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

5.7 Case 2: A take-it-or-leave-it contract

Consider a different situation for Angela and Bruno. As in Case 1, Bruno owns the land that Angela works on to produce grain. But in Case 2, the government—through laws administered by courts and enforced by police—protects Angela from being forced to work, as well as Bruno’s property rights as a landowner. Furthermore, the legal system will enforce contracts between Bruno and Angela. Angela’s rights improve her alternative options: she has greater structural power in her relationship with Bruno.

We will consider two kinds of contracts that Bruno could offer to Angela: an employment contract, and a tenancy contract. In both cases, it is a take-it-or-leave-it offer: Angela can accept or reject it. She has the right to say no, and Bruno would be punished by the government if he used violence, or threats of violence, to make Angela accept his offer. Their situation is similar to the self-interested players in the ultimatum game, with Bruno as the Proposer and Angela as the Responder.

Contracts are written or spoken agreements that are intended to be enforced by law. For a contract to be valid, both parties have to agree voluntarily, and both are required to provide something. For example, if a homeowner makes a contract with a painter, the painter will be required to paint the home, and the homeowner will be required to pay the agreed fee in a reasonable time period (as specified in the contract).

Contracts play a central role in the economy. One of the key factors in explaining historical differences in economic growth between countries is the variation in the ability of their institutions to secure property rights and enforce contracts at a low cost. Effective contract enforcement requires a well-functioning judiciary (that is, a system of courts) that resolves cases in a reasonable time and is predictable and accessible to the public.

Units 6 and 9 discuss employment contracts, and contracts between lenders and borrowers, in more detail. They examine the important cases where contracts are incomplete: that is, the contract does not cover everything that matters to the parties involved: for example, how hard the employee works, or the risks taken by the borrower with the funds they have borrowed. Here we simplify by assuming that Angela cannot vary how hard she works in an hour.

An employment contract

In an employment contract, Bruno can specify Angela’s hours of work, and the wage (in bushels of grain) that she will be paid. This is a take-it-or-leave-it offer, so she does not have the option to ask for different terms of employment. If she rejects, she would leave Bruno’s land and find other work. Her reservation option is the utility she would receive in this alternative. Angela’s reservation option in Case 2 is much better than in Case 1: her utility from finding work elsewhere is higher than the utility she could expect if she attempted to revolt or escape.

Figure 5.13 shows Angela’s reservation indifference curve in Case 2. It is IC2, above IC1, which represents her alternative as a forced labourer.

The decisions Bruno has to make are similar to those in Case 1: how many hours Angela should work, and how much grain she should receive (as a wage, this time). The difference is that her alternative option is better. Unless he offers a contract that gives Angela at least as much utility as that indicated by IC2, she will say no.

There are 2 diagrams. In Diagram 1, 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. A downward-sloping, convex curve intersects the feasible frontier in two points and is labelled IC1. Another downward-sloping, convex curve intersects the feasible frontier in two points, lies above the reservation indifference curve at all points, passes through point L (16, 23) and is labelled IC2. At A and L, the slope of the feasible frontier and IC2 respectively is the same. In Diagram 2, 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 40. A hump-shaped curve peaks starts at (8, 0), peaks at (16, 22) and terminates at (23.5, 0). This is Bruno’s share when force is allowed and alternative option is escape or revolt.  A second hump-shaped curve above the first one starts at (7, 0), peaks at (16, 31), and termainates at (24, 0). This is Bruno’s share when force is not allowed and alternative option is different employment.
Fullscreen

Figure 5.13 Bruno’s employment contract offer.

Angela’s reservation indifference curves: There are 2 diagrams. In Diagram 1, 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) and (24, 0) and is labelled feasible frontier. A downward-sloping, convex curve intersects the feasible frontier in two points and is labelled IC1. Another downward-sloping, convex curve intersects the feasible frontier in two points, lies above the reservation indifference curve at all points and is labelled IC2. The area between IC2 and the feasible frontier is the feasible set. In Diagram 2, 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 40. A hump-shaped curve peaks starts at (8, 0), peaks at (16, 22) and terminates at (23.5, 0). This is Bruno’s share when force is allowed and alternative option is escape or revolt.  A second hump-shaped curve above the first one starts at (7, 0), peaks at (16, 31), and termainates at (24, 0). This is Bruno’s share when force is not allowed and alternative option is different employment.
Fullscreen

Angela’s reservation indifference curves

The feasible frontier is the same as in Case 1, but Angela’s reservation indifference curve is higher: it is IC2 rather than IC1. Bruno’s feasible set of allocations is smaller.

