Robinson Crusoe economy

The Robinson Crusoe economy , based on the novel " Robinson Crusoe " by Daniel Defoe from 1719, describes a one-person economy with a consumer, a producer and two goods that are available.

The novel is about a shipwrecked Scottish sailor who is stranded on a desert island for several years.

This history is often used as an international economic model, as it simplifies the economic relationships of reality and one can infer more complex models from this model. For example, to infer several actors from a one-person economy with only one agent. The model is also used in public finance , growth theory, and microeconomics , on which this article is based.

In the Robinson Crusoe economy, Robinson is believed to be the only economic agent. This means that he maximizes his profit as a producer and maximizes his utility as a consumer. It is also assumed that all goods must be produced or found from existing supplies on the island and that the island is separated from the rest of the world and therefore there is no trade. However, the prevailing conditions change when another person is introduced - Friday - as is also the basis of the novel.

Robinson as a hermit

Transformation curve from Robinson to goods coconut and fish

Robinson as a hermit means that he is stranded alone on an island and has to work to survive. Here it is assumed, for example, that there are only coconut palms and fish live in the sea. In addition, Robinson can work a certain number of hours a day catching fish and gathering coconuts.

For example, Robinson can work 8 hours a day. So he knows he can catch 20 fish a week if he doesn't collect coconuts. Conversely, he can collect 40 coconuts if he refrains from fishing. Of course, Robinson can also vary between the two goods. For example, he can catch 10 fish and collect 20 nuts. These combinations of goods can be plotted graphically in a coordinate system. The Y-axis is limited to the amount of nuts and the amount of fish to the X-axis.

At point A, Robinson only catches fish for seven days and thus receives 20 fish and zero coconuts.

In point B, Robinson only collects coconuts for a week and receives 40 nuts and zero fish. The connection between these two points creates the production possibility curve , which is also called the transformation curve. All points that lie on this straight line are realizable and efficient . All points that lie below this straight line are realizable, but inefficient. All points above this straight line cannot be realized.

Which combination Robinson decides on depends on his preferences , i.e. whether he would rather eat more coconuts or more fish.

The slope of the curve reflects the marginal rate of the transformation curve , that is, how much Robinson gets from one good if he foregoes the other good, also known as opportunity costs .

The shape of this curve depends on the technology used and the number of resources.
Above all, however, it depends on the type of economies of scale in the technology. If the returns to scale are decreasing, the transformation curve is concave , but if the returns to scale are increasing, the curve is convex towards the origin. However, if there are constant returns to scale and only one input is used in production, the transformation curve is linear .

Derivation of the linear transformation curve: the
relationship between the product (e.g. good fish = ) and the amount of work required for this is represented by a production function ${\displaystyle x_{1}}$ ${\displaystyle A_{1}}$

(1). ${\displaystyle {x_{1}}={\frac {A_{1}}{a_{1}}}}$

${\displaystyle a_{1}}$ = Consumption coefficient that indicates how much working time Robinson needs to create a unit of total production.

Example: 1 fish = (1/20) week

The maximum possible total production of nuts (good ) for a given total production can then be calculated as the quotient of the remaining working time input ([B] - A ₁ ) using the consumption coefficient of: ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle a_{2}}$

(2). ${\displaystyle {x_{2}}={\frac {[B]-A_{1}}{a_{2}}}}$

By transforming equation (1.) one obtains for : ${\displaystyle A_{1}}$

(3). ${\displaystyle {A_{1}}=x_{1}a1\,}$

Inserting (3.) and (2.) one obtains:

(4.) ${\displaystyle {x_{2}}={\frac {[B]-x_{1}a_{1}}{a_{2}}}\,}$

By transforming one obtains the last equation:

(5.) ${\displaystyle {x_{2}}={\frac {[B]}{a_{2}}}-{\frac {a_{1}}{a_{2}}}\ x_{1}}$

Optimal work from Robinson

Robinson’s optimal work

The more Robinson works, the less free time Robinson has. Robinson thus chooses between work and leisure.
Robinson has two options: generate income by collecting coconuts (shown on the Y-axis) or by consuming leisure time (shown on the X-axis). As with the choice between two goods, Robinson has to decide between:

1. To work more and thereby collect more coconuts and thus to forego more free time or
2. vice versa, to consume more free time and to forego coconuts.

