Question

Consider a demand and supply model where households pay a tax T for every unit they buy. Demand and supply are therefore given by Q^d=D(P+T) and Q^s=S(P). Calculate (∂Q^*)/∂T and (∂P^*)/∂T and show that both are negative (Q^* and P^* are equilibrium quantity demanded and equilibrium price respectively.)

Answer 1

Let the demand be Q = a - bP and supply be Q = c + dP. With a tax in place, demand becomes Q = a - b(P + T). Find the equilibrium quantity and price after tax

a - b(P + T) = c + dP

a - c = b(P + T) + dP

a - c = bP + bT + dP

This gives P* = (a - c - bT)/(d + b)

Quantity at equilibrium is Q* = c + d*(a - c - bT)/(d + b)

= (cd + bc + ad - cd - bdT)/(d + b)

= (ad + bc - bdT)/(d + b)

Now find (∂Q^*)/∂T and (∂P^*)/∂T

(∂Q^*)/∂T = (∂(ad + bc - bdT)/(d + b))/∂T

= -bd/(d + b)

< 0 because both b and d are > 0.

(∂P^*)/∂T = (∂(a - c - bT)/(d + b))/∂T

= -b/(d + b)

< 0 because both b and d are > 0.

Hence (∂Q^*)/∂T and (∂P^*)/∂T both are negative