How Bruno chooses the wage: There are 2 diagrams. In Diagram 1, 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) and (24, 0) and is labelled feasible frontier. A downward-sloping, convex curve intersects the feasible frontier in two points and is labelled IC1. Another downward-sloping, convex curve intersects the feasible frontier in two points, lies above the reservation indifference curve at all points and is labelled IC2. At 12 hours of free time, Angela produces 54 bushels of grain and keeps 36 of these. The vertical distance between the feasible frontier and IC2 is Bruno’s profit. The vertical distance between the horizontal axis and IC2 is Angela’s wage. In Diagram 2, 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 40. A hump-shaped curve peaks starts at (8, 0), peaks at (16, 22) and terminates at (23.5, 0). This is Bruno’s share when force is allowed and alternative option is escape or revolt.  A second hump-shaped curve above the first one starts at (7, 0), peaks at (16, 31), and termainates at (24, 0). This is Bruno’s share when force is not allowed and alternative option is different employment.
Fullscreen

How Bruno chooses the wage

Suppose that Bruno asks Angela to work for 12 hours, to produce 54 bushels of grain. To maximize his own amount of grain, he will choose the wage that gives her reservation utility. He will pay her 36 bushels, and make a profit of 18 bushels.

Bruno will choose an allocation on IC2: There are 2 diagrams. In Diagram 1, 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) and (24, 0) and is labelled feasible frontier. A downward-sloping, convex curve intersects the feasible frontier in two points and is labelled IC1. Another downward-sloping, convex curve intersects the feasible frontier in two points, lies above the reservation indifference curve at all points and is labelled IC2. In Diagram 2, 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 40. A hump-shaped curve peaks starts at (8, 0), peaks at (16, 22) and terminates at (23.5, 0). This is Bruno’s share when force is allowed and alternative option is escape or revolt.  A second hump-shaped curve above the first one starts at (7, 0), peaks at (16, 31), and termainates at (24, 0). This is Bruno’s share when force is not allowed and alternative option is different employment.
Fullscreen

Bruno will choose an allocation on IC2

Whatever the level of working hours, Bruno will choose the wage so that he gets the whole amount of grain between IC2 and the feasible frontier.

Bruno does best where MRS = MRT: There are 2 diagrams. In Diagram 1, 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. A downward-sloping, convex curve intersects the feasible frontier in two points and is labelled IC1. Another downward-sloping, convex curve intersects the feasible frontier in two points, lies above the reservation indifference curve at all points, passes through point L (16, 23) and is labelled IC2. At A and L, the slope of the feasible frontier and IC2 respectively is the same. In Diagram 2, 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 40. A hump-shaped curve peaks starts at (8, 0), peaks at (16, 22) and terminates at (23.5, 0). This is Bruno’s share when force is allowed and alternative option is escape or revolt.  A second hump-shaped curve above the first one starts at (7, 0), peaks at (16, 31), and termainates at (24, 0). This is Bruno’s share when force is not allowed and alternative option is different employment.
Fullscreen

Bruno does best where MRS = MRT

Just as before (in Case 1), Bruno gets the most grain where the slope of Angela’s reservation indifference curve (her MRS) is the same as the slope of the feasible frontier. He offers a contract specifying eight hours of work, and a wage of 23 bushels.

Again, Bruno will choose an allocation on Angela’s reservation indifference curve in which Angela’s MRS on her reservation is equal to the MRT for grain production. He offers a contract that pays 23 bushels for eight hours of work. She produces 46 bushels of grain, so Bruno also gets 23 bushels—less than in Case 1, because Angela’s alternative option is better.

Bruno’s choice of hours is the same in Case 1 and Case 2, and the same as Angela’s choice as an independent farmer. This happens because, in our model, Angela’s MRS at a given level of hours is the same on every indifference curve. It would not happen if she had different preferences. But it always happens that—whether Angela or Bruno chooses the hours—they do best at a point where MRS = MRT.

The joint surplus

gains from trade, gains from exchange
The benefits that each party gains from a transaction compared to how they would have fared without the transaction.
joint surplus
The sum of the economic rents of all involved in an economic interaction.

People engage voluntarily in economic interactions when there are potential benefits for both parties, compared with their reservation options: in other words, economic rents or gains from exchange. There are rents available from an interaction between Angela and Bruno: his reservation option is zero bushels of grain, and the amount of grain she can produce by farming his land is greater than the amount that would give her reservation utility. The total amount of rent available in Figure 5.13 is the vertical distance between the feasible frontier and Angela’s reservation indifference curve. This additional grain is the potential joint surplus from the interaction.

When he offers Angela an employment contract, Bruno sets her working hours so that MRS = MRT. This is where the joint surplus (the vertical distance) is maximized: it is 23 bushels.