Point A shows the optimal trade-off between demand for leisure time and supply for work, which is the tangential point of the indifference curve and the straight line. The indifference curve shows the combination of goods which have the same benefit and thus make the consumer indifferent between the choice of these combinations. It has the same slope as the marginal rate of substitution , which measures the exchange ratio of goods. The exchange ratio reflects how many fish Robinson has to do without, for example, in order to receive one more unit of coconuts.

Derivation of the indifference curve:
In general the utility function reads . The resulting full differential is set to zero: ${\displaystyle U=U(x_{1},x_{2})}$

(1). ${\displaystyle \mathrm {d} U={\frac {\partial \;U(x_{1},x_{2})}{\partial x_{1}}}\mathrm {d} x_{1}+{\frac {\partial \;U(x_{1},x_{2})}{\partial x_{2}}}\mathrm {d} x_{2}=0}$

When rearranged, this gives the slope of the indifference curve:

(2). ${\displaystyle {\frac {\mathrm {d} x_{2}}{\mathrm {d} x_{1}}}=-{\frac {\frac {\partial U(x_{1},x_{2})}{\partial x_{1}}}{\frac {\partial U(x_{1},x_{2})}{\partial x_{1}}}}}$

This makes it clear that the slope of the indifference curve is determined from the ratio of marginal utility .

Production function and indifference curve

The tangent point of Robinson's production function and indifference curve with decreasing returns to scale

The production function reflects the relationship between Robinson's labor input and the number of coconuts received for it. This is a concave function because the extra gain in coconuts decreases as the hours worked. This is due to the marginal product of labor .

At point (A) the indifference curve touches the production curve. Here is the optimal balance between work and leisure, given the technology, for collecting coconuts.

At this point of equilibrium, the marginal rate of substitution between leisure and coconuts must be equal to the marginal product of labor :

${\displaystyle GRS=GPA}$

The dual role of Robinson Crusoe

The dual role can be explained on the basis of the assumption that Robinson Crusoe is no longer a producer and a consumer at the same time. He wants to produce on one day and only consume the next day. In order to be able to coordinate this optimally, he founds a labor market and a coconut market.
In addition, Robinson is the only shareholder in his company that uses profit maximization. This means he looks at the demand for labor and compares it with the amount to be produced.
As a shareholder, he makes profits, which he uses as a consumer to buy the company's products.

producer

Robinson's production function and iso-profit line after coconuts and labor

Robinson Crusoe decides the day before how much work he wants the next day and how many coconuts he wants to produce.
He can produce the amount of coconuts that he sells for 1 / piece. The coconuts are a numéraire good. ${\displaystyle C}$

${\displaystyle w}$refers to the hourly wage rate for collecting coconuts. represents the hours worked. In combination, this gives the profit level .
${\displaystyle L}$
${\displaystyle \Pi }$

${\displaystyle \Pi =C-wL}$

If you rearrange the equation , you get the iso profit line , the geometric location of all combinations of quantities of work and coconuts that bring the company the same profit. ${\displaystyle C}$

${\displaystyle C=\Pi -wL}$

The profit is maximized at the point at which the iso-profit line affects the production function. This means that the marginal product of labor at this point must equal the wage rate:
${\displaystyle GPA=w}$

consumer

Indifference curve and budget line from Robinson to coconuts and work

Robinson Crusoe is viewed as a consumer. He decides between work and leisure, which thus determines his possible consumption.
But he can also decide not to work at all, since he also takes on the role of shareholder and thus has a basic set of equipment available that he produced the day before (in the role of producer). This is due to the fact that work is a bad thing for Robinson. This means the more he consumes work, the lower his utility. On the contrary, coconut is a good and thus explains the slope of the indifference curve. ${\displaystyle \Pi }$

