A company has a Cobb-Douglas production function f(x,y)= 20x^0.33y0.67 where x is the utilization of labor and y is the utilization of capital. Determine the number of units of product produced when 1728 units of labor and 27,000 units of capital are used.
20 (1728)^.33 (27000)^.67
I can do this with my calculator
20 (11.7)(931) = 217,886
or use logs
.33 log 1728 = .33 * 3.24 = 1.0684
10^1.0684 = 11.7
etc
Thank you very much
To determine the number of units of product produced when 1728 units of labor and 27,000 units of capital are used, we need to substitute these values into the given Cobb-Douglas production function.
The production function is f(x, y) = 20x^0.33y^0.67, where x represents labor and y represents capital.
Substituting the given values into the production function:
f(1728, 27000) = 20 * (1728^0.33) * (27000^0.67)
To simplify the calculations, let's break it down into separate parts:
(1728^0.33) ≈ 12.318
(27000^0.67) ≈ 137.627
Now substitute these values back into the expression:
f(1728, 27000) = 20 * 12.318 * 137.627
Multiplying these values:
f(1728, 27000) ≈ 32,000 units of product
Therefore, when 1728 units of labor and 27,000 units of capital are used, approximately 32,000 units of product are produced according to the Cobb-Douglas production function.