4 - Economics examples

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Linear Functions and Graphs

Section 4: Practise questions

Question 1

Draw the demand function $Q=40-2P$ and the corresponding inverse demand function.

The demand function $$Q = 40 - 2p$$ Here, the slope $b=-2$ and vertical intercept $(0,40)$
And the Inverse demand function, $$P = 20 - 0.5Q$$ Here, the slope is $-0.5$ and vertical intercept $(0,20)$

Figure 8

Question 2

A firm knows that its inverse demand function is $P = a - bQ$. The firm can sell $20$ units when the price is $$9$, and $40$ units when the price is $$6$. What is the slope of its inverse demand function?

The slope of the inverse demand function is: $$-b = \frac{$9 - $6}{20 - 40} = -\frac{3}{20}$$

Question 3

Assume that a consumer has a budget of $$B$ to spend on two goods $X$ and $Y$ and that price of $X$ is $P_X$ and price of $Y$ is $P_Y$ a unit. (a) If the consumer spends the entire budget $$B$ to buy two goods, write the equation of the consumer’s budget line. (b) What is the slope of the budget line if $Y$ is measured on the vertical axis? What are the intercepts? (c) What happens to the budget line if the budget increases from $$B$ to $$B'$? (d) What happens to the budget line if $P_x$ increases other things remaining constant?

(a) The equation of the budget line

(b) If $Y$ is measured on the vertical axis

$$P_{X}X + P_{Y}Y = B$$

$$P_{Y}Y = B - P_{X}X$$ $$\qquad Y = \frac{B}{P_{Y}} - \frac{P_{X}}{P_{Y}}X$$

The slope of the budget equation: ${{-P_X}\over {P_Y}}$
Here, the vertical intercept is $(0,{{B}\over {P_Y}} )$ and the horizontal intercept is $({{B}\over {P_X}} ,0)$.

(c) If the budget increases from $$B$ to $$B'$, the vertical intercept increases ${{B}\over {P_Y}}$ to ${{B'}\over {P_Y}}$ which shifts the new budget line parallel to the right of the initial budget line. A change in the budget when other factors remain constant does not affect the slope of the budget line.

(d) If $P_X$ increases, the slope of the budget line which is ${{-P_X}\over {P_Y}}$ changes, and the budget line becomes steeper. This changes the horizontal intercept $({{B}\over {P_X}} ,0)$, but the vertical intercept remains unchanged. Hence, if the price of a good changes while other factors remain constant, the slope of the budget line also changes.

Question 4

A consumer has an income of $$200$ to spend on two goods, $X$, and $Y$. Prices of $X$ and $Y$ are $$20$ and $$5$ respectively. (a) If the consumer spends the whole $$200$ on $X$ and $Y$, draw his budget line. (b) What happens to his budget line if the price of $Y$ rises to $$10$? (c) What happens if the budget falls to $$80$?

(a) The general form representing the budget equation is $$P_{X}X + P_{Y}Y = B$$ If $P_X=$20,P_Y=$5$ and $B=$200$, the budget equation can be simplified as:

Slope of the budget line
= -4

$$20X + 5Y = 200$$ $$\qquad\qquad\qquad 5Y = 200 - 20X$$ $$\qquad\qquad\quad Y = 40 - 4X$$

Slope of the budget line: $-4$
Vertical intercept: $(0,40)$
Consider $Y=40-4X$ as the initial budget line

(b) If the price of Y rises to $10, the budget equation becomes

Slope of the budget line
= -2

$$20X + 10Y = 200$$ $$\qquad\qquad\qquad 10Y = 200 - 20X$$ $$\qquad\qquad\qquad Y = 20 - 2X$$

Compared to the initial budget line, the new budget line becomes flatter and the vertical intercept also moves down.

Same slope but the intercept is different.

$$20X + 5Y = 80$$ $$\qquad\qquad\qquad 5Y = 80 - 20X$$ $$\qquad\qquad\qquad Y = 16 - 4X$$

Compared to the initial budget line, the vertical intercept of the new budget line changes but the slope does not change when only income changes.
The following figure presents all three budget lines.

Figure 9

Question 5

A consumer has an income of $$2400$ to spend on two goods, $X$, and $Y$. Prices of $X$ and $Y$ are $$30$ and $$50$ respectively. (a) If the consumer spends $$2400$ to consume $X$ and $Y$, draw his budget line. (b) What happens to his budget line if the price of $X$ doubles? (c) What happens if the budget increases by $50 $%?

(a) The budget equation is $$P_{X}X + P_{Y}Y = B$$ If $P_X=$30,P_Y=$50$ and $B=$2400$, the budget equation is $$30X + 50Y = 2400$$ $$\qquad\qquad\qquad 50Y = 2400 - 30X$$ $$\qquad\qquad\qquad \color{red}{Y = 48 - \frac{3}{5}}X$$ The slope of the budget line: $-\frac{3}{5}$
Vertical intercept: $(0,48)$
Consider $Y=48-{{3}\over 5} X$ as the initial budget line.

(b) If the price of X rises to $$60$, the new budget equation is

Price of $X$ doubles

$$\color{red}{60}X + 50Y = 2400$$ $$\qquad\qquad\qquad 50Y = 2400 - 60X$$

Slope changes

$$\qquad\qquad\qquad Y = 48 - \color{red}{\frac{6}{5}}X$$ In this example, the new budget line starts from the same vertical intercept as the initial budget line, but the new budget line is steeper.

(c) If the budget increases by $50$%, the budget line becomes: $$30X + 50Y = \color{red}{3600}$$ $$\qquad\qquad\qquad 50Y = 3600 - 30X$$

Vertical intercept changes but the slope remains the same.

$$\qquad\qquad\qquad Y = \color{red}{72} - \frac{3}{5}X$$ The budget line shifts parallel to the right of the initial budget line. Vertical intercept of the new budget line is bigger than the vertical intercept of the initial budget line, but the slope does not change.
Figure 10 presents all three budget lines.