With the basic equipment at the point and the slope , you get the budget line from Crusoe. The optimal point is obtained at the point at which the indifference curve touches the budget line. Here he decides how much to work and how many coconuts to consume, assuming the given wage rate. Accordingly, the marginal rate of substitution must be equal to the wage rate: ${\displaystyle (0,\Pi )}$ ${\displaystyle w}$

${\displaystyle GRS=w}$

balance

The equilibrium with the production optimum and the consumption optimum

→ Main article : Balance (economic theory)

In the equilibrium of the producer theory, the demand for coconuts is equal to the supply of coconuts. In the equilibrium of consumer theory, the demand for work equals the supply of work.

If you add the two together, you get the equilibrium point at which the slope of the indifference curve equals the slope of the production function:

${\displaystyle GRS_{F,K}=w}$, where stands for leisure and for the coconuts ${\displaystyle F}$ ${\displaystyle K}$

${\displaystyle GPA=w}$

Put together this results in:

${\displaystyle GRS_{F,K}=GPA}$

The result put together means that a competitive equilibrium can exist. Accordingly, there are prices for inputs and outputs that maximize a company's profit, as well as the benefit of an individual. At this point, supply and demand for a good are equal.

Two-person economy

Transformation curves from Robinson and Friday for the goods coconuts and fish

Friday is also stranded on the lonely island, which leads to important economic changes. Now Robinson can not only collect coconuts or fish, but also Friday.
Friday can do a lot better than Robinson. For example, he collects a maximum of 60 fish or 60 coconuts per week.
This also means that Freitag has a different production possibility curve, which is above Robinson's transformation curve, since his productivity is higher for both goods.
In such a situation one speaks of an absolute cost advantage .

In addition, the transformation curve has a flatter slope, which means that the opportunity costs of the two are also different.
Robinson and Freitag can now specialize in one task each using the principle of comparative cost advantage . This means that everyone should create the good that they can produce relatively cheaply.

The comparative costs of fishing:
In order to catch one more fish with a given total working time, you must:

• Robinson do without 2 coconuts
• Friday but only on 1 coconut

Thus, Friday has a comparative advantage in fishing compared to Robinson.

The comparative cost of one unit of coconut is

• with Robinson on ½ fish
• on Friday for 1 fish.

Robinson thus has a comparative cost advantage when collecting coconuts compared to Friday.

From this one derives the following: Everyone should produce that in which he has a comparative cost advantage. In our case, Robinson should collect coconuts and catch the fish on Friday.
In order to determine the optimal point of the goods collected by both of them, it is assumed that Friday before he was stranded on the island consumed 30 fish and 30 coconuts and would like to maintain this level. Together with the consumption of Robinson, who consumed ten fish and 20 coconuts, without a division of labor they would together distort 40 fish and 50 coconuts.

Consumption and production of Robinson and Friday without division of labor:

	Robinson	Friday	total
coconuts	20th	30th	50
fishes	10	30th	40

In the case of a division of labor according to the principle of the comparative cost advantage, this consumption level should at least be maintained.
In the division of labor, Friday specializes in fishing. However, Friday only needs 2/3 of his working week to catch 40 fish, which is why he collects an additional 20 coconuts. Robinson specializes in collecting coconuts and thus receives 40 coconuts.
In total, the two of them collected ten more coconuts than before.

Production of Robinson and Friday with division of labor:

	Robinson	Friday	total
coconuts	40	20th	60
fishes	0	40	40

Now in the example the profit from the division of labor of 10 coconuts is divided fairly. Both now have 5 more coconuts.

With the associated division of labor, Robinson and Friday can trade with each other. The quantities of goods to be traded result from the difference between the quantities of goods produced and consumed.

