Econ 101, Fall 2006
Dr. Schroeter
Homework #3: solution outline
PartI. Tradable pollution permits
- With price at a level above $100/permit, each firm would find it cheaper to completely abate its current pollution level than to pay for permission to pollute. As the price falls below $100/permit, firm A will demand 80 permits because purchasing a permit for each unit of pollution is now cheaper than abatement. As the price falls below $75/permit, firm B will add its demand for 100 permits, bringing aggregate demand to 180. As the price falls below $50/permit, firm C will add its demand for 120 permits, bringing aggregate demand to 300. The resulting demand curve is depicted in Figure 1.
- The supply of permits, perfectly inelastic at a quantity of 150, is shown in Figure 1. The equilibrium price, determined by the supply and demand intersection, is $75/permit.
Firm A values each permit at $100 (= firm A's marginal abatement cost). The
price is $75/permit. So firm A will demand 80 permits to allow it to maintain its
current pollution level. This means that firm A will buy 30 additional permits.
Figure 1. Supply and demand for tradable pollution permits
Firm C values each permit at $50 (= firm C's marginal abatement cost). The price is $75/permit. So firm C will not demand any permits. Because it was given 50 permits to start, it will sell these because the market price of permits is greater than the value that firm C places on each permit. It will sell 30 permits to firm A.
Firm B values each permit at $75 (= firm B's marginal abatement cost). Because the market price is exactly equal to the value that firm B places on a permit, firm B will be indifferent as to the number of permits it uses. (Purchasing an extra permit costs $75. The alternative is to abate 1 unit of pollution at the margin, and this also involves a cost of $75.) In order to clear the market, firm B will buy 20 permits (from firm C) because C needs to sell 20 permits (after it sells 30 permits to A).
- The total cost of reducing pollution to 150 units/year under the government's tradable pollution permit policy:
Firm A uses 80 permits and engages in no pollution abatement. Cost = $0/year.
Firm C uses no permits and therefore must completely abate its initial pollution level of 120 units/year. Cost = 120 x 50 = $6000/year.
Firm B uses 70 permits which means that it must abate 30 units/year of its initial pollution level of 100 units/year. Cost = 30 x 75 = $2250/year.
Total cost = 0 + 6000 + 2250 = $8250/year.
- The total cost of reducing pollution to 150 units/year if the government simply issues 50 non-tradable pollution permits to each firm:
Firm A uses 50 permits which means that it must abate 30 units/year of its initial pollution level of 80 units/year. Cost = 30 x 100 = $3000/year.
Firm B uses 50 permits which means that it must abate 50 units/year of its initial pollution level of 100 units/year. Cost = 50 x 75 = $3750/year.
Firm C uses 50 permits which means that it must abate 70 units/year of its initial pollution level of 120 units/year. Cost = 70 x 50 = $3500/year.
Total cost = 3000 + 3750 + 3500 = $10250/year.
Part II. The cost curves of a firm
Table 1. Cost of producing widgets.
Labor(workers/day) / Output (widgets/day) / FC
($/day) / VC
($/day) / TC
($/day) / AFC
($/widget) / AVC
($/widget) / ATC
($/widget) / MC
($/widget)
0 / 0 / 400 / 0 / 400
1 / 50 / 400 / 150 / 550 / 8.00 / 3.00 / 11.00 / 3.00
2 / 96 / 400 / 300 / 700 / 4.17 / 3.13 / 7.29 / 3.26
3 / 138 / 400 / 450 / 850 / 2.90 / 3.26 / 6.16 / 3.57
4 / 176 / 400 / 600 / 1000 / 2.27 / 3.41 / 5.68 / 3.95
5 / 210 / 400 / 750 / 1150 / 1.90 / 3.57 / 5.48 / 4.41
6 / 240 / 400 / 900 / 1300 / 1.67 / 3.75 / 5.42 / 5.00
7 / 266 / 400 / 1050 / 1450 / 1.50 / 3.95 / 5.45 / 5.77
8 / 288 / 400 / 1200 / 1600 / 1.39 / 4.17 / 5.56 / 6.82
9 / 306 / 400 / 1350 / 1750 / 1.31 / 4.41 / 5.72 / 8.33
10 / 320 / 400 / 1500 / 1900 / 1.25 / 4.69 / 5.94 / 10.71
Note: The last column gives marginal cost estimates for output ranges 0 to 50, 50 to 96, 96 to 138, etc.
Figure 2. Average fixed cost, average variable cost, average total cost, and marginal cost of producing widgets.
Note: In graphing the MC curve, each value of marginal cost was plotted against the quantity given by the midpoint of the quantity range to which the marginal cost value applies. For example, in Table 1 the first entry for marginal cost is 3.00. This figure is associated, neither with 0 units of output nor with 50 units of output, but with the changein output from 0 to 50 units. The marginal cost figure of 3.00 was plotted against the midpoint of this range, 25 (=(0+50)/2). This technique gives a more accurate representation of the MC curve.
Yes, the MC and ATC curves exhibit the expected relationship. Namely, they satisfy the "average-marginal rules." For example, where MC < ATC (as it is at an output of 50 widgets/day, for example), ATC is decreasing with respect to output (the ATC curve slopes down). Also, where MC > ATC (at 288 widgets/day, for example), ATC is increasing with respect to output (the ATC curve slopes up). Finally, MC cuts ATC at the lowest point of ATC.