Calculating and interpreting welfare loss

Overview

Welfare loss triangle vanish at the socially optimum quantity.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   In a market, demand is P = 30 - Q and supply is P = 10 + Q, with P in £ and Q in thousands of units per week. Calculate the equilibrium price and quantity, consumer surplus, producer surplus and community surplus.

   Model answer

   Set 30 - Q = 10 + Q, so 20 = 2Q and Q = 10 thousand units. Substituting into supply, P = 10 + 10 = £20. The demand curve meets the vertical axis at £30, so consumer surplus = 0.5 × 10 × (£30 - £20) = £50 thousand. The supply curve meets the vertical axis at £10, so producer surplus = 0.5 × 10 × (£20 - £10) = £50 thousand. Community surplus = £50 thousand + £50 thousand = £100 thousand per week.

2. 2

   A market has demand P = 30 - Q and supply P = 10 + Q, with P in £ and Q in thousands of units per week, and production carries an external cost of £6 per unit. Find the socially optimum quantity and the value placed on the last unit there.

   Model answer

   Marginal social cost is marginal private cost plus the external cost, so P = 10 + Q + 6, that is P = 16 + Q. The optimum is where marginal social benefit equals marginal social cost: 30 - Q = 16 + Q, so 14 = 2Q and Q* = 7 thousand units. Reading the demand curve at that quantity gives P = 30 - 7 = £23, which is the value society places on the last unit produced at the optimum.

3. 3

   A market has demand P = 30 - Q and supply P = 10 + Q, with P in £ and Q in thousands of units per week, and an external cost of production of £6 per unit. Calculate the welfare loss at the free market quantity.

   Model answer

   The free market produces where 30 - Q = 10 + Q, giving Q = 10 thousand units. Marginal social cost is P = 16 + Q, so the optimum is where 30 - Q = 16 + Q, giving Q* = 7 thousand units. At the market quantity marginal social benefit is 30 - 10 = £20 while marginal social cost is 16 + 10 = £26, a vertical gap of £6. The triangle has base 10 - 7 = 3 thousand units and height £6, so welfare loss = 0.5 × 3 × £6 = £9 thousand per week.

4. 4

   Why does the welfare loss triangle vanish at the socially optimum quantity?

   Model answer

   Because at the optimum marginal social benefit equals marginal social cost, so the two curves that form the sides of the triangle intersect exactly there. The triangle is generated by the divergence between those curves over the range between the optimum and the actual quantity; if the actual quantity is the optimum, that range has zero width and the area is zero. Community surplus measured against the social curves is then at its maximum, which is what allocative efficiency means.

5. 5

   Once an externality exists, why is community surplus no longer maximised at the market equilibrium?

   Model answer

   Because the demand and supply curves that generate the market equilibrium no longer measure social benefit and social cost. The relevant total is the surplus measured against the marginal social benefit and marginal social cost curves, and that total peaks where those two curves cross, not where the private ones cross. The market quantity therefore sits on the wrong side of the peak, and the difference between the maximum attainable surplus and the surplus actually achieved is the welfare loss.

6. 6

   How can you tell from a diagram alone whether a market over-produces or under-produces?

   Model answer

   Compare the intersection of the private curves with the intersection of the social curves. If the social intersection lies to the left of the private one the market quantity exceeds the optimum, so the market over-produces or over-consumes; this happens when marginal social cost lies above marginal private cost, or marginal social benefit lies below marginal private benefit. If the social intersection lies to the right, the market falls short of the optimum. The direction of the vertical gap between the private and social curves therefore fixes the direction of the error.

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