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# Calculating the Elasticity Values in the Given Case

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Suppose the estimated demand equation of good X looks as follows:

QX = 10,000 - 2 PX + 3 PY - 4.5M
Where Px is the price of the product itself
PY is the price of another (related) good
M is buyers' income.

Suppose currently PX = \$100, PY = \$50, and M = \$2,000.
a. What is the price elasticity of demand of good X?

b. What is the cross-price elasticity of good X with respect to the price of good Y?

c. what is the income elasticity of good X?

d. Estimate the demand function (representing the demand curve) of this product.

e. Suppose the local government decides to raise the sales tax on product X, causing the price to rise by 10%. Will sales of the product X rise or fall, and by what percentage amount?

f. If the seller of the product (X) wants to increase her sales quantity by 10% through a price-change, what should she do to price - increase? decrease? By what percentage amount?

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#### Solution Preview

a. What is the price elasticity of demand of good X?

First we calculate the quantity demanded at given parameters.
QX = 10,000 - 2 PX + 3 PY - 4.5M
Put PX=\$100, PY=\$50 and M=\$2000
QX=10000-2*100+3*50-4.5*2000=950

We are given that
QX = 10,000 - 2 PX + 3 PY - 4.5M
Differentiate QX with respect to PX, we get
dQX/dPX=-2

Price elasticity of demand=(dQX/dPX)*(PX/QX)=-2*(100/950)= -0.21

b. What is the cross-price elasticity of good X with respect to the price of good Y?
We have calculated in part (a) that QX=950 at given parametes.

We are given that
QX = 10,000 - 2 PX + 3 PY ...

#### Solution Summary

Solution describes the steps to estimate the price elasticity of demand, cross-price elasticity of demand and income elasticity of demand in the given case. It also studies the impact of price changes on quantity demanded.

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## Managerial Economics

Brown guitar company hires you to consult and you estimate the demand for guitars to be Q=9000-6P. The supply of guitars is given by Q= -3000+9.

1 What is the equilibrium price and quantity of guitars?

2 What is the price elasticity of demand at the equilibrium price and quantity?

3 What is the price elasticity of supply at the equilibrium price and quantity?

If a per-unit excise tax of \$ 90.00 per guitar is levied on the consumers, what price would consumers pay after the tax is levied ? What proportion of tax will be paid by the supplies of guitars? How many guitars will be sold after tax is imposed? How much consumer surplus do consumers get after the tax? What is the dead weight loss created by tax?

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