Stadium Algebra
PW = 20 – W / 2000 {Gotta lower price to Wolverton fans
Δ PW = – ΔW / 2000 a bit to sell them another seat.}
TRW = PW W
Δ TRW = Output Effect +
Price Effect
+ {Revenue when you
sell another seat}
- {Reduced price on all the
seats you’ve already sold}
= (20 – W / 2000) ΔW – W (ΔW / 2000)
MRW = ΔTRW / ΔW = 20 – W / 2000 – W / 2000
MRW = 20 – 2 W / 2000
Want MRW = MC = 10 so 20 – W/1000 = 10 = PM
Solution: W = 10,000 ; PW = 15 ; TRW = 150,000
M = 30,000; PM = 10 ; TRM = 300,000
The Mantra: MR = MC
Total Revenue = Price x Quantity
TR = Px
Change in Total Revenue = TR
TR = P x + x P
TR = Output Effect + Price Effect
Output effect: sell more at same price
as before … but to sell more
Price effect: must lower price on all
the units you’re already selling
Poiuyt Algebra
P(x) = 6 – 3 x / 5000
TC(x) = 1000 + x + x2/5000
= 1000 + (1 + x/5000)x
ΔP = - 3 Δx / 5000
= Fixed + {Avg Variable}x
TR = Px = (6 – 3 x / 5000)x = FC + {AVC } x
ΔTC ={AVC}Δx + x Δ{AVC}
ΔTR = P Δx + x ΔP
=output effect + cost effect
=(6–3x/5000) Δx – 3x Δx/5000
MR = ΔTR / Δx
MR=6 – 3x/5000 – 3x/5000
MR = 6 – 6 x / 5000
Δ{AVC} = + Δx / 5000
ΔTC=(1+x/5000)Δx + x Δx/5000
ΔTC = (1 + 2x/5000)Δx
MC=ΔTC/Δx
MC = (1 + 2x / 5000)
Poiuyt Solution
Equate marginal revenue to marginal cost:
MR = 6 – 6x / 5000 = 1 + 2x / 5000 = MC
8 x / 5000 = 5
x = 25000 / 8 = 3125
P(3125) = 6 – (3/5000)3125 = 6 – 1.875
P = 4.125
TR = Px = (4.125)(3125) = 12,891
TC(3125) = 1000 + (1 + 3125/5000)(3125)
= 1000 + (1.625)(3125) = 1000 + 5078 = 6,078
Profit = TR – TC = 12891 – 6,078 = 6,813
Poiuyts and Qwerts: Substitute Goods
• A firm produces and sells poiuyts and querts (which
are left-handed poiuyts)
TC = 10xp + 20xq + 1000
• Sales of querts drive down the price of poiuyts
• Sales of poiuyts drive down the price of qwerts
Pp = 90 – xp/100 – xq/300
Pq = 120 – xq/100 – xp/150
When there is no capacity constraint, produce each
good to the point where its marginal profit = 0.
Note: MRp = Pp + xp {ΔPp / Δ xp} + xq {ΔPq / Δ xp}
MRq = Pq + xq {ΔPq / Δ xq} + xp {ΔPp / Δ xq}
where ΔPp/ Δxp, ΔPq/Δxp, ΔPq/Δxq, ΔPp/Δ xq are all negative.
The Mantra: MR = MC
Total Revenue = Price x Quantity
TR = Px
Change in Total Revenue = TR
TR = P x + x P
TR = Output Effect + Price Effect
Output effect: sell more at same price
as before … but to sell more
Price effect: must lower price on all
the units you’re already selling
Puzzle 3: Allocating Overhead
Equal allocation of overhead
Widgets
Gidgets Gadgets
Sales
Variable cost
120
70
160
90
70
55
Contribution
Overhead
50
40
70
40
15
40
Net profit
10
30
-25
Shut down gadgets ???
Does Gadgets’ overhead go away???
Puzzle 4: Export Freedonia
Steel?
PFreedonia = $ 680
ACFreedonia = $ 400
PWorld = $ 375