Assume a perfectly competitive market with MWTP(Q)=53–Q and MC(Q)=15+Q. What is the change is producer surplus resulting from a price floor at $46?

Round to two decimal places and do not include a currency symbol. If your answer is -$1.275, enter -1.28. Also, you may need to assume that the highest MWTP get the good or that the lowest MC supply the good.

Question

Assume a perfectly competitive market with MWTP(Q)=53–Q and MC(Q)=15+Q. What is the change is producer surplus resulting from a price floor at $46?

Round to two decimal places and do not include a currency symbol. If your answer is -$1.275, enter -1.28. Also, you may need to assume that the highest MWTP get the good or that the lowest MC supply the good.

BlackTom AI · Accepted Answer

We start by parsing the problem: in a perfectly competitive market, MWTP(Q) = 53 − Q and MC(Q) = 15 + Q. A price floor of 46 is imposed, and we are asked for the change in producer surplus.

First, find the market equilibrium without the price floor:
- Set MWTP = MC: 53 − Q = 15 + Q.
- This gives 2Q = 38, so Q* = 19.
- The equilibrium price is P* = MWTP(Q*) = 53 − 19 = 34 (also equal to MC(Q*) at Q*).

Second, determine the quantities under the price floor P_f = 46:
- If price is 46, the quantity supplied is where MC(Q) = 46: 15 + Q = 46 ⇒ Q_s = 31.
- The quantity demanded at price 46 is determined by MWTP = 46: 53 − Q_d = 46 ⇒ Q_d = 7.
- With a price floor above the market-clearing price, actual traded quantity is the smaller of supply and demand, so Q_traded = min(31, 7) = 7 units.

Now compute producer surplus before and after the floor.
- Producer surplus in the no-floor equilibrium: PS_0 is the area above the MC curve and below the price line P* = 34, from 0 up to Q* = 19. It can be computed as the integral of (P* − MC(Q)) from Q = 0 to Q = 19:
  PS_0 = ∫_0^{19} [34 − (15 + Q)] dQ = ∫_0^{19} (19 − Q) dQ = [19Q − Q^2/2]_0^{19} = (361 − 180.5) = 180.5.
- Producer surplus with the price floor: PS_floor is the area above the MC curve and below the price floor, from Q = 0 to Q_traded = 7, since only 7 units are sold in the market under the price floor. It is given by:
  PS_floor = ∫_0^{7} [46 − (15 + Q)] dQ = ∫_0^{7} (31 − Q) dQ = [31Q − Q^2/2]_0^{7} = (217 − 24.5) = 192.5.

Finally, the change in producer surplus due to the price floor is:
Change PS = PS_floor − PS_0 = 192.5 − 180.5 = 12.0.

Thus, the producer surplus increases by 12.00 (in the same monetary units as price and costs).

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