Question

1. You have decided to go into the goat feed business. You’ve been mixing it for your own goats and now neighbors and friends want the same mix. So you need to sit down and do a little economics to figure out whether this is a profit opportunity or not. The production function for your special mix of feed is:

                                    Y = X₁^1/2 X₂^1/4

Where Y = lbs of feed mix

X₁ = lbs. of oats

X₂ = lbs. of secret mineral/vitamin mixture

P_y   = $80 per lb.

Px₁ = $4 per lb.

Px₂ = $3 per lb.

1. You are thinking of starting by producing about 100 lbs of feed this month. What is the least-cost level of the two inputs to produce this amount?

2. Using the information from the Least-Cost problem above, continue your analysis and determine what level of inputs would maximize profits?

Answer 1

Solution:

We are given the following:

Production function: Y = X1^1/2*X2^1/4

The price of output Y, Py = $80 per lb; input costs are: Px1 = $4 per lb, Px2 = $3 per lb

a) Given the prices of inputs, we know the total cost = Px1*X1 + Px2*X2

TC = 4*X1 + 3*X2

Also, require to produce 100 lbs of feed, that is, Y = 100

So, 100 = X1^1/2*X2^1/4

Thus, we have to find the input combination which minimizes TC = 4*X1 + 3*X2, subject to 100 = X1^1/2*X2^1/4

Accordingly, forming the Lagrangian, we get

L = 4*X1 + 3*X2 + d*(100 - X1^1/2*X2^1/4) ; where d is the Lagrangian multiplier

So, finding the partial derivatives and solving the first order conditions: = 0 and = 0

= 4 - d*(1/2)*X1^-1/2*X2^1/4

So, with FOC we have, 4 - d*(1/2)*X1^-1/2*X2^1/4 = 0

d = 8*X1^1/2/X2^1/4

= 3 - d*(1/4)*X1^1/2*X2^-3/4

So, again with FOC we have 3 - d*(1/4)*X1^1/2*X2^-3/4 = 0

d = 12*X2^3/4/X1^1/2

From above, we have 8*X1^1/2/X2^1/4 = 12*X2^3/4/X1^1/2

2*X1 = 3*X2

That is, X1 = 1.5*X2

Using the constraint then, we have 100 = (1.5*X2)^1/2*X2^1/4

On solving this, we get X2 = 3.54 units

So, X1 = 1.5*3.54 = 5.31 units

Thus, to produce 100 lbs of feed, minimum cost incurred = 4*5.31 + 3*3.54 = 21.24 + 10.62 = $31.86

b) Finding the level of inputs that maximize profits:

Profit = total revenue - total cost

Total revenue = Py*Y = 80*X1^1/2*X2^1/4

So, profits, W = 80*X1^1/2*X2^1/4 - 4*X1 - 3*X2

Again solving for the first order conditions (FOCs): = 0 and = 0

= 80*(1/2)*X1^-1/2*X2^1/4 - 4

So, using FOC we have, 80*(1/2)*X1^-1/2*X2^1/4 - 4 = 0

X2^1/4 = 0.1*X1^1/2 OR X2^3/4 = 0.001*X1^3/2... (*)

And, = 80*(1/4)*X1^1/2*X2^-3/4 - 3

Using FOC we have 80*(1/4)*X1^1/2*X2^-3/4 - 3 = 0

So, we have 20*X1^1/2/3 = X2^3/4 ... (**)

So, from (*) and (**), we can write: 0.001*X1^3/2 = 20*X1^1/2/3

X1 = 20/(3*0.001) = 6,666.67 units

So, X2 = (20/(3*0.001))²*0.001^4/3= 4,444.44 units