Comparative statics

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Figure 2.15 A shift of the demand curve (schedule).

the inverse demand function or the inverse supply function: it should be easy to see that p^∗= 4.

2.10.1 Comparative statics

Suppose that the inverse demand function for apples has changed to the new one as follows:

p= 14 − (q^D)². (2.29)

Figure 2.15 depicts two demand schedules: Equations (2.27) and (2.29). You can see the shift in the demand schedule to the right. Note that at each and every price, more apples will be demanded.

Various events may cause the shift. An increase in buyers’ income may be one of them.

Keeping other things constant, with more income buyers may want to buy more apples than before at each and every price of an apple. Another event that may give rise to the rightward shift in the demand schedule is an increase in the price of bananas (not apples).

Suppose everything (including the price of apples and buyers’ income) remained the same except for the price of bananas. If apples were close substitutes for bananas, people would substitute away from bananas and would consume more apples (again at each and every price of an apple).

One way to capture the effect of a change in the price of a substitute (or a change in buyers’ income levels) is to change a parameter value of the inverse demand function.

What we are interested in is to know how this change in a parameter value affects the equilibrium level of price and quantity. In general, we call this exercise comparative static analysis: it examines how the equilibrium values of the variables in question might change if one of the parameters in the question changed. We conduct comparative static analysis below and see how the equilibrium price and quantity should be affected.

p 14 8

0 q

shortage

1 2 3

p = 8 − q²

p = q + 2

p^** = 5 p^* = 4

p = 14 − q²

Figure 2.16 Comparative statics.

Question Explain why the equilibrium price cannot stay the same if there is the change in the demand as explained above. Obtain the new equilibrium price.

Solution If the price of apples stayed the same as the original equilibrium price, which is 4, there would be an excess demand (or a shortage) of apples. This leads apple buyers to bid the price up. Apple sellers who realise the shortage will also raise the price. The increase in the price of apples will continue until the excess demand becomes zero, i.e.

until it reaches the new equilibrium price.

The new system of equations is

p= q + 2 p= 14 − q² Substituting the second equation into the first one yields:

14− q²= q + 2.

Rearranging this equation gives the following quadratic equation:

q²+ q − 12 = 0.

We can factorise it as follows:

(q+ 4)(q − 3) = 0.

Therefore, q= 3, −4. We ignore the negative solution and hence q^∗∗ = 3. The new equilibrium price is p^∗∗ = 5. This exercise is described in Figure 2.16.

Exercise 2.12 Comparative statics.

43 2.11 Logic

2.11 Logic

In the course of your study of economics and finance, you will be required to construct your argument in a logical manner, especially to prove that your argument is correct. Well, it may sound trivial but doing so is not always easy. In closing this chapter, we briefly study logic and proofs. What is discussed in this section may seem simple and easy in the beginning, but please don’t be fooled by that. We shall cover some subtle issues that require you to think carefully.

2.11.1 Statements

In the 2006 FIFA World Cup (WC2006), Australia played Japan in the group stage. I was watching it on TV with my Australian friends and I was very excited when Japan scored first in the first half. It looked (to me) as if Japan was never going to lose, so I said to my friends, ‘I think the match won’t finish with 1-0, the final score will be 3-1’. What I meant to say – but didn’t say – was ‘3-1 to Japan’, and my friends seemed to interpret it that way. Well, in the end, to my disappointment, Australia scored three goals in the second half and my friends were happy to tell me that ‘3-1, you were right!’.

‘Australia defeated Japan in WC2006.’

This is an example of a statement. When a declarative (objective) sentence can be classified as either ‘true’ or ‘false’, but not both, we call it a statement. While the above statement is indeed true, what about the following?

‘This textbook is great!!’

In an everyday conversation, you might call it a statement, but whether you think my textbook is great is your personal (subjective) opinion. Therefore, in a mathematical sense, it is not regarded as a statement. While I hope the above opinion is popular amongst the readers, I shall introduce another one.

‘10 000 000 000 000 000 001 is a multiple of 7.’

Don’t worry if you cannot immediately tell whether it is true or false; perhaps none of us can! But we know that it can not be both true and false. So long as we know that we can classify it as either ‘true’ or ‘false’, we can call it a statement.

Some statements are sometimes true but false other times. For example, if we confine xto be real numbers,

‘x² = 4’

is true when x= ±2, but false otherwise (so the above statement is false if it is meant to be for all the real xs).

Let us denote one statement by P and another statement by Q. We say that two statements P and Q are logically equivalent if P is true (false) exactly when Q is true (false, respectively). When P and Q are logically equivalent, we write P ≡ Q. For

example,

‘x is Japan’≡ ‘x is beaten by Australia in WC2006’⁷

‘2x− 1 = 0’ ≡ ‘x = 1