Technology

Marginal Product and Returns-To-Scale

  • Q: Can a technology exhibit increasing RTS even if all of its MPs are diminishing?
  • A: Yes!
    • A Cobb-Douglas production function where y = x12/3x22/3 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

Technical Rate-of-Substitution, or Slope of Isoquants

  • 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(x1,x2)The change in output is dy=∂y∂x1dx1+∂y∂x2dx2Along the isoquant, dy=0, so an isoquant must satisfy ∂y∂x1dx1+∂y∂x2dx2=0∂y∂x2dx2=−∂y∂x1dx1dx2dx1=−∂y/∂x1∂y/∂x2=−MP1MP2Thus, given some production function y=f(x1,x2),TRS=−MP1MP2=−∂y/∂x2∂y/∂x1y = 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} \\

Well-Behaved Technologies

  • 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 and Short Run

  • 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
Consider the following long run production function: y=x11/3x21/3In the short run, good 2 might be limited to 1, so the function is y=x11/311/3=x11/3Good 2 might be limited to 8 instead: y=x11/381/3=2x11/3There can be multiple SR production functions depending on the constraint on good 2.\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.}

Profit Maximization

Economic Profits

  • A firm uses inputs j = 1, …, m to make products i = 1, …, n
    • Output levels are y1, …, yn
    • Input levels are x1, …, xn
    • Product prices are p1, …, pn
    • Input prices are w1, …, wn
    • The firm can choose the output and input levels, but the product and input prices must be taken as given
      • This is known as a competitive market