Consumption of Robinson and Friday with division of labor:

	Robinson	Friday	total
coconuts	25th	35	60
fishes	10	30th	40

Trade between Robinson and Friday with division of labor:

The composite transformation curves of Robinson and Friday

	Robinson	Friday
coconuts	Exported 15	Imported 15
fishes	Imported 10	Exported 10

If both work together in a two-person economy, the cornerstones A and B result from the complete specialization of Robinson and Freitag on one good each. In point A the two would have a maximum output of 100 coconuts and in point B of 80 fish. At point C Friday would catch 60 fish and Robinson would collect 40 coconuts. If the two of them want to distort more than 60 fish, Robinson would have to catch fish and avoid coconuts. "The slope from point A to point C gives the comparative cost of fish production from Friday (-1), that between C and B , the comparative cost of Robinson (-3)."

Pareto efficiency

Pareto efficient allocation in the Robinson Crusoe economy with the transformation curve and Egeworth diagram at one point

→ Main article : Pareto efficiency

In general, the Pareto-efficient allocation describes the allocation in which one no longer receives a unit of one good without foregoing part of the other good. In addition, this is the allocation that exploits the profits. Coconuts and fish units
are available for Robinson units . At each point on the transformation curve, one can draw an Edgeworth diagram to show the possible consumption bundles. The Pareto efficient bundle is obtained at the tangent point of the Robinson and Freitag indifference curves. In general, the indifference curve is defined as the combination of quantities of goods between which a household is indifferent. The marginal rates of substitution are always the same at this point: ${\displaystyle c}$ ${\displaystyle f}$

${\displaystyle GRS_{R}=GRS_{F}}$

All these Pareto-optimal solutions are shown on the contract curve.
In addition, the marginal rate of substitution of a consumer must be equal to the marginal rate of transformation so that Pareto efficiency prevails:

${\displaystyle GRS_{R,F}=GRT_{R,F}}$

(From each fish and coconuts)

See also

• Welfare economics
• Welfare theorem

literature

• R. Varian, Hal (2007): Fundamentals of Microeconomics. Seventh edition, R. Oldenbourg Verlag Munich Vienna, ISBN 0-393-93424-1 .
• Bofinger, Peter (2011): Fundamentals of Economics. An Introduction to the Science of Markets. third edition, Pearson Studium ISBN 978-3-8273-7354-0
• Bernanke, Ben; McDowell, Moore; Thom, Rodney; Pastine, Ivan; Frank, Robert (2012): Principle of Economics. third edition, Mc Graw Hill Education, ISBN 0-07-713273-4 , ISBN 978-0-07-713273-6

Web links

• Daniel McFadden's course at Harvard University
• Yossi Spiegel's course at Tel Aviv University

Individual evidence

1. ↑ Robinson Crusoe - Lag Blume - 2006 - Retrieved June 19, 2015.
2. ↑ ^{a b c} Peter Bofinger: Fundamentals of Economics - An Introduction to the Science of Markets. In: Pearson Education No. 3, 2011, pp. 32-39.
3. ↑ Peter Bofinger: Fundamentals of Economics - An Introduction to the Science of Markets. In: Pearson Education No. 3, 2011, p. 34.
4. ^ Daniel McFadden 1975, 2003, Department of Economics, University of California - Retrieved June 19, 2015.
5. ↑ ^{a b} [1] Robinson Crusoe example -Yossi Spiegel- Retrieved on June 19, 2015.
6. ↑ Peter Bofinger: Fundamentals of Economics - An Introduction to the Science of Markets. In: Pearson Education No. 3, 2011, pp. 87-89
7. ↑ ^{a b c d e f g} Prof. Hal R. Varian: Grundzüge der Mikroökonomik In: R. Oldenburg Verlag Munich Vienna No. 7, 2007, pp. 702–707
8. ↑ Prof. Hal R. Varian: Grundzüge der Mikroökonomik In: R. Oldenburg Verlag Munich Vienna No. 7, 2007, pp. 391–393.
9. ^ Prof. Hal R. Varian: Grundzüge der Mikroökonomik In: R. Oldenburg Verlag Munich Vienna No. 7, 2007, pp. 716–718