A competitive firm uses three factors of production.Its production function is f (x, y, z)= (x + y)^1/2z ^1/2.Originally the factor prices were wx = $1, wy = $2, and wz =$3.The prices of factors x and z decreased to half of their previous levels, but the price of factor y stayed constant.The cost of production A)decreased by more than 1/2. B)decreased by 1/3. C)decreased by exactly 1/2. D)stayed constant. E)decreased by less than 1/3.

Question

Quizplus · Accepted Answer

The cost function for this production function is
C = wx * x + wy * y + wz * z 
= $1 * x + $2 * y + $3 * z 
When the prices of factors x and z decrease to half, the new cost function becomes:
C' = $0.5x + $2y + $1.5z 
To find the percentage change in cost, we can use the formula:
(Old Cost - New Cost) / Old Cost * 100%
= (($1x + $2y + $3z) - ($0.5x + $2y + $1.5z)) / ($1x + $2y + $3z) * 100%
= (0.5x + 0.5z) / (x + 2y + 3z) * 100%
= 0.5 / (1 + 2y/x + 3z/x) * 100%
Since y is constant, the percentage change in cost is inversely proportional to x and z. Therefore, the total percentage change will be minimized when x and z have equal weights in the production function. That is, x = z = 1 and y = 0. Plugging these values into the formula, we get:
(Old Cost - New Cost) / Old Cost * 100%
= 0.5 / (1 + 0 + 0.5) * 100%
= 1/2 * 100%
= 50%
Therefore, the cost of production decreases by exactly 1/2 or 50%. The answer is C.

A Competitive Firm Uses Three Factors of Production