Notes on Production Function and Short-Run/Long-Run Concepts

Production Function: Short Run vs Long Run

  • Production involves transforming inputs into physical output; the output is a function of inputs. The production function shows how output responds to different levels of inputs during a given period. It is a technical relationship, not an economic one.

  • General form: Q = f(X1, X2, X_3, \dots)

    • Here, Q is the level of output, and X1, X2, X_3, \dots are inputs (factors).

  • With two inputs (labor and capital): Q = f(L, K)

    • Often, labor (L) is treated as a variable factor and capital (K) as a fixed factor in the short run.

  • Time-based categorisation of production function:

    • Short Run Production Function

    • Long Run Production Function

Short Run vs Long Run Production Function

  • Short Run:

    • A period in which output can be increased by increasing the input of a variable factor, while some factors remain fixed.

    • If one factor is fixed and the other is variable: Q = f(L, \overline{K})

    • Example wording from transcript: Fix capital at 10 units, variable labor at 10 units to produce 50 units; increasing labor to 15 units increases output to 60 units (60 = f(15L, 10K¯)).

  • Long Run:

    • A period in which all inputs can be varied; no inputs are fixed.

    • If the inputs are Labor and Capital and both are variable: Q = f(L, K)

    • Example wording: With 5 units of capital and 10 units of labor, output is 50; increasing to 10 units of capital and 20 units of labor yields 100.

Distinction: Short Run vs Long Run (Summary)

  • Short run: not all inputs can be changed; law of variable proportions applies.

  • Long run: all inputs can be varied; law of returns to scale applies.

  • Short run uses Total Product (TP), Average Product (AP), and Marginal Product (MP) curves.

  • Long run uses Isoquant curves to show combinations of inputs that yield a given output.

Total Product (TP), Average Product (AP), and Marginal Product (MP)

  • Total Product (TP): total output produced with a given amount of a variable input (holding other factors fixed).

    • TP increases at increasing rate at first, then at diminishing rate, reaches a maximum, then may decline.

  • Average Product (AP): output per unit of the variable input; AP = TP / L (or Q / L).

    • Shape: inverse U, rises initially, reaches a maximum, then declines.

  • Marginal Product (MP): additional output from an extra unit of the variable input; MP = ΔTP / ΔL (or MPL = ΔQ / ΔL).

    • MP is also typically inverse U-shaped: rises first, then declines after a point.

Elasticity of Production (output elasticity with respect to a variable input)

  • Definition: Elasticity of Production with respect to labor (E_P) is the percentage change in output divided by the percentage change in the input (labor):


    • E_P = \frac{\%\Delta Q}{\%\Delta L} = \frac{(\Delta Q / Q)}{(\Delta L / L)} = \frac{\Delta Q}{\Delta L} \cdot \frac{L}{Q} = \frac{MPL}{APL}

  • Interpretation: measures responsiveness of output to a change in input; E_P = MPL / APL.

  • Relationships:

    • AP = Q / L; MPL = ΔQ / ΔL.

    • Elasticity can be written as E_P = \frac{MPL}{APL} = \frac{\Delta Q/\Delta L}{Q/L} = \frac{MPL \cdot L}{Q}.

Law of Variable Proportions (LVP) and Assumptions

  • Also called the Law of Diminishing Marginal Returns.

  • Statement: When more and more units of a variable input are added to a given fixed input, total output initially increases at an increasing rate, then at a decreasing rate, reaches a maximum, and finally may fall.

  • Assumptions:
    1) Constant technology: technology remains unchanged; improvements would alter marginal/average productivity.
    2) Short-run: some factors fixed (K fixed, L variable).
    3) Homogeneous factors: units of the variable input are the same type and quality.
    4) Changeable input ratio: no fixed proportion between inputs; inputs can be substituted.
    5) Prices of inputs do not change.
    6) Output measured in physical units.

Three Stages of Production in the Short Run

  • Stage I: Increasing MP (and increasing TP at an increasing rate).

    • TP rises; MPL increases; APL also increases; MP > APL.

    • End of Stage I occurs where MP = AP (at the point where APL is maximized).

  • Stage II: Diminishing MP (still positive TP growth, but at decreasing rate).

    • TP reaches its maximum at the end of Stage II (MP = 0 at the end).

    • MPL declines; APL also declines but remains positive.

    • Causes:

    • Scarcity of fixed factor in the short run.

    • Indivisibility of fixed factors.

  • Stage III: Negative Returns to the variable input (TP declines; MP negative).

    • APL remains positive but tends toward zero.

  • Practical takeaway: Firms operate in Stage II because Stage I has rising MP and TP but expansion is continued; Stage III yields negative MP and falling TP, which is undesirable.

