More on RTS and Start of Profit Maximization

Technology

Q: Can a technology exhibit increasing RTS even if all of its MPs are diminishing?
A: Yes!
- A Cobb-Douglas production function where y = x₁^2/3x₂^2/3 has two diminishing MPs with increasing RTS
Recall that the MP tracks the rate-of-change in output by increasing one input, while RTS tracks the rate-of-change in output by increasing all inputs proportionally

The slope of an isoquant at any given budnle represents the rate at which one input must be substituted for another in order to maintain the same level of output
- This slope is known as the technical rate-of-substitution, or TRS

y = f(x_1, x_2) \\ \text{The change in output is } dy = \frac{\partial y}{\partial x_1} dx_1 + \frac{\partial y}{\partial x_2} dx_2 \\ \text{Along the isoquant, } dy = 0 \text{, so an isoquant must satisfy } \frac{\partial y}{\partial x_1} dx_1 + \frac{\partial y}{\partial x_2} dx_2 = 0 \\ \frac{\partial y}{\partial x_2} dx_2 = - \frac{\partial y}{\partial x_1} dx_1 \\ \frac{dx_2}{dx_1} = - \frac{\partial y / \partial x_1}{\partial y / \partial x_2} = -\frac{MP_1}{MP_2} \\ \text{Thus, given some production function } y = f(x_1, x_2), TRS = -\frac{MP_1}{MP_2} = -\frac{\partial y / \partial x_2}{\partial y / \partial x_1} \\

A well-behaved technology is both monotonic and convex
- Monotonicity: More of any input generates more output
- Convex: If the input bundles x’ and x’’ both provide y units of output, then the mixture tx’t(1-t)x’’ provides at best y units of output, for any 0 < t < 1
  - Strict Convexity is when the mixture is greater than y, implying that the TRS increases (less negative) as x₁ increases, AKA diminishing TRS
  - Weak Convexity is when the mixture is greater than or equal to y

Weakly Convex Techonology

Long run means that a firm is unrestricted in its choice of all input levels
Short run means that a firm is restricted in its choice of at least one input levels
- The long run can be thought of as a firm “choosing” its short run circumstances
- The long run will always be more optimized than the short run, as there are no constraints

\text{Consider the following long run production function: } y = x_1^{1/3}x_2^{1/3} \\ \text{In the short run, good 2 might be limited to 1, so the function is } y = x_1^{1/3} 1^{1/3} = x_1^{1/3} \\ \text{Good 2 might be limited to 8 instead: } y = x_1^{1/3} 8^{1/3} = 2x_1^{1/3} \\ \text{There can be multiple SR production functions depending on the constraint on good 2.}