Unit 5 The rules of the game: Who gets what and why

5.5 Institutions, and the case of the independent farmer

As a starting point—our baseline case—it is helpful to think about what Angela would choose to do if she owned the farm herself. How many hours would she work, and how much grain would she consume? Then we consider three different institutional settings in which Bruno owns the land. For each case, we identify the total amount of grain produced, hours worked by Angela, and how much grain Angela and Bruno each get. We will then explore how the outcomes differ depending on the rules of the game.

Figure 5.6 summarizes how the rules of the game differ between the cases:

property rights
Legal protection of ownership, including the right to exclude others and to benefit from or sell the thing owned. Property rights may cover broadly-defined goods such as clean water, safety, or education, if these are protected by the legal system.
  • How Angela’s work hours are determined: by her, by Bruno, or by the two bargaining
  • Angela’s alternatives (her next best opportunity): attempted escape from Bruno’s coercion, taking a different job
  • The part played by the government: enforcing Bruno’s coercion over Angela, protecting her personal autonomy but enforcing Bruno’s property rights, facilitating bargaining between the two subject to approval by an elected state.
An independent farmer
Baseline case: Angela owns the land
Angela owns the land herself. The government protects her right to exclude others from the land (or its produce).
Angela decides: how many hours to work and how much grain to produce and consume.
A landowner and a farmer
Bruno owns the land and Angela farms it
What happens depends on the institutional setting
Case 1
Forced labour
Case 2
A take-it-or-leave it contract
Case 3
Bargaining in a democracy
The rules of the game
Bruno can force Angela to work for him producing grain which he owns.

Bruno decides: how many hours Angela must work, and how much grain she can consume.

Angela decides: Obey, or attempt escape or, with other farmers, revolt (in both latter cases risking death).
Bruno offers Angela a contract (either an employment contract, or a farm tenancy) which she can accept or reject. There is no scope to negotiate over the contract terms. Bruno cannot physically threaten Angela, and she can refuse the offer and seek work elsewhere. If necessary, the government will protect Bruno’s property rights and enforce a contract.

Bruno decides: what contract to offer Angela.

Angela decides: Accept or reject Bruno’s contract. Also, if she accepts a tenancy, how many hours to work.
Bruno offers Angela a contract. Angela can accept the contract, reject the contract, or negotiate for alternative contract terms. Angela and other farmers are able to vote, enabling them to elect a government that will pass legislation limiting the maximum number of hours worked and stipulating a minimum wage equal to what she would have earned in Case 2.

Bruno and Angela bargain: Angela, and others, vote in order to improve their options. Then she and Bruno negotiate a contract.

Figure 5.6 The rules of the game in different institutional settings.

Baseline case: Private ownership by an independent producer

We begin with the case in which Angela owns the land that she farms. She chooses her own working hours, and consumes the grain she produces. There is no other character in this case—only Angela, so this is not a social interaction and the question of income distribution does not arise.

Land tenure institutions

Ownership of land by individuals or households is a form of private property, meaning that the owner can exclude others from the use of the land or enjoyment of its products and is free to sell or give away the property. Private ownership is one of many land tenure institutions, that is, rules (written or informal) that govern who can use and buy or sell land and the conditions under which use, sale, and purchase can take place. Aside from private ownership, other forms of land tenure include communal tenure (for example, where members of a community may have the right to graze cattle on a common pasture); open access, where specific rights are not assigned to anyone and no-one can be excluded (such as oceans and some forests); and state ownership, where property rights are assigned to some authority in the public sector such as the national or state government.

Angela’s ownership of the land means that she can exclude others from using it or from receiving any of its produce. The government will enforce this right if necessary, penalizing anyone who attempts to violate it.

Angela’s decisions as an independent producer

When Angela can choose how to run the farm for herself, consuming all the grain she produces, she faces a constrained choice problem just like the one we analyse for Karim in Unit 3. She wants to find a point in the feasible set of combinations of free time and consumption that gives her the highest possible utility.