By setting the wage, he chooses how the joint surplus of 23 bushels will be shared between himself and Angela: that is, how much rent each will get. For example, he could choose to share it equally, taking 11.5 bushels for himself, and giving her a wage of 34.5 bushels (23 for her reservation utility, plus a rent of 11.5).

Since Angela’s rent is zero, you may wonder whether she would actually be willing to participate after all. But we know she would participate if she got very slightly more. Rather than repeatedly saying, ‘Angela’s wage is 23 bushels of grain plus a little bit more,’ we assume that she is willing to participate if she gets her reservation utility. This is a common assumption in economic models.

But Bruno is self-interested and wants to maximize his own economic rent. And he can make a take-it-or-leave-it offer—so he has all of the bargaining power. He sets the wage as low as possible, at allocation L, where Angela gets only her reservation utility. Her rent is zero, and he gets the whole of the joint surplus: a rent for himself of 23 bushels.

A tenancy contract

What happens if, rather than employing her, Bruno offers Angela a tenancy: he specifies the rent that she must pay him to use the land, and leaves it to her to decide how to use it?

In this situation, it appears that Bruno has less power over Angela. The contract does not allow him to determine how much she works. But he can still make a take-it-or-leave-it offer, and as before he will do best if he can ensure that Angela gets no more than her reservation utility.

We know from Figure 5.13 that, of all the allocations on the reservation indifference curve, Bruno gets the most grain when Angela works for eight hours. So he would like the allocation to be point L where—under an employment contract—he could receive 23 bushels of grain.

Perhaps surprisingly, he can achieve exactly the same outcome with a tenancy contract, by setting the rent payment for the land at 23 bushels. Figure 5.14 shows the situation that then faces Angela. She is free to choose her hours of work. The dotted line shows how much grain she can consume—23 bushels less than she produces, depending on the hours she chooses. The highest indifference curve she can reach is IC2, which just touches her frontier for grain consumption at point L.

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) and (24, 0) and is labelled feasible frontier. Another downward-sloping, concave curve connects points (0, 40), L (16, 23) and (22, 0) and is labelled Angela’s feasible frontier for grain consumption. A downward-sloping, convex curve is labelled IC2, is tangential to Angela’s feasible frontier for grain consumption and intersects the feasible frontier in two points. The vertical distance between the the two feasible frontiers at 16 hours of free time is the land rent and is equivalent to 23 bushels of grain.
Fullscreen

Figure 5.14 Angela’s choice of hours under the tenancy contract.

The best Angela can do if she has to pay a land rent of 23 bushels is to work for eight hours, and consume 23 bushels of grain (point L). This gives her the reservation level of utility, so she will be willing to accept Bruno’s offer. (Any other choice of hours would give her less than reservation utility.) Bruno has achieved exactly the same outcome for himself, and Angela, as under the employment contract.

Rents and efficiency in Case 2

The allocation from a take-it-or-leave-it contract is L, where Angela gets 16 hours of free time and she and Bruno both get 23 bushels of grain. Angela has better outside options than in Case 1, and she is better off as a result: her working hours are the same, but her income is higher.

Bruno’s rent is lower than in Case 1, where the rules of the game allow him to use force. Although the outcome is better for Angela because she has more structural power, she still gets zero economic rent because the ability to make a take-it-or-leave-it offer gives Bruno all the bargaining power.

The Pareto criterion does not rank the outcomes in the two cases: Angela is better off at allocation L, while Bruno is better off in Case 1 at D. Both allocations are Pareto efficient. At allocation L, there is no change that could make either Bruno or Angela better off without making the other worse off. To understand why, we can use exactly the same argument that we used in the previous section for allocation D.

But if you regarded the outcome at D as unfair, you may think allocation L, with an equal distribution of grain between the two, should be preferred.

Given the institutions governing the interaction, and the Pareto efficiency of the outcome, Angela cannot negotiate to improve her situation beyond allocation L. Any alternative contract she suggested would make Bruno worse off.

Angela’s hours of free time 16  
Angela’s bushels of grain 23  
Bruno’s bushels of grain 23  
Angela’s economic rent (bushels) 0 She gets the same level of utility as in the best alternative of working elsewhere
Bruno’s economic rent (bushels) 23 His best alternative is 0 (if she refuses his offer)

Figure 5.15 The outcome of Case 2.

Question 5.4 Choose the correct answer(s)

Figure 5.13 shows the outcomes when Angela is forced to work for Bruno (Case 1) and when Angela has the choice to accept or reject Bruno’s contract (Case 2). Based on this information, read the following statements and choose the correct option(s).

