Question #72860

Assume that the demand for a product X is:
Qdx = 4,500 – 0.5Px + Py – 6Pz + 0.05M,
where Px is unit price of product X,
Py is unit price of product Y,
Pz is unit price of product Z, and
M is average income of consumers of product X.
Determine the relationship between (i) products X and Y and (ii) products X and Z, i.e., determine whether Y and Z are substitutes or complements for product X. Provide a justification for each case.
Determine whether product X is a normal or an inferior good. Explain.
Given that Py = $4,760, Pz = $85, and M = $75,000, derive the inverse demand function for product X. Clearly show your steps and calculations.
Graph the demand curve for product X.
Determine the size of the consumer surplus at $10,500 per unit price of X. Clearly show your steps and manual calculations.

Expert's answer

Assume that the demand for a product X is:

Qdx = 4,500 – 0.5Px + Py – 6Pz + 0.05M,

Product Y is a substitute (if Py increases, then Qdx increases) and product Z is a complement for product X (if Pz increases, then Qdx decreases).

Product X is a normal good, because if M increases, then Qdx increases too.

If Py = $4,760, Pz = $85, and M = $75,000, then the inverse demand function for product X is:

Qdx = 4,500 – 0.5Px + 4,760 – 6*85 + 0.05*75,000 = 13,520 - 0.5Px.

The size of the consumer surplus at $10,500 per unit price of X is:

CS = 0.5*10,500*(13,520 - (13,520 - 0.5*10,500)) = 5,250*5,250 = $27,562,500.

Qdx = 4,500 – 0.5Px + Py – 6Pz + 0.05M,

Product Y is a substitute (if Py increases, then Qdx increases) and product Z is a complement for product X (if Pz increases, then Qdx decreases).

Product X is a normal good, because if M increases, then Qdx increases too.

If Py = $4,760, Pz = $85, and M = $75,000, then the inverse demand function for product X is:

Qdx = 4,500 – 0.5Px + 4,760 – 6*85 + 0.05*75,000 = 13,520 - 0.5Px.

The size of the consumer surplus at $10,500 per unit price of X is:

CS = 0.5*10,500*(13,520 - (13,520 - 0.5*10,500)) = 5,250*5,250 = $27,562,500.

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