Like Karim, she will choose the point where the feasible frontier reaches the highest possible indifference curve. Work through Figure 5.7 to find how much grain she will produce, and how much free time she will take.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64), A (16, 46) and (24, 0) and is labelled feasible frontier. There are four parallel, downward-sloping, convex curves, which are labelled, from the lowest to the highest one: IC3, IC-star, and IC5. IC3 intersects the feasible frontier in two points. IC-star is tangential to the feasible frontier at point A where MRS = MRT. IC5 lies above the feasible frontier at all points.
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Figure 5.7 Independent farmer Angela’s choice between free time and grain.

The feasible frontier: In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64), A (16, 46) and (24, 0) and is labelled feasible frontier.
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The feasible frontier

The diagram shows Angela’s feasible frontier, determined by her production function.

The best Angela can do: In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64), A (16, 46) and (24, 0) and is labelled feasible frontier. There are four parallel, downward-sloping, convex curves, which are labelled, from the lowest to the highest one: IC3, IC-star, IC4 and IC5. IC3 intersects the feasible frontier in two points. IC-star is tangential to the feasible frontier at point A. IC4 and IC5 lie above the feasible frontier at all points.
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The best Angela can do

To find the combination she prefers, we add her indifference curves to the diagram. The highest indifference curve she can reach is IC*, at point A, where she would have 16 hours of free time, work for eight hours, and produce 46 bushels of grain.

MRS = MRT for maximum utility: In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64), A (16, 46) and (24, 0) and is labelled feasible frontier. There are four parallel, downward-sloping, convex curves, which are labelled, from the lowest to the highest one: IC3, IC-star, and IC5. IC3 intersects the feasible frontier in two points. IC-star is tangential to the feasible frontier at point A where MRS = MRT. IC5 lies above the feasible frontier at all points.
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MRS = MRT for maximum utility

At point A, the indifference curve is tangent to the feasible frontier. So her two trade-offs balance out: her marginal rate of substitution (MRS) between grain and free time (the slope of the indifference curve) is equal to the MRT (the slope of the feasible frontier).

The diagram shows that Angela will choose point A, with 16 hours of free time and 46 bushels of grain, where her two trade-offs balance: the trade-off she is willing to make between grain and free time (her MRS) is equal the trade-off she is constrained to make by her technology (the MRT).

Thinking about the trade-offs gives you another way to understand why this is the best she can do. Suppose, for example, that she had chosen to have more than 16 hours of free time. Then her feasible frontier is steeper and her indifference curves are flatter than at A, so MRT > MRS. This means she could transform an hour of free time into more grain than the least amount she would be willing to accept for the loss of free time. So she can increase her utility by reducing her free time. Similarly, if she had chosen to have less free time, where MRT < MRS, she could increase her utility by taking more free time and having less grain.

We can think of the combination of free time and grain at point A as a measure of Angela’s standard of living. Figure 5.8 summarizes the outcome.

Angela’s hours of free time 16
Angela’s bushels of grain 46
Bruno’s bushels of grain n/a (Bruno is not a character in this scenario)

Figure 5.8 The outcome in the baseline case.

This is the best she can do as an independent farmer. In the following sections, we consider each of the three cases in which Bruno owns the land. As you might expect, Angela’s standard of living will be lower when she works for Bruno.

Question 5.2 Choose the correct answer(s)

The figure below shows Angela’s feasible frontier and some of her indifference curves. Based on this information, read the following statements and choose the correct option(s).

