Question

The Ontario Brassworks produces brazen effronteries. As you know brass is an alloy of copper and zinc, used in fixed proportions. The production function is given by: f(x1,x2) = min{x1,2x2}, where x1 is the amount of copper it uses and x2 is the amount of zinc that it uses in production.

(a) Illustrate a typical isoquant for this production function in a graph. (b) Does this production function exhibit increasing, decreasing or constant returns to scale?
c) If the firm wanted to produce 10 effronteries, how much copper would it need? How much zinc would it need?
(d) If the firm faces factor prices (1, 1) what is the cheapest way for it to produce 10 effonteries? How much will this cost?
(e) If the firm faces factor prices (w1,w2), what is the cheapest cost to produce 10 effronter- ies?
(f) If the firm faces factor prices (w1,w2), what will be the minimal cost of producing Y ef- fronteries?

Answer 1

(a) The isoquant for the production function f(x1, x2) = min{x1, 2x2} appears as a straight line with a slope of 1/2, showing the trade-off between copper (x1) and zinc (x2).

(b) This production function exhibits decreasing returns to scale.

(c) To produce 10 effronteries, the firm would require 10 units of copper and 5 units of zinc.

(d) When facing factor prices (1, 1), the cheapest way to produce 10 effronteries is by using 10 units of copper and 5 units of zinc, costing $15.

(e) The cheapest cost to produce 10 effronteries when facing factor prices (w1, w2) is w1*10 + w2*5.

(f) When facing factor prices (w1, w2), the minimal cost of producing Y effronteries is w1*Y + (w2/2)*Y.

Step-by-step explanation:

(a) The isoquant depicts the combinations of inputs (copper and zinc) that produce the same level of output. For f(x1, x2) = min{x1, 2x2}, the isoquant is a straight line with a slope of 1/2, representing the fixed proportions of the alloy.

(b) Decreasing returns to scale occur when an increase in inputs leads to a proportionately smaller increase in output. In this case, doubling both inputs more than doubles the output, indicating decreasing returns to scale.

(c) To produce 10 effronteries, the production function's minimum requirement is 10 units of copper (x1) and 5 units of zinc (x2), based on the function f(x1, x2) = min{x1, 2x2}.

(d) When facing factor prices (1, 1), the firm minimizes costs by using the input combination that satisfies the production function: 10 units of copper and 5 units of zinc, resulting in a total cost of $15 (1*10 + 1*5).

(e) With factor prices (w1, w2), the cheapest cost to produce 10 effronteries is determined by the total cost equation: w1*10 + w2*5.

(f) Extending this logic for Y effronteries and factor prices (w1, w2), the minimal cost becomes w1*Y + (w2/2)*Y, as the production function's fixed proportionality necessitates half the units of zinc compared to copper.

Answer 2

(a) The isoquant for the production function f(x1,x2) = min{x1,2x2} would be a series of L-shaped lines where x1 is the amount of copper and x2 is the amount of zinc. (b) The production function exhibits decreasing returns to scale. (c) To produce 10 effronteries, the firm would need 10 units of copper and 20 units of zinc. (d) If the firm faces factor prices (1, 1), the cheapest way to produce 10 effronteries is to use 10 units of copper and 10 units of zinc, costing a total of 20.

Step-by-step explanation:

(a) The isoquant can be illustrated by plotting various combinations of copper (x1) and zinc (x2) that satisfy the production function f(x1,x2) = min{x1,2x2}. This would result in L-shaped isoquant curves. (b) The production function exhibits decreasing returns to scale because doubling both inputs does not double the output.

(c) To find the amount of copper and zinc needed to produce 10 effronteries, we set f(x1,x2) = min{x1,2x2} equal to 10. Since the function takes the minimum of x1 and 2x2, we choose x1 = 10 and x2 = 5, resulting in 10 effronteries. (d) If the factor prices are (1, 1), the firm minimizes cost by choosing the cheapest input, which is 10 units of copper and 10 units of zinc, costing a total of 20.

(e) For factor prices (w1,w2), the firm minimizes cost by comparing the prices of copper and zinc. If w1 < 2w2, the firm uses only copper, and if w1 > 2w2, the firm uses only zinc. (f) The minimal cost of producing Y effronteries with factor prices (w1,w2) would depend on the specific values of w1 and w2 and the optimal combination of copper and zinc, but the general approach involves comparing the relative prices of the two inputs.