Business Application: Hiring an Assistant: Suppose you are a busy CEO—with lots of consumption but relatively little leisure. I, on the other hand, have only a part-time job and therefore lots of leisure with relatively little consumption.
A: You decide that the time has come to hire a personal assistant—someone who can do some of the basics in your life so that you can have a bit more leisure time.
(a) Illustrate our current situation in an Edgeworth Box with leisure on the horizontal and consumption on the vertical axis. Indicate an endowment bundle that fits the description of the problem and use indifference curves to illustrate a region in the graph where both of us would benefit from me working for you as an assistant.
(b) Next, illustrate what an equilibrium would look like. Where in the graph would you see the wage that I am being paid?
(c) Suppose that anyone can do the tasks you are asking of your assistant — but some will do it cheerfully and others will do it with attitude. You hate attitude — and therefore would prefer someone who is cheerful. Assuming you can read the level of cheerfulness in me, what changes in the Edgeworth box as your impression of me changes?
(d) How do your impressions of me — i.e. how cheerful I am — affect the region of mutually beneficial trades?
(e) How does increased cheerfulness on my part change the equilibrium wage?
(f) Your graph probably has the new equilibrium (with increased cheerfulness) occurring at an indifference curve for you that lies below (relative to your axes) the previous equilibrium (where I was less cheerful). Does this mean that you are worse off as a result of me becoming more cheerful?
B: Suppose that my tastes can be represented by u(c,ℓ) = 200lnℓ+c while yours can be represented by u(c,ℓ,x) = 100x lnℓ+ c where ℓ stands for leisure, c stands for consumption and x stands for cheerfulness of your assistant. Suppose that, in the absence of working for you, I have 50 leisure hours and 10 units of consumption while you have 10 leisure hours and 100 units of consumption.
(a) Normalize the price of c as 1. Derive our leisure demands as a function of the wage w.
(b) Calculate the equilibrium wage as a function of x.
(c) Suppose x = 1. What is the equilibrium wage, and how much will I be working for you?
(d) How does your MRS change as my cheerfulness x increases?
(e) What happens to the equilibrium wage as x increases to 1.2? What happens to the equilibrium number of hours I work for you? What if I get grumpy and x falls to 0.4?