Suppose the demand function for good X is given by: Q_dx = 15-0 5P_x - 0.8 P y where Q_dx is the quantity demanded of good X, P_x is the price of good X, and P_y is the price of good Y, which is related to good X. a. Using the midpoint method, if the price of good Y is $10 and the price of good X decreases from $5 to $3, what is the price elasticity of demand for good X? Is the demand elastic, unitary elastic, or inelastic? b. Good X and Good Y are related as c. Using the midpoint method, if the price of good X is $10 and the price of good increases from $8 to $10, the cross price elasticity of demand is about

a. Price elasticity of demand for good X is -0.80; and the demand is inelastic.

b. Good X and Good Y are related as complements.

c. The cross price elasticity of demand is about -2.57%.

Explanation:

Note: There is an error in the demand function for good X. This is therefore corrected by restating the function as follows:

Q_dx = 15 - 0.5P_x - 0.8 P_y …………………………… (1)

a. Using the midpoint method, if the price of good Y is $10 and the price of good X decreases from $5 to $3, what is the price elasticity of demand for good X? Is the demand elastic, unitary elastic, or inelastic?

From the question and equation (1), we have:

Old price of good X = Old P_x = $5

New price of good X = New P_x = $3

New quantity demanded of good X = New Q_dx = 15 - (0.5 * 3) - (0.8 * 10) = 5.50

Old quantity demanded of good X = New Q_dx = 15 - (0.5 * 5) - (0.8 * 10) = 4.50

Ordinarily, the formula for calculating the price elasticity of demand is as follows:

Price elasticity of demand = Percentage change in quantity demanded of good X / Percentage change in price of good X (1)

Where, based on the midpoint formula, we have:

Percentage change in quantity demanded of good X = {(New quantity demanded of good X - Old

quantity demanded of good X) / [(New quantity demanded of good X + Old quantity demanded of good X) / 2]} * 100 = {(5.50 - 4.50) / [(5.50 + 4.50) / 2]} * 100 = 20%

Percentage change in price = {(New price of good X - Old price of good X) / [(New price of good X + Old Price of good X) / 2]} * 100 = {(3 - 5) / [(3 + 5) / 2]} * 100 = -25%

Substituting the values into equation (1), we have:

Price elasticity of demand for good X = 20% / -25% = -0.80

Therefore, the price elasticity of demand (based on the midpoint formula) when price decreases from $5 to $3 is -0.80.

Since the absolute value of the price elasticity of demand for good X i.e. |-0.80| is less than one, the demand is inelastic.

b. Good X and Good Y are related as

From equation (1) above, the coefficient P_Y is -0.80 which shows that it has a negative sign.

The negative sign indicates Good X and Good Y are related as complements. This implies that as price of Good Y falls, the quantity demand of Good X increases.

c. Using the midpoint method, if the price of good X is $10 and the price of good Y increases from $8 to $10, the cross price elasticity of demand is about

From the question and equation (1), we have:

Old price of good Y = Old P_y = $8

New price of good Y = New P_y = $10

New quantity demanded of good X = New Q_dx = 15 - (0.5 * 10) - (0.8 * 10) = 2

Old quantity demanded of good X = New Q_dx = 15 - (0.5 * 10) - (0.8 * 8) = 3.60

Ordinarily, the formula for calculating the cross price elasticity of demand is as follows:

Cross price elasticity of demand of goods X and Y = Percentage change in quantity demanded of good X / Percentage change in price of good Y (2)

Where, based on the midpoint formula, we have:

Percentage change in quantity demanded of good X = {(New quantity demanded of good X - Old

quantity demanded of good X) / [(New quantity demanded of good X + Old quantity demanded of good X) / 2]} * 100 = {(2 - 3.60) / [(2 + 3.60) / 2]} * 100 = -57.1428571428572

Percentage change in price of good Y = {(New price of good Y - Old price of good Y) / [(New price of good Y + Old Price of good Y) / 2]} * 100 = {(10 - 8) / [(10 + 8) / 2]} * 100 = 22.2222222222222

Substituting the values into equation (2), we have:

Cross price elasticity of demand of good X and Y = -57.1428571428572 / 22.2222222222222 = -2.57142857142857

Rounding to 2 decimal places, we have:

Cross price elasticity of demand of good X and Y = -2.57

Therefore, the cross price elasticity of demand is about -2.57%.

Note: This confirms that the relationship between Good X and Good Y is complement because the cross-price elasticity between them is negative. That is, an increase in the price of Good Y makes consumer to buy less of Good X which is a complement.