An orange grower has discovered a process for producing oranges that requires two inputs. The production function is Q = min{2x1, x2}, where x1 andx2 are the amounts of inputs 1 and 2 that he uses. The prices of these two inputs are w1 = $5 and w2 = $2,respectively. The minimum cost of producing 140 units is therefore
A) $980.
B) $630.
C) $1,400.
D) $280.
E) $700.

Question

Quizplus · Accepted Answer

To find the minimum cost of producing 140 units, we need to minimize the cost function:

C = w1x1 + w2x2

subject to the production function:

Q = min{2x1, x2}

In other words, we need to find the values of x1 and x2 that give us a production quantity of at least 140 units while minimizing the cost.

Let's start by solving the production function for x2:

x2 = 2x1

If we substitute this into the cost function, we get:

C = w1x1 + w2(2x1)
C = 5x1 + 2(2x1)
C = 9x1

Now we can rewrite the production function in terms of x1 only:

Q = min{2x1, 2x1} = 2x1

So to produce 140 units, we need:

2x1 = 140

x1 = 70

Substituting this into the cost function, we get:

C = 9(70) = $630

Therefore, the minimum cost of producing 140 units is $630, which is choice (B).

An Orange Grower Has Discovered a Process for Producing Oranges