  • With a take-it-or-leave-it offer, Bruno’s economic rent is equal to the joint surplus.
  • Both Bruno and Angela are better off in Case 2 compared to Case 1.
  • When Bruno offers a contract (Case 2), Angela accepts because she receives an economic rent.
  • Angela works longer under voluntary exchange than under coercion.
  • Bruno’s reservation option is to receive nothing. Under the contract, Bruno receives the whole of the surplus: the amount in excess of what Angela needs to be willing to work. So this is his economic rent.
  • Bruno is better off in Case 1, while Angela is better off in Case 2.
  • Under the contract, Bruno offers an allocation Angela is only just willing to accept. She is indifferent between D and her reservation option, so her economic rent is zero.
  • Angela’s working hours remain the same in both cases.

The rules of the game illustrated by Case 2 make Angela better off and Bruno worse off than under the forced labour rules of the game of Case 1. Of course, Angela was better off in the baseline case than in either of Case 1 or 2 because she owned the land and could consume all of the grain she produced.

Extension 5.7 The outcome of a tenancy contract

In the main part of this section, we showed that the outcome Bruno would most prefer if he offered Angela an employment contract could also be achieved by offering a tenancy contract. Continuing the analysis from Extensions 5.4 and 5.5, we now analyse the tenancy contract directly, as a constrained choice problem for Angela and also Bruno, and again solve it using calculus. We consider both the general case and the example discussed in Extension 5.5.

Bruno, the landlord, wants to maximize the amount of grain he receives, and Angela’s preferences and technology are the same as in Extension 5.5 (the case of the independent farmer). Bruno offers Angela a tenancy contract, according to which she pays an amount of rent, \(c_0\), in exchange for the right to farm the land.

Angela’s preferences are represented by a quasi-linear utility function \(u(t,\ c) = v(t) + c\), where \(t\) represents her daily hours of free time, \(c\) the number of bushels of grain she consumes per day, and the function \(v\) is increasing and concave. She will accept the contract offer only if it gives her at least her reservation level of utility, which we will denote by \(u_0\).

The feasible frontier for producing grain, \(y=g(24-t)\), tells us how many bushels of grain she can produce, \((y)\), given the amount of free time she takes. As before, the production function \(g(24-t)\) is increasing and concave. But since she has to pay rent, this is no longer her feasible frontier for consumption.

To analyse what will happen, we will first determine how many hours Angela will choose to work if she accepts a contract with rent, \(c_0\). We will then consider whether the contract offer gives her sufficient utility to make it acceptable. Lastly, we will determine the rent that Bruno will set when he offers her the contract.

If Angela accepts the contract offer, the amount of grain left for her to consume after paying rent, \(c_0\), to her landlord Bruno will be \(y-c_0\). How many hours will she choose to work? Her constrained choice problem is almost the same as in the case of the independent farmer, except that her feasible frontier for consumption is now:

\[c= g(24-t) -c_0\]

The tenant farmer’s constrained choice problem

Choose \(t\) and \(c\) to maximize \(u(t,\ c)\), subject to the constraint \(c=g(24-t)-c_0\).

We will solve this problem by finding the MRS and MRT. Since \(c_0\) is a constant (Angela has to pay the same amount of rent whatever choice she makes), the slope of the feasible frontier is the same as before:

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

Her preferences are unchanged, and since they are quasi-linear, they do not depend on her consumption, \(c\). The slope of her indifference curves is again given by:

\[MRS=v'(t)\]

So the first-order condition remains the same as when she was an independent farmer:

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

She will choose exactly the same amount of free time as before, and produce the same amount of grain. Her choice of \(t\) does not depend on the rent, \(c_0\). But her consumption, given by:

\[c= g(24-t) -c_0\]

will be lower.

The reason this happens is that paying rent and getting lower consumption doesn’t affect her marginal rate of substitution between consumption and free time, because of her quasi-linear preferences.

Quasi-linearity makes the tenancy contract quite easy to analyse. But we could use the approach described in this extension to solve the problem for other utility functions. In Extension 5.9, we will calculate the MRS and MRT in the case of a Cobb–Douglas utility function. The MRS depends on \(c\) as well as \(t\). So in this case, paying rent to Bruno would affect Angela’s free time as well as her consumption.

If she had different preferences, paying rent might make her more willing to give up free time to raise her consumption. In that case, her choice of free time as a tenant would differ from her choice as an independent farmer.