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64), A (16, 46) and (24, 0) and is labelled feasible frontier. There are four parallel, downward-sloping, convex curves, which are labelled, from the lowest to the highest one: IC3, IC-star, and IC5. IC3 intersects the feasible frontier in two points, one of which is point B which lies at a lower amount of hours of free time and a higher amount of bushels of grain than point A. IC-star is tangential to the feasible frontier at point A where MRS = MRT. IC5 lies above the feasible frontier at all points and passes through point C, which corresponds to 16 hours of free time and to more bushels of grain than point B.
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  • At point B, the amount of grain that Angela is willing to give up for an additional hour of free time is greater than at point A.
  • At point B, Angela could move to a higher indifference curve by increasing her chosen hours of free time.
  • At point B, the opportunity cost of grain production is lower than the amount of grain that Angela is willing to give up for an additional hour of free time.
  • Angela could choose point C instead of point A because the MRS is the same as at point A, but the utility level is higher.
  • At point B, the slope of Angela’s indifference curve (her MRS) is steeper than at point A, due to diminishing marginal utility (of free time).
  • If Angela moved to the right along the feasible frontier, away from point B, she would increase her utility until she reached point A. For example, at 12 hours of free time, the indifference curve that cuts the feasible frontier would be higher than the indifference curve passing through point B (this indifference curve is not shown in the diagram).
  • The opportunity cost of grain production is the MRT. The amount of grain that Angela is willing to give up for an additional hour of free time is given by the MRS. At point B, the feasible frontier is less steep than IC3, which indicates that MRT < MRS.
  • Angela would not choose point C because it is unattainable with the given technology.

Extension 5.5 Angela’s choice of working hours

Following on from Extension 5.4, we apply calculus methods for solving constrained choice problems to analyse Angela’s choice of working hours as an independent farmer with quasi-linear preferences. The methods were explained in Extension 3.5, which you may need to re-read before reading this extension. We consider both the general case, and a particular example.

As an independent farmer, Angela divides her day between work and free time. Her work produces an amount of grain, \(y\), which she also consumes. Her daily hours of free time are denoted by \(t\) and the number of bushels of grain she consumes per day by \(c\).

We explained in Extension 5.4 that:

  • She has convex and quasi-linear preferences, so her utility function can be written in the form:
\[u(t,\ c) = v(t) + c\]

where the function \(v\) is increasing and concave, and her marginal rate of substitution (MRS) is \(v'(t)\).

  • Her feasible frontier for producing grain is:
\[y=g(24-t)\]

where \(g(24-t)\) is her production function, which is also increasing and concave, and her marginal rate of transformation (MRT) is \(g'(24-t)\).

And since she can consume all the grain she produces, \(c=y\). So her feasible frontier for consumption is \(y=g(24-t)\), again with \(\text {MRT}=g'(24-t)\).

Like Karim in Section 3.5 and the student in Section 3.7, Angela faces a constrained choice problem.

The independent farmer’s constrained choice problem

Choose \(t\) and \(c\) to maximize \(u(t,\ c)\) subject to the constraint \(c=g(24-t)\).

The problem can be solved by either of the two methods explained in Extension 3.5: that is, by applying the condition \(\text{MRT}=\text{MRS}\), or by using the constraint to substitute for \(c\) in the objective function \(u\), and differentiating. Either way, the first-order condition gives us the equation:

\[v'(t) = g'(24-t)\]

The utility-maximizing choice of \(t\) must satisfy this equation. Once we know \(t\), we can find the corresponding value of \(c\) from the constraint:

\[c=g(24-t)\]

An example

Suppose that in Angela’s quasi-linear utility function \(u(t,\ c) = v(t) + c\), the function \(v(t)\) is given by:

\[v(t) = 4\sqrt{t}\]

You can verify by differentiating that this function is increasing and concave, as required for convex quasi-linear preferences.

Secondly, assume that Angela’s production function is \(y=2\sqrt{2h}\), where \(h\) is hours of work. Then the equation of her feasible frontier is:

\[c = 2\sqrt{2(24-t)}\]

Again, you can verify, by differentiating, that the feasible frontier is downward-sloping and concave. The marginal rates of transformation and substitution are:

\[\text{MRT} = -g'(t) = \frac{2}{\sqrt{48 -2t}}\ \text{ , } \quad \text{MRS} = v'(t) = \frac{2}{\sqrt{t}}\]

The first-order condition, \(\text{MRT} = \text{MRS}\), is: \(\frac{2}{\sqrt{48-2t}}=\frac{2}{\sqrt{t}}\), which we can solve to obtain \(t=16\); the corresponding value of \(c\) is \(c=2\sqrt{48-32}=8\).

