Income and preferences

Warren Buffett and income elasticity

Why does one of the richest people in the world live in an old house?

You have perhaps heard of Warren Buffett. He is regularly on the list of the world's richest people. The popular version of how he became rich is that he has an eye for finding undervalued stocks, which he buys up and holds in the long term. He is a so-called value investor.

A microeconomist might make a more hard-nosed assesment. Warren Buffett has an eye for companies and industries that have natural monopolies and market power. In the wall-street vocabulary of Wall Street, he looks for companies with "wide moats" that can keep out competition. This includes power companies with regional monopolies. Banks and insurance companies with large economies of scale. It also includes companies with strong brands, like Coca Cola. We will talk more about market power and monopolies in a later lesson.

What we want to focus on now are some of Warren Buffett's lifestyle choices. Even though he is one of the world's richest people, he is known for still living a quite modest lifestyle.

Instead of fine french cuisine and expensive wine, Buffett prefers a burger and a Coca Cola (though these choices might reflect his ownership stakes in both Coca Cola and Burger King.)

He also lives in the modest house in the mid-sized city, Omaha, Nebraska, which he bough as a young man in 1958. Warren Buffett would fit in well among people in the FIRE-movement.

We find it surprising that Warren Buffett has not bought a new, larger house, because we usually think of housing as a good that we buy more of, or higher quality of when we increase our income. Housing is what we call a normal good: A good we buy more of (either in quality or quantity) when we get more income.

On the other hand, fast food and soda are things we might think of as inferior goods: Goods we buy less of when we get more income. But as Warren Buffett shows, these definitions are necessarily specific to the individual.

In this lesson, we take a closer look at modelling the relationship between income and consumption.

A change in income

Income elasticity: How much more will we buy when we get rich.

If we wish to know how much more (or less) we will buy of a good when our income rises, this is called income elasticity.

For example, if we call \(x_f^*\) our optimal amount of frozen pizza, then we can call the income elasticity of frozen pizza \(E_f\)

$$E_f = \frac{\frac{dx_f^*}{x_f^*}}{\frac{dm}{m}}= \frac{dx_f^*}{dm}\frac{m}{x^*_g}$$

In plain English, income elasticity asks: If we increase our income by 50%, ((\frac{dm}{m}=0,50)\), how much does does our consumption of frozen pizza change?

Problems

  1. Assume that frozen pizza is an inferior good for you. The government decides to cut the amount of financial aid it provides to students and your income therefor drops. Will you expect your consumption of frozen pizza to increase or decrease?

    By definition of an inferior good, we woudld expect you to increase your consumption of frozen pizza when your income decreases. Perhaps you decide to eat out less, and eat more frozen pizzas at home.

  2. You have a friend who loves frozen pizza. He says that the only thing that stops him from eating more is not having enough money. Is frozen pizza a normal or inferior good for your friend?

    It sounds like frozen pizza is a normal good for your friend.

  3. You earn 1,000 EURO per month and eat 10 frozen pizzas in a typical month. Then you lose your part-time job and your income falls to 750 EURO. After this fall in income, your consumption of frozen pizzas rises to 15. What is your income elasticity for frozen pizza?

    Our income has gone down by 25% (-0.25), while our consumption of frozen pizza has increased by 50% (0,50). Our income elasticity is (-0.50/0.25=-2).

    4. Lets assume that we can write our indifference curve between apples and bananas in the form of a Cobb-Douglas equation:

    $$U = x_a^{0,5}x_b$$

    where \(x_a\) and \(x_b\) are the quantities of apples and bananas.

    We have a fruit budget to spend on bananas and apples for 50 EUR per month. Apples cost 0.5 euro and bananas cost .8 euro.

  4. What is the optimal consumption of apples and bananas?

    Our conditions for an optimum are that the slopes of the budget line and indifference curve are the same and that the budget constraint holds.

    $$p_a/p_b = U_a/U_b$$ $$p_a x_a + p_b x_b = m$$

    Since \(U = x_a^{0,5}x_b\)

    Taking the derivative of U with \(x_a\):

    $$U_a = 0,5x_a^{-0,5}x_b$$

    The derivative of U with b \(x_b\):

    $$U_b = x_a^0,5$$

    The first condition becomes:

    $$\frac{p_a}{p_b} = \frac{0,5x_a^{-0,5}x_b}{x_a^{0,5}}$$ $$\frac{p_a}{p_b} = \frac{0,5x_b}{x_a^{0,5}x_a^{0,5}}$$ $$\frac{p_a}{p_b} = \frac{0,5x_b}{x_a}$$ $$x_a = \frac{p_b}{p_a} 0,5 x_b$$

    We set our equation for x_a in our budget constraint:

    $$m=p_a x_a+p_b x_b$$ $$m=p_a (\frac{p_b}{p_a} 0,5x_b)+p_b x_b$$ $$m=p_b 1,5 x_b$$ $$x_b^* = \frac{m}{1,5 p_b}$$

    (Remember that * represents optimal values)

    So we can insert \(x_b^*\) in our equation for \(x_a\):

    $$x_a^* = \frac{p_b}{p_a} 0,5 x_b^*$$ $$x_a^* = \frac{p_b}{p_a} 0,5 (\frac{m}{1,5 p_b})$$ $$x_a^* = \frac{m}{3 p_a}$$

    Now we can insert our amounts for income and prices:

    $$x_b^* = \frac{50}{1,5*0.8} \approx 42$$ $$x_a^* = \frac{50}{3*0.5} \approx 33$$

  5. Your rich aunt Olga wants to support your economics education and gives you a monthly stipend. You use 20 euro of this to increase your budget for fruit (apples and bananas.) What becomes your new optimal amounts for apples and bananas? Are bananas a normal or inferior good?

    We can insert our new budget (m=70) and calculate new optimal values:

    $$x_b^* = \frac{70}{1,5*0.8} \approx 58$$ $$x_a^* = \frac{70}{3*0.5} \approx 47$$

    Bananas are a normal good.