \( P_F/P_C = 2/1 \), he can gain more by specializing in fishing because he can trade the fish for more coconuts than he would be able to produce with an equivalent effort.

Thus, adjusting his production according to world prices (\( F^* = 2C^* \) due to the price ratio), Robinson can maximize his gains from trade. This principle applies broadly, from individual decision-making, as seen with Robinson, to entire nations' trade policies.

Graphical analysis of production

Graphical analysis of production helps visualize the relationships between input and output, production possibilities, and the effects of trade and consumption. By plotting the production function, we can see the trade-offs and constraints faced. For Robinson's case, plotting the respective production functions for fish and coconuts illustrates the impact of the 200-hour constraint on his production possibilities.

The graphical representation also allows us to analyze how Robinson's production point changes when he's allowed to trade at world prices. It shows his production possibilities before and after adjusting to world prices, highlighting the advantages of trade and the potential increase in consumption beyond his initial production possibilities frontier.