Question

Assume the following market is a pure competitive and all firms are identical with the same costs functions: TC=100 +80*q+q^2 MC=80+2q The market demand is P=150-Q_d The equilibrium price in short run is $100. Note that Q is the market quantity and q is the quantity produced by a single firm. Calculate the output that minimizes average total cost (ATC). (2 points) What is the breakeven price for this firm in long run? How many firms will be in the long run? (3 points) Find the market supply equation 〖P = a + b*Q〗_S, where (a, b) are constant to be determined. And classify the supply as elastic, inelastic, etc. (3 points) What is the shutdown price for this firm in short run? (2 points) Assume a new technology causes the costs functions to change as follow: Fixed Cost: FC= $144 Average variable cost: AVC = 48 + q Marginal cost: MC = 48 + 2q Based on this information and holding everything else constant, how many firms will be in the lon

Assume the following market is a pure competitive and all firms are identical with the same costs functions:

                                                           TC=100 +80*q+q2

                                                           MC=80+2q

The market demand is                          P=150-Qd   

The equilibrium price in short run is $100.

Note that Q is the market quantity and q is the quantity produced by a single firm.

1. Calculate the output that minimizes average total cost (ATC). (2 points)
2. What is the breakeven price for this firm in long run? How many firms will be in the long run? (3 points)
3. Find the market supply equation P = a + b*QS, where (a, b) are constant to be determined. And classify the supply as elastic, inelastic, etc. (3 points)
4. What is the shutdown price for this firm in short run? (2 points)

Answer 1

To calculate the output level which minimizes ATC we need to calculate first the ATC and from there we can take first order derivation of ATC with respect to q.

TC = 100 + 80q + q^2

Therefore, ATC = TC/q = (100 + 80q + q^2)/q

Or, ATC = 100/q + 80 + q.

Now at the minimum point of ATC F.O.C with respect to q is 0.

Therefore, dATC/dq = d/dq(100/q + 80 + q)

Or , -100/q^2 + 1 = 0

Or, 100/q^2 = 1

Or, q^2 = 100

Or, q = 10 ( as q = -10 is not possible).

Therefore we can say for q = 10 , ATC is minimum. (Ans).

Now we need to calculate break even price for this firm in the long run.

We know in Break-Even point the firm's TR = TC. It means firm sets it's price equal to minimum of ATC.

At q = 10 , ATC = 100/q + 80 + q = 100/10 + 80 +10 = 100.

Now at q= 10, TC = ATC*q = 100*10 = 100.

Therefore we can say Break-Even price is 100. As at break even point P = Minimum ATC = $100. Break even price in the long run is $100. (Ans).

Now we will calculate how many firms was there in the long run.

Demand function is P = 150 - Q , if price is $100 then we have 100 = 150 - Q , or, Q = 150 - 100 = 50.

Therefore in the long run price is 100 and Q = 50.

If a single firm produce 10 units i.e if q = 10, then the number of firms in the long run is 50/10 = 5.

In the long run there will be 5 firms. (Ans).

Market supply curve can be determined by MC curve.

MC = 80 + 2q , this MC is already showing upward sloping for any level of q.

So, supply curve for a firm is P = 80 + 2q. Now if there is 5 firms in the industry then the supply curve is P = 80 + 2/5Q

Or, P = 80 + 0.4Q. This is the market supply curve.

Intercept is 80 because when q= 0 then Q = 5*q = 0. So when Q = 0 , P = $80.

Now we determine slope for a single firm slope is 2. But for 5 firms slope will be 2/5 = 0.4.

For single firm when q = 1 , P = $82 , when q = 2 , P = $84

Now for 5 firms there will be Q = 5q = 5*1 = 5 , when P = $82

When P = $84 , Q = 5*q = 5*2 = 10

P = 80 + 0.4Q is the market supply curve. (Ans).

Supply equation is P = 80 + 0.4Q

Or, 0.4Q = P - 80

Or, Q = 2.5P - 200

Now dQ/dP = 2.5

Now at P = 100 and Q = 50 elasticity is (dQ/dP)*(P/Q) = 2.5*100/50 =5.

In Equilibrium elasticity of supply is 5. Therefore we can say supply is elastic as elasticity is greater than 1. Supply is elastic (Ans).

For shut down price we need to calculate minimum of AVC.

TVC = 80q + q^2

AVC = TVC/q = (80q + q^2)/q = 80 + q

Now AVC is upward sloping i.e if q increases AVC increases

AVC is minimum when q= 0. At q = AVC = $80.

Therefore shutdown price is $80 in short run. (Ans).

With new technology when FC = $144

AVC = 48 + q , MC = 48 + 2q

From AVC we get TVC. TVC = AVC*q = (48 + q)*q = 48q + q^2

TC = 144 + 48q + q^2

ATC =TC/q = 144/q + 48 + q

Now , dATC/dq = -144/q^2 + 1 = 0

Or, - 144/q^2 = - 1

Or, q^2 = 144

Or, q = 12.

Now everything else constant there will be 50/12 = 4.5 firms