Question 5 (20 marks) (a) i) A firm has production function y - f(x,z) -ax+Bz, for...

Solution:

Returns to scale depicts how the output or scale of production changes as all inputs are changed by the same proportion. If the output changes by more (less) than the common factor, we say the production function exhibits increasing (decreasing) returns to scale. If it changes by the same factor, we say the function exhibits constant returns to scale. Denoting alpha by a and beta by b for ease of writing.

(a) i) Production function: y = f(x,z) =  ax + bz, y >= 0

If the inputs x and z are increased by the common factor t,

y' = f(tx, tz) = a(tx) + b(tz)

y' = atx + btz = t(ax + bz) = t*f(x,z)

So, we have y' = ty or f(tx,tz) = tf(x,z) implying that output also increases by same factor, t. Thus, we can conclude that the returns to scale is constant.

ii) Production function: f(x1,x2) = x1^a*x2^a

Increasing the inputs x1 and x2 by a common factor t gives us:

f(tx1,tx2) = (tx1)^a*(tx2)^a = t^a*t^a(x1^ax*2^a)

f(tx1,tx2) = t^2a(x1^a*x2^a) = t^2a*f(x1,x2) (using the property of exponents: A^X*A^Y = A^X+Y)

For the returns to scale in production to be increasing, we know that we require 2a > 1 (so that output increases by more than the increase in both inputs).

So, required range of 'a' is: a > 1/2 or a belongs to (1/2, infinity).

iii) y = f(x,z) = (xz)/(x+z), for y >= 0

y' = f(tx,tz) = (tx)(tz)/(tx + tz) = t²xz/(t(x+z))

y' = t*[xz/(x+z)] = t*f(x,z)

So, again we have y' = ty implying that the output increases by same factor as the inputs, exhibiting constant returns to scale.

b) i) For a competitive firm, total cost function is: c(y) = 2 + (y²/3)

Under perfect competition, a firm sells at market price, P which equals the marginal cost, MC, of the firm.

So, given the cost function as above, MC =

MC = 2*y/3

So, for 30 units of output, that is y = 30, market price, P = MC(30) = 2*30/3 = $20.

ii) c(0) = 0

c(y) = 16 + 2y², y > 0

In the long run, a firm produces positive profit from where it earns a normal profit. Normal profit in economics is the 0 accounting profit, which is achieved where total revenue, TR = total cost, TC. (this is because, in short run when a firm earns positive accounting profit, more and more firms get attracted and enter the market, till the price falls due to too much supply, and finally every firm earns a 0 profit in long run, under perfect competition).

Total revenue = price*output

So, price = TR/y

Further, since we want profits = 0, so TR = TC

Dividing both sides by output, y: TR/y = TC/y

Price = Average total cost: this condition is also known as minimum efficient scale, at which a firm starts production in long run.

So, this is the minimum price at which firm is prepared to produce a positive output.

ATC = c(y)/y = 16/y + 2y

So, minimum price = (16/y + 2y)

Note that the condition of positive output level from shutdown point: Average variable cost, AVC = MC(=price) is the short run condition, while in the question, we are asked for a long run condition).