5.7.1 ANGELA’S CHOICE OF WORKING HOURS WHENSHE PAYS RENTIn Leibniz 5.4.2 we solved Angela’s constrained optimizationproblem for the decisions she made as an independent farmer.Now we look at the case in which Bruno owns the land, andAngela has to pay rent in order to produce grain. We will see thatAngela’s quasi-linear preferences have an important implicationfor the effect of the rent on her decision.

As before, Angela’s preferences are represented by a quasi-linear utilityfunction:

where represents her daily hours of free time and the number of bushelsof grain she consumes per day. The function is increasing and concave.Her feasible frontier shows how the number of bushels of grain she canconsume per day, , depends on the free time she takes.

When Angela was an independent farmer, she could consume all thegrain she produced. Her feasible frontier was , where and

, and her constrained optimization problem was to choose andto maximize , subject to the constraint , giving us the first-order condition:

Her optimal choice of free time, , is the value of that satisfies thisequation. Remember that her MRS is and her MRT is ; theoptimal choice is where MRS = MRT.

How does the problem change when Angela pays rent?Suppose she has to pay bushels of grain per day to her landlord Bruno

for the right to operate her farm. This means that Angela’s consumption ofgrain is less than her production. If she takes hours of free time sheproduces bushels of grain as before, but now her consumption will be:

In this case, her constrained optimization problem is to choose and tomaximize , subject to the constraint

LEIBNIZ

1

Figure 1 Allocation of time with and without payment of rent.

But the first-order condition for her optimal choice of free time isexactly the same as before:

so she will choose exactly the same amount of free time as she did in thecase when she did not have to pay rent. Of course, this means that she willproduce the same amount of grain as before, although her consumption willbe lower by the amount of rent she has to pay.

Why does this happen? When Angela has to pay rent, she takes hours offree time and pays to Bruno, so her consumption will be lower. But therate at which she can transform free time into consumption for herselfdoesn’t change. An additional hour of work will give her just as much addi-tional grain as when she didn’t have to pay rent, so her MRT is still .And the lower consumption doesn’t affect her marginal rate of substitutioneither, because of her quasi-linear preferences: her MRS is irrespectiveof her consumption of grain.

The effect of payment of rent is shown in Figure 1. Angela’s originalfeasible frontier as an independent farmer, , is the upper red curve.Her optimal choice of free time and consumption in that case is at P. Whenshe pays a rent of , her feasible frontier for consumption and free time isshifted vertically downward by units. This is the lower red curve in thediagram: .

Angela’s new optimal point is Q: she continues to have hours of freetime per day, and her daily consumption of grain is now bushels.You can see the effect of quasi-linearity in the figure: quasi-linear indiffer-ence curves have the same slope as you move up or down a vertical line.

We have shown that Angela will choose the same amount of free timewhatever the level of the rent . She makes the same choice if the rent iszero—the independent farmer case—and she would still make the samechoice if were negative: for example, if she had no landlord and receiveda subsidy from the government for working on her farm.

LEIBNIZ

2

The result that payment of rent does not change Angela’s allocation oftime means that any distributional conflict between her and Brunoconcerns only how much grain each of them gets to consume. Because ofher quasi-linear preferences, Angela’s hours of work are determinedindependently of how the grain is distributed, so the size of the pie and howit is divided are entirely separate issues. The assumption of quasi-linearityis made in Unit 5 to enable us to think about these two issues separately.

Read more: Sections 17.1 to 17.3 of Malcolm Pemberton and Nicholas Rau.2015. Mathematics for economists: An introductory textbook, 4th ed.Manchester: Manchester University Press.

5.7.1 ANGELA’S CHOICE OF WORKING HOURS WHEN SHE PAYS RENT

3