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Determining long run and short run equilibrium

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Q5. Ten competitive widget makers each have the following cost structure:
Ci = 0.005qi2 + 4qi + 200 i = 1, 2,....,10
(a) What is the short-run supply curve for each firm?
(b) What is the industry supply curve?
(c) If market demand is:
Q = 6,500 - 500p
What are short-run equilibrium market price and quantity?
(d) At the price determined in (c), is the firm breaking even, making profits or losses?
(e) In the long-run, firms enter the industry -assume constant costs over long-run-what will be market price and quantity?
(f) How many firms will there be in the industry?

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Solution:

(a) What is the short-run supply curve for each firm?

Ci = 0.005qi^2 + 4qi + 200
Marginal cost of each firm = MCi=dCi/dqi=0.01qi+4
In equilibrium MC = Price =p
pi = 0.01qi+4
0.01qi=-4+p
qi= -400+100p

(b) What is the industry supply curve?
There are ten firms in market
Total supply will be Qs=10*qi=(-400+100p)*10
Qs =-4000+1000p

(c) If market demand is:
Q = 6,500 - 500p
What are short-run equilibrium market price and quantity?

For equilibrium Qd=Qs
6500-500p=-4000+1000p
10500=1500p
p=7

Qd=6500-500*7=3000
Qs=-4000+1000*7=3000
Equilibrium Price =7
Equilibrium Quantity = 3000

(d) At the price determined in (c), is the firm breaking even, making profits or losses?

At equilibrium, output of each firm =-400+100*7=300
Total cost of each firm= 0.005qi^2 + 4qi + 200=0.005*300^2+4*300+200=1850
Total Revenue = output*price=300*7=2100
Total Revenue > Total cost, firms are making profit in short run.
Profit of each firm = 2100-1850=250

(e) In the long-run, firms enter the industry -assume constant costs over long-run-what will be market price and quantity?

Ci = 0.005qi^2 + 4qi + 200
Average Cost = TC/q = 0.005qi+4+(200/qi)
We will find minimum of average cost
d(ATC)/dq=0.005-200/qi^2
Put d(ATC)/dq=0
0.005 -(200/qi^2)=0
0.005=200/qi^2
qi^2=200/0.005=40000
qi=200

Minimum Average Total Cost = 0.005*200+4+(200/200)=6

Output of each firm = 200
Price = Minimum ATC =6
Quantity demanded = 6,500 - 500p=6500-500*6=3500

(f) How many firms will there be in the industry?

Number of firms = Total Demand/ output of each firm = 3500/200=17.5
There will be 18 firms.

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