Strict case: positive slope Non-differentiable: for all x1 (strict case: > for all x1 > y(x1) ≥ y(x0) ≥ x0 y(x1) y(x0) x0) Exception: phyotoxic effects, nutrient toxicity, etc. Concavity: as use more input, eventually extra output generated by extra input begins to decrease: Law of Diminishing Marginal Product [Curvature] , strict case Non-differentiable: for all and and 0 1 (Concave)  x0 x1 ≤ ≤ for all and and 0 1 (Strict Case)  x0 x1 ≤ ≤ Concave, differentiable Concave, non-differentiable Strictly concave linear section: only concave, not strictly concave Quasi-concavity: less strict than concavity, allows convex regions, so sigmoidal functions acceptable. Allows biologically relevant cases that will still be economically meaningful For all = and = for 0 1 (quasi-concave)  f(x0) t f(x1) t, ≥ ≥ ≤ ≤ For all = and = for 0 <  < 1 (strict case) f(x0) t f(x1) t, ≥ ≥ Quasi-concave not quasi-concave Convex section okay convex section make sit non-quasi-concave Technical Terms Another term for output (yield or meat) Total Product: from last unit of input” Marginal Product: “Output Derivative formulation: = MP Discrete formulation: = MP = output per unit of input” Average Product: AP “Average MP at = Slope of the TP function at = x0 x x0 AP at = Slope of the line from origin to TP function at = (Prove graphically) x0 x x0 AP: Input use efficiency: nutrient use efficiency, water use efficiency, etc. Numerical Example:
Strict case: positive slope Non-differentiable: y(x1) ≥ y(x0) for all x1 ≥ x0 (strict case: y(x1) > y(x0) for all x1 > x0) Exception: phyotoxic effects, nutrient toxicity, etc. Concavity: as use more input, eventually extra output generated by extra input begins to decrease: Law of Diminishing Marginal Product [Curvature] , strict case Non-differentiable: for all x0 and x1 and 0 ≤  ≤ 1 (Concave) for all x0 and x1 and 0 ≤  ≤ 1 (Strict Case) Concave, differentiable Concave, non-differentiable Strictly concave linear section: only concave, not strictly concave Quasi-concavity: less strict than concavity, allows convex regions, so sigmoidal functions acceptable. Allows biologically relevant cases that will still be economically meaningful For all f(x0) ≥ = t and f(x1) ≥ = t, for 0 ≤  ≤ 1 (quasi-concave) For all f(x0) ≥ = t and f(x1) ≥ = t, for 0 <  < 1 (strict case) Quasi-concave not quasi-concave Convex section okay convex section make sit non-quasi-concave Technical Terms Total Product: Another term for output (yield or meat) Marginal Product: “Output from last unit of input” Derivative formulation: MP = Discrete formulation: MP = Average Product: AP = “Average output per unit of input” MP at x0 = Slope of the TP function at x = x0 AP at x0 = Slope of the line from origin to TP function at x = x0 (Prove graphically) AP: Input use efficiency: nutrient use efficiency, water use efficiency, etc. Numerical Example:
Technical%20Aspects%20of%20Production.docx