Question

Qb)The current price in the market for apples is $0.10 per kg. At this price, 1 million kg are sold per year. The price elasticity of demand is –5 and the short run price elasticity of supply is 0.05. Solve for the equations of demand and supply, assuming that demand and supply are linear.

• Suppose the demand curve is given by : Qd = 10 – 2P + Ps
Where: P is the price of the product and Ps is the price of a substitute good. The price of Ps = $2

• Suppose P = $1. What is the price elasticity of demand
• What is the cross price elasticity

If the price of P becomes $2
• What is the price elasticity of demand

• There are two goods X and Y. Tabithas Utilities are given by U = 1/3X2 + 2 Y1/2. If the price of X is $2 and the Price of Y is $0.75 What is Tabithas optimum bundle?

Answer 1

1) First we should find the coefficients of the equations:

$b×0.1/1 = -5,$

b = -50 is the coefficient of the demand equation.

$b×0.1/1 = 0.05,$

b = 0.5.

Then we use the equation of a line:

$\frac{0.1 - p1} {1 - 0} = -50,$

P1 = 50.1.

$\frac{p - 50.1} {0.1 - 50.1} = \frac{q - 0} {1 - 0}$

So, the demand equation is P = 50.1 - 50Q.

$\frac{0.1 - p1} {1 - 0} = 0.5,$

P1 = -0.4.

P = 0.5Q - 0.4 is the supply equation.

2) Qd = 10 – 2P + Ps = 12 - 2P,

If P = $1, then Qd = 12 - 2×1 = 10.

The price elasticity is Ed = -2×1/10 = -0.2, so the demand is inelastic.

The cross price elasticity is:

Ecp = 1×1/10 = 0.1, so the goods are substitutes.

If the price of P becomes $2, then:

Qd = 12 - 2×2 = 8

Ed = -2×2/8 = -0.5, so the demand is inelastic too.

3) The bundle is optimal, when:

MUx/Px = MUy/Py,

MUx = TU'(X) = 2/3X,

MUy = 1/Y^0.5,

$\frac{2/3×X} {2} = \frac{1} {0.75Y^{0.5}} ,$

X = 4/Y^0.5.

We need to know the amount of income to find the exact quantities.