When a = 3/4 and b = 1/4 for the Cobb-Douglas production function, returns to scale are (a) constant, (b) increasing, (c) decreasing, or (d) first increasing and then decreasing.
Technological progress refers to (a) an increase in MPL and MPK, (b) the reduction in L and K to produce any level of output, (c) a shift of the isoquants toward the origin, or (d) all of the above.
The Correct Answer and Explanation is :
1. Returns to Scale in Cobb-Douglas Production Function
The Cobb-Douglas production function is generally written as:
[
Y = A L^a K^b
]
where:
- (Y) is output,
- (L) is labor,
- (K) is capital,
- (A) is a constant representing technology,
- (a) and (b) are the output elasticities of labor and capital, respectively.
To determine the returns to scale, we examine the effect of scaling both inputs, (L) and (K), by the same factor (t). When the inputs are scaled, the output becomes:
[
Y(tL, tK) = A (tL)^a (tK)^b = t^{a+b} A L^a K^b
]
If the sum of the exponents, (a + b), is:
- Equal to 1: Returns to scale are constant (i.e., scaling inputs by a factor of (t) results in scaling output by the same factor).
- Greater than 1: Returns to scale are increasing (i.e., scaling inputs by a factor of (t) results in a more than (t)-fold increase in output).
- Less than 1: Returns to scale are decreasing (i.e., scaling inputs by a factor of (t) results in less than (t)-fold increase in output).
In your case, with (a = \frac{3}{4}) and (b = \frac{1}{4}), the sum (a + b = \frac{3}{4} + \frac{1}{4} = 1), so the returns to scale are constant.
Answer: (a) constant
2. Technological Progress
Technological progress refers to advancements that make it possible to produce more output with the same amount of inputs. Technological progress could manifest in several ways:
- (a) An increase in MPL (Marginal Product of Labor) and MPK (Marginal Product of Capital): This indicates that each unit of labor and capital becomes more productive, which is often associated with technological improvements.
- (b) The reduction in L and K to produce any level of output: Technological progress can allow a firm to produce the same output using fewer inputs, reflecting more efficient production techniques.
- (c) A shift of the isoquants toward the origin: Isoquants represent combinations of inputs that yield the same output. A shift toward the origin implies that less labor and capital are needed to produce the same output, which also reflects technological progress.
- (d) All of the above: Technological progress encompasses all of these aspects, as improvements in technology can lead to increases in productivity (higher MPL and MPK), greater efficiency (less input required for the same output), and a shift in isoquants.
Thus, technological progress refers to all these phenomena.
Answer: (d) all of the above
Summary
- For the Cobb-Douglas production function with (a = 3/4) and (b = 1/4), returns to scale are constant because (a + b = 1).
- Technological progress involves increases in MPL and MPK, a reduction in inputs needed for the same output, and shifts in isoquants toward the origin. Hence, the correct answer is (d) all of the above.