Deadweight loss and consumption externalities

Question

here's the problem and my solution so far:

In this case, the Marginal Private Benefit (MPB) is given by the equation MPB = 100 - 4q and the Marginal Social Cost (MSC) is given by the equation MSC = 3q.

The market equilibrium occurs where MPB equals MSC, so we can set the two equations equal to each other and solve for q:

100 - 4q = 3q => 7q = 100 => q = 100 / 7 ≈ 14.29

However, there is a positive consumption externality of $40 per unit consumed. This means that the Marginal Social Benefit (MSB) is actually MPB + Externality = (100 - 4q) + 40 = 140 - 4q.

The socially optimal quantity occurs where MSB equals MSC, so we can set these two equations equal to each other and solve for q:

140 - 4q = 3q => 7q = 140 => q = 140 / 7 = 20

The Deadweight Loss (DWL) is the area of the triangle formed by the quantity difference (20 - 14.29) and the price difference ((140 - 420) - (100 - 414.29)). So,

DWL = 0.5 * (20 - 14.29) * ((140 - 420) - (100 - 414.29)) => DWL ≈ \$19.60

So, the deadweight loss that results in the market equilibrium is approximately \$19.60. '

I'm not too sure why this is incorrect. Can anyone give me a hand. Any help will be greatly appreciated.

Answer 1

The DWL triangle has (horizontally) height $20-100/7$ and (vertically) base $40$ (the marginal external benefit). So we have

$$DWL=\frac{1}{2}\left(20-\frac{100}{7}\right)(40)=\frac{1}{2}\frac{40}{7}(40)=\frac{80}{7}\approx 11.43$$

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In your calculation, you found the base of the triangle by finding:

1. $MSB$ at $q=20$: $140-4(20)$
2. $MPB$ at $q=100/7\approx 14.29$: $100-4(14.29)$

and then subtracting 2 from 1. You should have found the base of the triangle by finding:

1. $MSB$ at $q=100/7$: $140-4(100/7)$
2. $MPB$ at $q=100/7$: $100-4(100/7)$

and then subtracting 2 from 1. This just gives $40$, the marginal external benefit of consumption.