Question

Problem II: The demand curve for product X is given by QD x = 220 − PX + 3PY + 0.001I where PY is the price of a related good Y, and I is income. The supply curve for good X is given by QS X = 10 + 3PX. a) What is the marginal effect of an increase in PY on the equilibrium price of good X? (3/4) b) How much do we need to increase income, if we want people to trade 5 more units of product X?(6666.67) c) Assuming PY = 2 and I = 50, 000, what is the equilibrium price and quantity for good X? (Q=209.5; P=66.5) d) Assuming the same values for PY and I, find the value of PX that would result in a surplus of 100 units. Also find the value of PX that would result in a shortage of 100 units. (91.5 and 41.5)

I need the step by step solutions for each answer. Thank you

Answer 1

The demand curve for product X is given by QD = 220 − PX + 3PY + 0.001I. The supply curve for good X is given by QS = 10 + 3PX.

a) What is the marginal effect of an increase in PY on the equilibrium price of good X?

Find the equilibrium price

220 − PX + 3PY + 0.001I = 10 + 3PX

210 – 4PX + 3PY + 0.001I = 0

This gives PX = 210/4 + (3/4)PY + (0.001/4)I

Find derivative of PX with respect to PY which gives dPX/dPY = (3/4).

b) How much do we need to increase income, if we want people to trade 5 more units of product X?

Find the derivative of demand function with respect to income

dQ/dI = 0.001. When income is increased by $1, quantity rises by 0.001. To have change in Q = 5, we must have change in I = 5/0.001 = 5000. Hence income should increase by 5000.

c) Assuming PY = 2 and I = 50, 000, what is the equilibrium price and quantity for good X?

At PY = 2 and I = 50,000, equilibrium price is = 210/4 + (3/4)*2 + (0.001/4)*50000 = 66.5 and quantity is 10 + 3*66.5 = 209.50 units.

(d) Assuming the same values for PY and I, find the value of PX that would result in a surplus of 100 units.

When there is a surplus of 100 units, supply exceeds demand by 100

QS – QD = 100

10 + 4PX - 220– 3*2 - 0.001*50000 = 100

4PX = 366

This gives PX = 91.5

If there is a shortage, demand exceeds supply at

220 − PX + 3PY + 0.001I - 10 - 3PX = 100

210 – 4PX + 6 + 50 = 100

This gives PX = 166/4 = 41.5