Question

1. Let the production function beq Kili (K: capital, L: labor), the unit prices of capital and labor be both $1. (a) Find the cost function. 2b (b) If the firm with this production function is in a competitive market an many units of products should the firm produce? d the market price is S12, how 2. (a) Let the cost function be C(g)-F+mq G Is there economies of scale? (ii) If market price is equal to marginal cost, can the firm make profit? (b) Let the cost function be C(qi, q)-F+itqa (i) Is there economies of scope? (ii) If market prices are equal to marginal costs, can the firm make profit? 1+- 3. Consider a market with 2 identical firms. Each firm has cost function c(q)-q. The market demand is p-100-q. (i) Find the market equilibrium, price elasticity of demand at the equilibrium, and each firm's profit if (i) Find the arkot quilbrium and.gch fim's polit if the two firms choose output to maximize their joint profit. I(こ言に「-25. T'=aL (iii) What is the percentage change in the profit of each firm. from (i) to (i)? 4.0% 4. (a) There is only one firm in a market. This firm has 100 plants, each produces according to cost
these three questions 1、2and 3

Answer 1

1. a)The cost function is given as

TC = wL+ rK where w= unit labor price and r = unit capital price, L= number of labor, K= number of capital

The production function is given as

q= K^1/3 L^1/3

Taking partial derivative of production function with respect to L and K respectively we get,

Marginal product of capital =dq/dK = 1/3L^1/3 /K^2/3 and

Marginal product of labor= dq/dL= 1/3K^1/3 / L^2/3    

Under cost minimization condition,

Marginal product of labor/marginal product of capital = w/r

or, 1/3K^1/3 / L^2/3/1/3L^1/3 /K^2/3 = 1/1=1

or, K/L=1 or, L=K

Therefore, the cost function is

TC= wL+rK= L+K= 2K (Since w=1 and r=1)

b) Again putting L=k in the production function we get

q= K^2/3

or, K= q^3/2

Therefore, substituting value of K in total cost function we get,

TC= 2K= 2.q^3/2 .......(i)

and marginal cost is derived from taking partial derivative of cost function (i) with respect to q,

MC= 2*3/2q^1/2

In a competitive market condition,

              Marginal revenue = marginal cost = Price

or, 3q^1/2 = 12 or, q=16

2. i) The cost function is given as

C(q) = F +mq

Here F= fixed cost and m= variable cost that increases with the increases the output quantity. Therefore, the given cost function represent economies of scale.

ii) The cost function is

C(q) = F+mq

Therefore, marginal cost = d(C(q))/dq= m

By the given condition, marginal cost = Price(P)

i.e., P =m

Therefore, total revenue of the firm = mq

and profit($\pi$) = Total revenue-Total cost = mq-F-mq=-F which means the firm is incurring loss or no profit.

iii) Economies of scope is reduction in average cost of production of a firm by producing two or more distinct products with same inputs.

The production function is

C(q₁ , q₂ ) = F+q₁² + q₂² which represents cost function two distinct products q₁ and q₂ .

Therefore, the above function represents economies of scope.

iv) Marginal cost of product q₁ is

d(C(q₁ , q₂ ))/dq₁ = 2q₁ and

Marginal cost of product q₂ is

d(C(q₁ , q₂ ))/dq₂ = 2q₂

Therefore total revenue of the firm = 2q₁ .q₁ + 2q₂ .q₂ = 2(q₁² + q₂² )

Therefore, profit ($\pi$) = Total revenue- Total cost = 2(q₁² + q₂² )-( F+q₁² + q₂² )=q₁² + q₂² -F

3. The given question is Cournot duopoly example.

The cost function of each firm is

c(q) = q²

Let output of one firm =q₁ and output of second firm = q₂ so that q₁ + q₂ =q

If each firm separately produce output, then total cost of first firm is

c(q₁ ) = q₁² and marginal cost of first firm is

d (c(q₁ ))/dq₁ =2q₁

Similarly,

marginal cost of second firm is

d(c(q₂ ))/dq₂ = 2q₂

Again market demand function is given as

p =100-q = 100-(q₁ +q₂ )..........(ii)

From equation (ii), the total revenue of firm 1 is,

pq₁ =100q₁ - q₁² -q₁ q₂ and

total revenue of second firm is

pq₂ =100q₂ - q₂² -q₁ q₂

Under cournot nash equilibrium condition ,

d$\pi$₁/dq₁ = Marginal revenue of first firm -Marginal cost of first firm= 0

or, 100-2q₁ -q₂ - 2q₁ = 0

or, 100-4q₁ -q₂ = 0

or, 4q₁ +q₂ =100......(iii)

Similarly

d$\pi$₂/dq₂ =100-2q₂ -q₁ - 2q₂ = 0

or, 4q₂ +q₁ =100. .......(iv)

Solving (iii) & (iv), we get, q₁ = q₂ = 20 and p =100-(q₁ +q₂ )= 100-40=60

Again equilibrium market condition of competitive market is

Marginal revenue = Marginal cost

or, 100-2q= 2q or, q=25 and p=100-q=100-25=75

Therefore, change in price over two different equilibrium conditions= 25-20= 5

and change in quantity demanded over two different equilibrium conditions = 75-60 = 15

Therefore, price elasticity of demand = Change in quantity demanded / change in the price = 5/15= 1/3

Profits of first firm ($\pi$₁ ) = pq₁ - q₁² = 20*60-(20)² = 800

and profit of second firm ($\pi$₂ ) = 20*60-(20)² =800

ii) If the two firms collectively produce output q, then from part (i),

price (p) =75 and quantity (q) =25

Therefore, total revenue of both firms = pq = 75*25= 1875

joint profit of both firms = total revenue - total cost = 1875-(25)² = 1250 which will be equally divided by both firms.

i.e., $\pi$₁ = $\pi$₂ = 1250/2= 625

iii) percentage change in profit of each firm = (800-625)/800*100= 21.87%