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Perfect Rivalry At Symmetry Level

  Perfect Rivalry at Symmetry Level

Illustration 69

Presume the following demand and supply functions of a product are provided which is being manufactured under conditions of perfect rivalry. Determine the symmetry price and quantity.

                                    VD      =          2000 – 100P

                                    VS       =          500 + 50P

Solution

There are two different methods of solving for symmetry price and volume. First, we can determine the symmetry price and volume by using the symmetry condition like, VD = VS. Next, we can procure symmetry price and volume by using the qualitative outcomes of the demand and supply model, P = A – C and V = AD + BC
                              B + D                B + D

(1) As in symmetry,                            VD      =          VS.

                                                2000 – 100P    =          500 +50P

                                                            1500    =          150P

                                                            P          =          1500 / 150       =          10

Now, substituting the value of P in the Demand Equation,

                                                            VD      =          2000 – 100*10

                                                            VD      =          1000

Substitute Method

                                    P          =          A – C
                                                             B + D

A = 2000, B = 100, C = 500 and D = 50

                                    P          =          2000 – 500
                                                              100 + 50

                                                =          1500    =          10
                                                             150

                                    VD      =          AD + BC
                                                               B + D

                                                =          2000 * 50 + 100 * 500
                                                                        100 + 50

                                                =          100000 + 50000
                                                                        150

                                                =          150000            =          1000
                                                                 150

Therefore, both methods capitulates the same outcome.

Illustration 70

In the event if an industry is functioning under perfectly rivalled conditions in the market. It faces the following revenue and cost conditions:

                        TR       =          80V

                        TC       =          48 + 56V + V^2

Compute the symmetry level of productivity using both the first order and second order conditions of symmetry. Determine total profits made.

Solution

Profits are optimised when the industry equates marginal cost with MR and marginal cost is rising. Therefore, in order to procure the symmetry output we equate MC = MR.

                                    TR       =          80V

                                    MR      =          dTR     =          80
                                                             dV

                                    MC      =          dTC     =          56 + 2V
                                                             dV

At symmetry,
                                    MC      =          MR

                        56 + 2V           =          80

                                    2V       =          80 – 56

                                    V         =          24 / 2   =          12

Total Profits π                         =          TR – TC

                                                =          80V - 48 - 56V - V^2

                                                =          24V – 48 – V^2

Substituting the value of V as 12 in the above,

                                                =          24*12 – 48 – (12) ^2

                                                =          288 – 48 – 144

                                                =          96

Note that in order to ensure for the fulfilment of second order condition, we have to test whether MC is enhancing. For this we consider the derivative of MC i.e. second derivative of TC

Therefore,                    MC      =          dTC     =          56 + 2V
                                                             dV

                                                =          d^2TC =          + 2
                                                             dV^2

The positive symbol of the second derivative of TC entails that MC is enhancing.

Illustration 71

In a district there are a large number of industries selling a commodity and no single industry has any control over the price of commodity. The following total revenue and cost functions are provided for a single seller.

                                    TR       =          50V

                                    TC       =          5000 + 10V + 0.1V^2

Compute how many units of the commodity an industry will manufacture per annum if it aims at profit optimisation. Also determine the total profits made by it in the symmetry situation.

Solution

We ascertain MR and MC from the provided revenue and cost functions. Therefore,

                                    TR       =          50V

                                    MR      =          dTR     =          50
                                                             dV

note that since MR is invariable, price will be equal to it.

                                    TC       =          5000 + 10V + 0.1V^2

                                    MC      =          dTC     =          10 + 0.2V
                                                             dV

for profit optimisation,

                                    MC      =          MR

                        10 + 0.2V        =          50

                                    V         =          40 / 0.2

                                    V         =          200

Profits
                                    π          =          TR – TC

                                    TR       =          P.V

                                                =          50 * 200          =          10000

                                    TC       =          5000 + 10V + 0.1V^2

Substituting the value of V in the above, we get,

                                    TC       =          5000 + 10 * 200 + 0.1(200) ^2

                                                =          5000 + 2000 + 4000

                                                =          $11,000

Illustration 72

A rivalled industry has the following data

Productivity

Total Fixed Cost

Total Variable Cost

0

150

0

1

150

45

2

150

95

3

150

185

4

150

245

5

150

395

  1. If price = $50m how many units will the industry manufacture?
  2. What will be the level of profits or losses at this level of manufacture?
  3. Will the industry functions in short run?
  4. What happens in the long run?

Solution

  1. In order to procure the number of units of productivity, the rivalled industry will manufacture provided where ΔV = 1 or in other words, MC = TVCn – TVCn-1. We compute below MC at diverse levels of productivity.

Productivity

Total Variable Cost

Marginal Cost
TVCn – TVCn-1

0

0

-

1

45

45

2

95

50

3

185

60

4

245

60

5

395

150

In order to be in symmetry a perfectly rivalled industry will manufacture the number of units of commodity at which price = MC.

It will be seen from the above tablet that price of $50 equals marginal cost when it is manufacturing 2 units of productivity.

  1. With 2 units of productivity, the total cost = TVC + TFC = 95 + 150 = 245. Total revenue TR earned by manufacturing 2 units of productivity will be P * V = 50 * 2 = 100. As the total cost $245 surpass total revenue $100, the industry will be incurring losses equal to $245 - $100 = $145 through it will be minimising losses.
  1. As at 2 units of productivity, total revenue of $100 surpasses the total variable cost (95), the industry will carry on functioning in the short run.
  1. In the long run, some industries will get away the firm that is the basis for increasing the price of commodity. The industries that stay in the firm must earn zero economic profits i.e. for them price must be equal to the long run average cost.

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