Figure E5.4 shows the solution for this example. Angela chooses to have 16 hours of free time and work for eight hours per day. She consumes eight bushels of grain.

In this diagram, the horizontal axis shows Angela’s hours of free time, denoted t, ranging from 0 to 24, and the vertical axis shows bushels of grain, denoted c, ranging from 0 to 20. Coordinates are (hours of free time, bushels of grain). Angela’s feasible frontier is a downward-sloping concave curve that connects the points (0, 13.86) and (24, 0) and has the equation c = 2 times the square root of 2 times 24 minus t. Angela’s highest possibility utility level is denoted by a downward-sloping convex curve that is tangent to the feasible frontier at the point (16, 8).
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Figure E5.4 The solution of the constrained choice problem with utility \(u=4\sqrt{t}+c\) and feasible frontier \(c = 2\sqrt{2(24-t)}\).

It is clear from the diagram that this is a maximum point, but you can work out the second-order condition to check this mathematically.

More about the general case

In the example above, there is a single solution to the first-order condition, and it is a maximum point. Will this always happen, whatever the functions \(v\) and \(g\)? We consider three questions:

1. Might there be more than one solution?

The first-order condition for the general case is:

\[v'(t) = g'(24-t)\]

Since \(v(t)\) is concave, the left-hand side of the equation, \(v'(t)\), is a decreasing function of \(t\) (a downward-sloping line). And since \(g\) is concave, the right-hand side is an increasing function of \(t\) (it slopes upward). If the two lines cross, they can only cross once, and the crossing point gives us the solution. This tells us that if we can find a solution, it is the only one—it is unique.

2. Can we be sure there is a solution to the first-order condition?

It is possible that the functions \(g\) and \(v\) are such that two lines don’t cross for feasible values of \(t\)—that is, between \(t=0\) and \(t=24\). In that case, there is no feasible solution to the first-order condition, and Angela’s best option would either be \(t=0\) or \(t=24\), depending on which gives her higher utility (this is known as a ‘corner solution’).

Although corner solutions may occur in mathematical problems, they are not usually very interesting economically. If we found that Angela, the independent farmer, chose not to work at all, or to work all the time, this would suggest that the functions we had chosen for \(v\) or \(g\) were not very plausible representations of her utility or technology.

3. If there is a solution to the first-order condition, is it a maximum point?

You can find the second-order condition most easily from the substitution method. The second derivative of the objective is:

\[\frac{d^2u}{dt^2}= g''(24-t)+v''(t)\]

Since \(v\) and \(g\) are both concave functions, \(g''\) and \(v''\) are both negative, and this expression is negative for all values of \(t\). So a solution to the first-order condition must be a maximum.

It is the concavity of \(v\) and \(g\) which enables us to answer ‘yes’ to both the first and third questions above. When we model economic problems mathematically, we often use general functions (rather than specific ones as in the example) and make assumptions about concavity and convexity to simplify the mathematics. But we should ensure that we can justify these assumptions by thinking about their economic interpretation.

Exercise E5.2 Another example

Suppose Angela’s friend is also a farmer with the production function \(c(t) = 100 \text{ ln } (25-t)\) and the utility function \(u(t,c)= c + 75 \text{ ln } (t)\), where \(t\) is hours of free time per day and \(c\) is bushels of grain.

  1. Use both methods (substitution and MRS = MRT) to determine the amount of free time and bushels of grain that Angela’s friend would choose. (You should obtain the same answer in both cases.)
  2. Sketch and label a diagram like Figure E5.3 to illustrate your answer to Question 1 (for \(t\) = 1,…,24).

Read more: Sections 17.1 to 17.3 of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.