PAI 723 Economics for Public Decisions > Exercise 2

Solution

Part A

Question 1

At equilibrium, W2P=P and W2A=P, so W2P=W2A. Carrying out the algebra:

W2P = W2A

700 - 2*Q = 100 + 4*Q

600 = 6*Q

Q = 100

Substituting this into the W2P equation to find the equilibrium price:

P = W2P = 700 - 2*100 = $500

Checking the solution by inserting Q into the W2A equation to make sure that it is equal to $500 as well:

W2A = 100 + 4*100 = $500

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


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 W2P and W2A 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:

W2P at Q=50: 700 - 2*50 = $600

W2A 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 W2P at point A is $600, and the W2A at B is $300.

Part B

Question 1

The graph looks like this:


As before, the market equilibrium is where W2P=W2A. Since the supply curve is flat and W2A is always $20, the equilibrium P is easy to determine: it will be $20. The equilibrium quantity can be found from the demand curve:

W2P = 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-W2A 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 W2P at 1,600 and $20:

W2P 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 W2A 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 W2P is just equal to the price. For type 1, the algebra involved goes as follows:

W2Pi = 100 - 0.5*Qi

P = 100 - 0.5*Qi

0.5*Qi = 100 - P

Type 1 demand: Qi = 200 - 2*P

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

Type 2 demand: Qi = 100 - P

Question 2

The market demand is the sum of the quantities bought by all of the buyers. It's easiest to find the equation in two steps. For clarity, we can number the buyers 1 through 15, with buyers 1 to 5 being type 1 and buyers 6 to 15 being type 2. First, find the total demand for the 5 type-1 buyers by adding up their Q's; let Qt1t be the type-1 total demand:

Qt1t = Q1 + Q2 + ... + Q5

Qt1t = (200 - 2*P) + (200 - 2*P) + ... + (200 - 2*P)

Since all 5 buyers are identical, the sum can be written compactly as 5 times the individual demand:

Qt1t = 5(200 - 2*P)

Qt1t = 1,000 - 10*P

For type 2 buyers, the total will be Qt2t and the steps look like this:

Qt2t = Q6 + Q7 + Q8 + ... + Q15 (remember, those are the Q's for buyers 6 to 15)

Qt2t = (100-P) + (100-P) + ... + (100-P)

Qt2t = 10*(100-P)

Qt2t = 1,000-10*P

Combining the demands of the type 1 and type 2 buyers follows the same basic steps, Let Qd be the total quantity demanded:

Qd = Qt1t + Qt2t

Qd = (1,000 - 10*P) + (1,000 - 10*P)

Qd = 2,000 - 20*P

Question 3

Solving for the equilibrium price:

Qd = Qs

2,000 - 20*P = 20*P

2,000 = 40*P

P = $50

Finding the total quantity using the demand equation:

Qd = 2,000 - 20*50

Qd = 1,000

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

Qs = 20*$50

Qs = 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:

Qi = 200 -2*P

Qi = 200 - 2*$50 = 100

Consumer surplus, as usual, is the area under the W2P 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 W2P curve:

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

CS = $2,500

For type 2 buyers, the calculations are similar:

Qi = 100 - P

Qi = 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 01/31/2007