Prices and income

Tesla and the art of choosing your competition

Why Elon Musk succeeded where Ford, GM and BMW did not

Electrifying cars is seen as one of the large necessary steps in order to reduce CO2 emissions as well as other local pollutants. Already in the 1990's GM produced a small electric car. Norway also had a tiny 2-seater electric car called the buddy. Through the early 2000's, many of the traditional car companies produced electric cars, without much success.

Then came Tesla, originall founded by Martin Eberhard, but now best known for it's largest shareholder and CEO Elon Musk. In 2008 they started selling their first car, a sports car called the Roadster, which while it was an impractical car for most uses, it had acceleration that rivaled that of supercars from the likes of Ferrari and Lambourgini.

But Tesla had bad timing. 2008 was also when the Financial Crisis struck, and financing disappeared. Thanks to government loans, Tesla survived and was able to produce more practical and polished models: S, X, 3 and Y. For many years, Tesla had a near monopoly on high performance, high quality electric cars. This was partly a result of technology: Tesla had legitimate advantages in both battery and electric motor technology.

But Tesla's success also was a function of the types of customers they went after. Initially, they didn't go after customers that usually bought a Toyota or a Ford, but rather those that would otherwise have bought a Porsche. By creating a car that competed with a Porsche, they could charge a high price and absorb the extra cost of expensive batteries.

The point, which we will look more closely at in this lesson, is that when we consider the effect of price on what we buy, then we need to consider both the preferences of our customers as well as to consider the products we are competing against. To analyze such situations, we use the concepts of price elasticity and cross-price elasticity.

Price changes

Increase price of apples Reduce price of apples

Optimal consumption

We recall from lesson 5 that when the price of one of our goods change, then it can have several effects.

• Usually, when a the price of a good declines, then consumption of that good will increase. We measure the size of this effect by price elasticity, which we define below.
• But when the price of a good declines, then we may also change the amount we buy of other goods. We measure this effect by cross price elasticity.
• In the figure, when the price of apples goes down, then we also buy fewer bananas. We move our consumption from bananas to apples, since apples have become relatively cheaper.
• Ved å endre prisen på apples i figuren, viser vi at cross price elasticity er positiv: Det vil si at når prisen på apples går ned, så kjøper vi færre bananas - vi bytter fra bananas til apples.

Price elasticity and cross price elasticity

Price elasticity

We use price elacticity when we want to measure how much we change our consumption of a certain good after the price has increased.

Let us write the price elasticity for apples as \(e_a\):

$$e_a = \frac{\frac{dx_a^*}{x_a^*}}{\frac{dp_a}{p_a}} = \frac{dx_a^* p_a}{dp_a x_a^*}$$

Here $dx$ and $dp$ are the cahnge in quantity and prices

We can interpret this to say that if the price of apples changes by 1%, then the price of our (optimal) consumption changes by \(e_a\) %

But why not just measure the absolute changes in quantity? The price of apples goes down by 0.2 Euro, and we buy 3 apples more. Seems simple!

But if may be that we want to compare with other goods. Then we would literally be comparing apples to bananas. Instead, we convert to elasticities so that we can compare different goods.

Cross price elasticity

In the same way, we can measure how we change our consumption of apples when the price of bananas changes \(e_{ba}\) (litt dårlig notasjon her, men småskrift e her representerer bananas, mens storskrift e representerer elasitet. Boka bruker bare vare 1 and 2, som er enklere å skrive, men mindre intuitivt).

$$e_{ba} = \frac{\frac{dx_b^*}{x_b^*}}{\frac{dp_a}{p_a}} = \frac{dx_b^* p_a}{dp_a x_b^*}$$

Problems

1. Initially the price of frozen pizza is 4 euro and you usually buy 8 pizzas per month. Your price elasticity for frozen pizzas is -50% (-0,50). The price then increases to 5 euro. How many frozen pizzas will you buy? Show answer

   The price has changed by $(5-4)/4 = .25 = 25%$

   From the definition of price elasticity:

   $$-0,50 = \frac{\frac{dx^*_g}{8}}{0,25}$$ $$dx^*_g = -0,5*0,25*8 = -1$$

   You will eat one less pizza

2. The cross price elasticity between frozen pizza and kebab is 0.4. Does that mean you will eat more or less kebab when the price of frozen pizzas increase? Can you explain intuitively why the price elasticity of frozen pizza is negative, but the cross price elasticity to kebab is positive? What does this say about the relationship between kebab and frozen pizza? Show answer

   When the cross price elasticity is positive, that means that when the price of frozen pizza increases, you eat more kebab. The price elasticity of 0.4 says that a 1% increase in the price of frozen pizza will increase consumption of kebab by 0.4%

   Intuitively, we can imagine that when the price of frozen pizzas increase, then we eat more kebab (no price change). The relative price of kebabs has decreased, so we eat more. We can also say that frozen pizza and kebab are substitues.

3. When you eat frozen pizza, you always like to drink a pepsi. is the cross price elasticity positive or negative? Why? Show answer

   If the price of frozen pizza increases, and we eat less frozen pizza, then we will probably drink less Pepsi as well. Therefor the cross price elasticity is negative.

4. Let us say that we have a Cobb-Douglas utility function for bananas and apples:

   $$U=2x_B^{0,5}x_A^{0,5}$$

   The price of bananas is \(p_B\), and the price of apples is \(p_A\), while we write income as m.

   Can you derive the price elacticity for bananas (as a function of prices, quantities and income)?

   Show answer

   We need to start by finding a formula for our optimal amount of bananas.

   To find an optimal point, we start with our tangent condition (That the Marginal Rate of Substitution (MRS) is equal to the relative prices):

   $$MRS = \frac{p_B}{p_A} => \frac{U_B}{U_A} = \frac{p_B}{p_A}$$

   Where

   $$U_B = \frac{\partial U}{\partial x_B} = x_B^{-0,5}x_A^{0,5}$$ $$U_A = \frac{\partial U}{\partial x_A} = x_B^{0,5}x_A^{-0,5}$$

   And therefor:

   $$MRS = \frac{U_B}{U_A} = \frac{x_A}{x_B}$$

   So our tangent condition is:

   $$\frac{x_A}{x_B} = \frac{p_B}{p_A}$$

   We can write this as:

   $$x_A = \frac{p_B x_B}{p_A}$$

   We put this into our budget constraint:

   $$m = p_B x_B + p_A x_A$$ $$m = p_B x_B + p_A [\frac{p_B x_B}{p_A}]$$

   From this we can get an optimal consumption of bananas written as a function of prices and income:

   $$x_B^* = \frac{m}{2 p_B} = \frac{m P_B^{-1}}{2}$$

   Nå kan vi se på vår definisjon av priselasitet:

   $$e = \frac{dx_B}{dp_B}\frac{p_B}{x_B}$$

   We can solve this by seeing that \(\frac{dx_B^*}{dp_B}\) is the derivative for our equation for the optimal consume of \(x_B\), which we have just derived:

   $$x_B^* = \frac{m}{2p_B}= \frac{mP_E^{-1}}{2}$$ $$\frac{dx_B^*}{dp_B} = \frac{-mp_B^{-2}}{2} = \frac{-m}{2p_B^2}$$

   Then we put this in our definition of price elasticity:

   $$e_{AB} = \frac{-m}{2p_B^2}\frac{p_B}{x_B} = \frac{-m}{2p_Bx_B}$$