Question

have completely no idea for part a

3. Consider a firm that produces a single output using four input factors. The production function is given by: f(21, 22, 23, 2a) = min{x1, x2} + min{X3, 24} (a) (8 marks) Find the vector of conditional factor demands to produce one unit of output when the prices for the input factors are given by w = (1, 2, 4, 6). (b) (2 marks) Compute the cost function.

Answer 1

Solution:

Given the production function as: f(x1, x2, x3, x4) = min{x1, x2} + min{x3, x4}

w = (1, 2, 4, 6)

a) We solve this rather logically than economic calculations: With such production function, following are points to be noted:

x1 and x2 are complement factors, and x3 and x4 are complements to each other.

x1 (or x2) and x3 (or x4) are substitutes (so, replaceable) to each other.

Then, it should be that x1 = x2, and x3 = x4.

Further, being substitutes, the factor demanded should be the one which carries a lower cost.

In other words, we can now have the production function in terms of two factors: f(x1, x2, x3, x4) = min{x1, x2} + min{x3, x4}

f(x1, x3) = min{x1, x1} + min{x3, x3}

f(x1, x3) = x1 + x3

Then, marginal product of x1 = df(x1, x3)/dx1 = 1, and marginal product of x3 = df(x1, x3)/dx3 = 1

And cost of x1 = $1, while cost of x3 = $4

Then, MPx1/w1 = 1/1 = 1 > 1/4 = MPx3/w3

So, x1 would be used (think in general, x1 and x3 give out same return, and x1 costs lower than x3, so x1 would be preferred, indeed).

Then, for 1 unit of output, that is for f(x1, x2, x3, x4) = 1

1 = min{x1, x2} + min{x3, x4}

As x3 (or x4) will never be chosen for efficiency and cost minimization, 1 = min{x1, x2}

And, with x1 = x2, 1 = min{x1, x1} = x1

Then, we have x1 = 1

Vector for conditional factor demands is: (x1, x2, x3, x4) = (1, 1, 0, 0)

b) Total Cost = w1*x1 + w2*x2 + w3*x3 + w4*x4

With above vector, for 1 unit of output, total cost incurred = 1*1 + 2*1 + 4*0 + 6*0 = $3

(You may check for any other combination for production of one unit of output, it will incur a higher cost)

Total cost function can be derived as:

We already saw that given the production function and wage rates, x3 = x4 = 0 (always), and x1 = x2 = f(x1, x2, x3, x4)

Let's have number of units produced (or output) be denoted by y, that is y = f(x1, x2, x3, x4) as we generally assume.

So, x1 = x2 = y

Total cost = 1*y + 2*y + 4*0 + 6*0 = 3y

Required cost function C(y) = 3y

(Note: cost function is always denoted in terms of output, using the inputs, so we derive it in terms of y).