Question

The demand for good X is given by this equation:
QX =1.0–2.0PX +0.8I+1.5PY –3PZ+1.0A
Where PX, PY, and PZ represent the prices of goods X, Y, and Z;
I measures income per capita; and A is advertising. Currently:
PX =2.00,PY =2.50,PZ =1.00,I=4,andA=3.05.

Is good X a necessity or a luxury good? How do you know?

Calculate the cross elasticity of demand for X with respect to the price of good Z. Are goods X and Z substitutes or complements?

Answer 1

Answer on Question #39978 – Economics – Microeconomics

Assignment

The demand for good X is given by this equation:

$$QX = 1.0 - 2.0PX + 0.8I + 1.5PY - 3PZ + 1.0A$$

Where PX, PY, and PZ represent the prices of goods X, Y, and Z;

I measures income per capita; and A is advertising. Currently:

$$PX = 2.00, PY = 2.50, PZ = 1.00, I = 4, \text{ and } A = 3.05.$$

Is good X a necessity or a luxury good? How do you know?

Calculate the cross elasticity of demand for X with respect to the price of good Z. Are goods X and Z substitutes or complements?

Solution

The demand for good X is given by this equation:

$$QX = 1.0 - 2.0PX + 0.8I + 1.5PY - 3PZ + 1.0A,$$

where PX, PY, and PZ represent the prices of goods X, Y, and Z; I measures income per capita; and A is advertising.

$$PX = 2.00, PY = 2.50, PZ = 1.00, I = 4, \text{ and } A = 3.05.$$

**A.** Is good X a necessity or a luxury good? How do you know?

$$Qx = 1 - 2*2 + 0.8*4 + 1.5*2.5 - 3*1 + 1*3.05 = 4 \text{ units}$$

It is a luxury good, as its quantity is only 4 units.

**B.** Calculate the cross elasticity of demand for X with respect to the price of good Z. Are goods X and Z substitutes or complements?

$$E_{A,B} = \frac{P_{B,1} + P_{B,2}}{Q_{A,1} + Q_{A,2}} \times \frac{\Delta Q_A}{\Delta P_B} = \frac{\partial Q_A}{\partial P_B} \frac{P_B}{Q_A}$$

So, $Ex, z = k*Pz / Qx = -2*1/4 = -0.5$, where $k$ is coefficient before $Px$ as the derivative of $\Delta Q / \Delta P$. Two goods that complement each other show a negative cross elasticity of demand: as the price of good Y rises, the demand for good X falls. So, $x$ and $z$ are complements.