Question

Pedro is a college student who receives a monthly stipend from his parents of $1,000. He uses this stipend to pay rent for housing and to go to the movies (you can assume that all of Pedro’s other expenses, such as food and clothing have already been paid for). In the town where Pedro goes to college, each square foot of rental housing costs $2 per month. The price of a movie ticket is $10 per ticket. Let xdenote the square feet of housing, and let ydenote the number of movie tickets he purchases per month.

a) What is the expression for Pedro’s budget constraint?

b) Draw a graph of Pedro’s budget line.

c) What is the maximum number of square feet of housing he can purchase given his monthly stipend?

d) What is the maximum number of movie tickets he can purchase given his monthly stipend?

e) Suppose Pedro’s parents increase his stipend by 10 percent. At the same time, suppose that in the college town he lives in, all prices, including housing rental rates and movie ticket prices, increase by 10 percent. What happens to the graph of Pedro’s budget line?

Answer 1

Solution -

a) The budget line expresses available resources as total spending -- recall that spending for each good is the price per unit times the number of units. Let's call m the budget, p the price of x and P, the price of y. Then, we have m=px+Py. In our peculiar case, p=2, P=10 and m=1000.

b) Here, you'll need to answer c) before drawing your line.

c) They're basically asking you what are the coordinates (0, m/P) and (m/p ,0) -- that is, they want to know how much you can get for each if you send your whole budget on them. For our peculiar case, he could afford 100 tickets (1000/10) or 500 square ft of appartment (1000/2). As you can see, these are the coordinates where the curve crosses the axes -- it crosses y at (0, m/P) and x at (m/p, 0) -- and, once you have these two points, you can draw a line which will be your budget curve.

Of course, the budget equation could be hell of a lot more complex. I could add transfer payments and taxes that take effect only once you go beyond a certain point, for instance. But the intuition is always the same: all you need to do is to substitute the coordinates in the budget equation and isolate the quantity of your interest. Here, you'd get
m=px+Py
m-Py=px
(m-Py)/p=x

As y=0 when you spend everything on x, you have m/p=x.

d) What happens if P increases? How much x can you buy now if you don't buy any y? You can buy just as much. If you hesitate, look at the equation: m/p=x. X is not a function of P when y=0. So, what changed? Well, y=m/P and P increased. The new price is 3/2 times higher than the former -- 15/10=3/2 -- and so you can't afford as much y when x=0. Now, you can afford this:

y=m/(3/2)P=(m/P)(2/3)

The fraction applies to the denominator and, so, you multiply the original ratio by its inverse (2/3). Graphically, the budget line rotates. It meets the x axis where it used to, except that now it meets the y axis lower -- at 2/3 of its original height, to be precise. Intuitively, it's obvious: if you increase the price for a given budget, I won't be able to afford as much as I used to.

e) Let's multiply m, p and P by a constant a. If you really want to stick to your example, let a be a fraction.

am=apx+aPy

What happened? Nothing changed. You can divide everything by the constant. This is an interesting result. Economically, it means that only changes in the relative prices have real effects (graphically, the ratio of p/P must change for your budget line to rotate). However, I want to add some details here for your knowledge.

What if your wage increased exactly along the lines of the inflation rate, is your budget constraint identical? The answer is no, except for one case. To get this effect, you need every price of every good, as well as your income to grow at exactly the inflation rate -- i.e., the standard deviation for the growth rates of prices must be 0. As the inflation rate is a weighted average of price variations and that the above is very unlikely