PAI 723 Economics for Public Decisions > Exercise 2

Solution

Part A

Question 1

At the equilibrium, WTP = P and WTA = P, so WTP and WTA are equal to each other.  Since the market WTP and WTA were given (and there are no taxes or subsidies in the problem), the easiest way to solve the problem is to find the quantity where WTP = WTA. Carrying out the algebra:

WTP = WTA

700 - 2*Q = 100 + 4*Q

600 = 6*Q

Q = 100

Substituting this into the WTP equation to find the equilibrium price:

P = WTP = 700 - 2*100 = $500

Checking the solution by inserting Q into the WTA equation to make sure that it is equal to $500 as well:

WTA = 100 + 4*100 = $500

Thus, the equilibrium is P=$500, Q=100. The graph is shown below:

An alternative way to solve the problem is to construct the market demand and supply curves and then solve for the price that makes `Q^d` = `Q^s`. On the demand side, the algebra for that approach goes as follows:

`WTP = 700 - 2*Q^d`

`WTP = P`

`P = 700 - 2*Q^d`

`Q^d = 350 - 0.5*P`

On the supply side it looks like this:

`WTA = 100 + 4*Q^s`

`WTA = P`

`P = 100 + 4*Q^s`

`Q^s = 0.25*P - 25`

Finding the equilibrium price:

`Q^d = Q^s`

`350 - 0.5*P = 0.25*P - 25`

`375 = 0.75*P`

`P = 500`

`Q^d = 350 - 0.5*500 = 100`

`Q^s = 0.25*500 - 25 = 100`

Thus, the two approaches are equivalent.

Question 2

The surplus calculations are as follows:

CS = 0.5*100*($700-$500) = $10,000

PS = 0.5*100*($500-$100) = $20,000

SS = CS + PS = $30,000

Question 3

The regulation reduces the number of trades from 100 down to 50. The lost social surplus will be the gains from trade that would have been obtained on units beyond Q=50. The area can be found using the same technique as in Exercise 1: find the intersections of the WTP and WTA curves with the Q=50 line, use the difference in the y-coordinates as the base of the triangle, and use 100-50=50 as the height of the triangle. Carrying out the calculations:

WTP at Q=50: 700 - 2*50 = $600

WTA at Q=50: 100 + 4*50 = $300

Area of lost SS: 0.5*($600-$300)*50 = $7,500

Question 4

The diagram is shown below:

Point C is the original equilibrium, with P=$500 and Q=100. The coordinates of points A and B were calculated above: the WTP at point A is $600, and the WTA at B is $300.

Part B

Question 1

The graph looks like this:

As before, the market equilibrium is where WTP=WTA. Since the supply curve is flat and WTA is always $20, the equilibrium P is easy to determine: it will be $20. The equilibrium quantity can be found from the demand curve:

WTP = P

120 - 0.05*Q = $20

100 = 0.05*Q

Q = 2,000

Consumer surplus is straightforward:

CS = 0.5*2,000*($120-$20) = $100,000

There isn't any producer surplus at all in this case. For each one of the 2000 units purchased by consumers, producers are willing to accept $20 and receive exactly $20. Since P-WTA is 0, they don't get any producer surplus. Social surplus is thus equal to CS.

Question 2

The calculation is much like that from part A3 except easier since there isn't any PS. A 20% reduction in total trades would mean that Q would be 1,600 instead of 2,000. The lost CS is a triangle with base equal to 2,000-1,600 = 400 and with height given by the difference between WTP at 1,600 and $20:

WTP at 1,600: 120 - 0.05*1,600 = $40

Lost CS = 0.5*(2,000-1,600)*($40-$20) = $4,000

All of the lost surplus is CS; there's no loss of PS because there wasn't any PS initially. The producers are selling fewer units, but they are also saving money (their WTA per unit) so they are really hurt by the regulation. The situation is illustrated below:

Part C

Question 1

Each type of buyer will purchase the good until their WTP is just equal to the price. For type 1, the algebra involved goes as follows, where `Q_i^1` is the quantity demanded by type-1 person i:

`WTP_i^1 = 100 - 0.5*Q_i^1`

`P = 100 - 0.5*Q_i^1`

`0.5*Q_i^1 = 100 - P`

Type 1 demand: `Q_i^1 = 200 - 2*P`

The process is similar, but a bit easier, for type 2. The resulting demand equation for type-2 person i is:

Type 2 demand: `Q_i^2 = 100 - P`

Question 2

The market demand is the sum of the quantities bought by all of the buyers. Since there are 5 identical type-1 buyers and 10 identical type-2 buyers, the sum can be written as:

`Q_m = 5*Q_i^1 + 10*Q_i^2`

Inserting the demands from above:

`Q_m = 5*(200-2*P) + 10*(100-P)`

`Q_m = 2000 - 20*P`

If that isn't completely clear, here's another way to think about it. Writing out the summation in detail looks like this:

`Q_m = Q_1^1 + Q_2^1 + ... + Q_5^1 + Q_1^2 + Q_2^2 + ... + Q_10^2`

Inserting the equations above for `Q_i^1` and `Q_i^2` gives:

`Q_m = (200-2*P) + (200-2*P) + ... + (200-2*P) + (100-P) + (100-P) + ... + (100-P)`

Collecting similar terms together gives:

`Q_m = 5*(200-2*P) + 10*(100-P)`

`Q_m = 2000 - 20*P`

Question 3

Solving for the equilibrium price:

`Q^d = Q^s`

`2,000 - 20*P = 20*P`

`2,000 = 40*P`

`P = $50`

Finding the total quantity using the demand equation:

`Q^d = 2,000 - 20*50`

`Q^d = 1,000`

The equilibrium is thus P=$50, Q=1,000. As a check, it's good to compute `Q^s` to make sure that it's 1000 as well:

`Q^s = 20*$50`

`Q^s = 1,000`

Question 4

Now that we know the equilibrium price, we can find out what happens to each buyer by going back to their individual demand and willingness to pay curves. For type 1:

`Q_i^1 = 200 -2*P`

`Q_i^1 = 200 - 2*$50 = 100`

Consumer surplus, as usual, is the area under the WTP curve and above the price line. Note that the first 100 in the equation below is the quantity, and the second 100 is the y-intercept of the WTP curve:

CS = 0.5*100*($100 - $50)

CS = $2,500

For type 2 buyers, the calculations are similar:

`Q_i^2 = 100 - P`

`Q_i^2 = 100 - 50 = 50`

CS = 0.5*50*($100-$50)

CS = $1,250

Total consumer surplus is just the sum of the CS received by each of the buyers:

CS = 5*$2,500 + 10*$1,250

CS = $25,000

Question 5

Producer and total social surplus are straightforward:

PS = 0.5*1,000*$50 = $25,000

SS = CS + PS = $50,000

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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 09/12/2018