We will write \(t^*\) and \(c^*\) for the particular values of \(t\) and \(c\) that Angela will have under the contract—that is, the solution to the two equations above. Then she will accept the contract if, and only if, it gives her at least her reservation utility, \(u_0\), that is:

\[u(t^*, \ c^*)\geq u_0, \text{or equivalently } u(t^*, \ g(24-t^*)-c_0)\geq u_0\]

What level of rent \((c_0)\) will Bruno set when he offers her the contract? He cares only about the amount of grain he receives from Angela; he wants to maximize the rent, but also to ensure that Angela accepts the contract. So he too needs to solve a constrained choice problem.

The landlord’s constrained choice problem

Choose \(c_0\) to maximize \(c_0\) subject to the constraint, \(u(t^*,\ g(24-t^*)-c_0)\geq u_0\), where \(t^*\) is the level of free time Angela will choose.

To choose the level of the rent, Bruno needs to think ahead and work out what Angela will do, and what level of utility she will therefore receive under the contract he offers. However, this is quite straightforward because Angela has quasi-linear preferences, so \(t^*\) does not depend on the rent, \(c_0\).

There is only one variable to consider in the landlord’s problem: \(c_0\). Bruno wants \(c_0\) to be as high as possible. By examining the constraint, and remembering that Angela’s utility is increasing in consumption, this means her utility must be as low as possible. So he should choose \(c_0\) so that she gets only her reservation utility. His best choice is the value of \(c_0\) that satisfies:

\[u(t^*, \ g(24-t^*)-c_0)= u_0 \Rightarrow v(t^*)+g(24-t^*)-c_0=u_0 \Rightarrow c_0=g(24-t^*)-u_0\]

Our analysis using general functions, \(v\) and \(g\), has given us a set of equations for Angela’s free time and consumption, and Bruno’s income, when they both do the best they can for themselves. We will now return to the example we used in Extension 5.5, for which we can solve the equations fully.

An example

Suppose as before (in the case of the independent farmer) that Angela has quasi-linear utility function, \(u(t,\ c) =4\sqrt{t} + c\), and production function, \(y=2\sqrt{2h}\), where \(y\) is output and \(h\) is hours of work. Suppose also that her reservation level of utility, \(u_0\), is 21.

When she has to pay rent, the equation of her feasible frontier is:

\[\begin{align*} c = 2\sqrt{2(24-t)} -c_0 \end{align*}\]

We know from our analysis above (but you can check again if you wish) that her MRS, MRT, and choice of free time are the same as before:

\[MRS=MRT \Rightarrow \frac{2}{\sqrt{48-2t}}=\frac{2}{\sqrt{t}} \Rightarrow t^*=16\]

But the corresponding level of consumption is:

\[\begin{align*} c^*=2\sqrt{2(24-t^*)} -c_0=8-c_0 \end{align*}\]

and her utility is

\[\begin{align*} v(t^*)+c^*= 4\sqrt{16} + 8-c_0 = 24-c_0 \end{align*}\]

Bruno’s best choice for the rent gives Angela her reservation utility. So it satisfies:

\[24-c_0=21 \Rightarrow c_0=3\]

Figure E5.5 shows the solution for this example. It is the equivalent of Figure 5.14 in the main part of this section. A is the point chosen by Angela as an independent farmer, where \(t=16\) and \(c=8\). It is the point on the feasible frontier for production where MRS = MRT. And it is above her reservation indifference curve.

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 original 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 corresponding to that feasible frontier is denoted by a downward-sloping convex curve that is tangent to the feasible frontier at the point A (16, 8). Angela’s feasible frontier under a tenancy contract with a rent of 3 bushels of grain is a downward-sloping concave curve that connects the points (0, 10.86) and (24, 0). Angela’s reservation utility level is 21, represented by a downward-sloping convex curve that is tangent to the tenancy contract feasible frontier at the point L (16, 5).
Fullscreen

Figure E5.5 The outcome under a tenancy contract.

As a tenant, her feasible frontier for consumption is shifted down by the amount of the rent. She chooses a point on this frontier where MRS = MRT; since she has quasi-linear preferences, she still chooses \(t=16\). Bruno maximizes his rent by setting it so that she then gets her reservation utility. He chooses a rent of three bushels of grain; this shifts Angela’s feasible frontier downwards so that the best she can do is at point L on her reservation indifference curve.

Exercise E5.3 The landlord’s problem

(Note: This exercise continues the example from Exercise E5.2. You may want to solve the example in Exercise E5.2 before starting this exercise.)

Suppose that Angela’s friend is now a tenant and has a reservation utility of 200 bushels of grain.

  1. Assume the landlord wants to maximize the amount of rent he can charge her. How much should her landlord charge her in rent?
  2. Sketch your answer using a diagram similar to Figure E5.5 (with \(t = 1,…,24\)).

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.