  • Elasticity across stages:

    • Stage 1: EP > 1 (MP > AP).

    • End of Stage 1: AP = MP, EP = 1.

    • Stage 2: 0 < EP < 1 (MP > 0 but MP < AP).

    • End of Stage 2: MP = 0, EP = 0, TP at maximum.

    • Stage 3: EP < 0 (MP < 0).

Practical Notes on Stage Boundaries and Decision Rules

  • Stage II is considered the optimal operating range for a rational firm in the short run.

  • The level of input usage in Stage II depends on:

    • The amount of output the firm sells.

    • The price of the product.

    • The cost of the variable input.

Types of Production Function (Functional Forms)

  • 1) Linear Production Function without intercept

    • One-input case: y = aL

    • With intercept: Q = a + bL

    • Average product: APL = \frac{Q}{L} = \frac{a}{L} + b

    • Marginal product: MP = \frac{\Delta Q}{\Delta L} = b

  • 2) Power Function: Q = A L^{\alpha}

    • If (\alpha > 1), MP is increasing.

    • If (\alpha < 1), MP is decreasing.

    • If (\alpha = 1), MP is constant.

    • Notes: The exponent is the elasticity of production; MP declines as input increases when (\alpha\neq 1).

    • Examples: Q = 10 L^{1.5},\; Q = 10 L^{0.5},\; Q = 10 L^{1}

  • 3) Quadratic Production Function: Q = a + bL - cL^{2}

    • The negative sign on the last term implies diminishing marginal returns.

    • Does not allow increasing MP after a certain point (MP eventually declines).

    • Elasticity is not constant along the curve; declines with input magnitude.

    • MP is never increasing throughout the domain.

  • 4) Cubic Production Function: Q = a + bL + cL^{2} - dL^{3}

    • Can exhibit both increasing and decreasing MP.

    • Elasticity of production varies along the curve.

    • MP decreases at an increasing rate in later stages.

Technological Change and Production Function

  • Technological progress that increases productivity shifts the TP curve upward.

  • It also shifts MP and AP curves upward.

  • Consequences of better technology:

    • For the same inputs, more output can be produced.

    • The same output can be produced with fewer inputs.

    • Short-run cost curves shift downward (lower costs for given output).

Short-Run Cost (Conceptual Link)

  • Technological progress affects short-run costs by shifting cost curves downward due to higher productivity.

Example: Production Function with a Table (Before and After Technical Progress)

  • Given baseline labor-input data and TP:

    • L: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

    • TP: 0, 10, 30, 60, 80, 95, 108, 112, 112, 108, 100

  • After a 20% productivity improvement (same inputs yields 20% higher TP at each unit of labor):

    • TP (after): 0, 12, 36, 72, 96, 114, 129.6, 134.4, 134.4, 129.6, 120

  • Compare APL (Average Product) before vs after:

    • APL_before at L = 1,2,3,4,5,6,7,8,9,10: 10, 15, 20, 20, 19, 18, 16, 14, 12, 10

    • APL_after at L = 1,2,3,4,5,6,7,8,9,10: 12, 18, 24, 24, 22.8, 21.6, 19.2, 16.8, 14.4, 12

  • Marginal Product (MP) before vs after:

    • MP_before (ΔTP/ΔL) for L from 1 to 10: 10, 20, 30, 20, 15, 13, 4, 0, -4, -8

    • MP_after: 12, 14, 36, 24, 18, 15.6, 4.8, 0, -4.8, -9.6

  • These data illustrate that technical progress raises TP and shifts AP/MP upward; elasticity patterns and MP behavior change accordingly.

Exercise Illustrations (from the transcript)

  • Example 1 (Power, Quadratic, Cubic): simplified forms and their MP behavior summarize how MP can rise or fall depending on the functional form.

  • Example 2 (Technology and cost): technological progress shifts traceable curves and reduces costs in the short run.

Summary of Key Takeaways

  • Production function formalizes the relationship between inputs and output; in the short run, some inputs are fixed and one input is variable (commonly labor).

  • TP, AP, and MP describe output, average productivity, and marginal productivity with respect to the variable input.

  • The Law of Variable Proportions explains the typical three-stage pattern of production in the short run due to fixed factors, including increasing returns in Stage I, diminishing returns in Stage II, and negative returns in Stage III.

  • Elasticity of production connects MP and AP to output responsiveness via E_P = \frac{MPL}{APL}\,.

  • Different functional forms (Linear, Power, Quadratic, Cubic) capture various MP behaviors and elasticity properties; in particular, quadratic and cubic forms can model diminishing or initially rising MP.

  • Tech progress shifts TP, MP, and AP upward and lowers short-run costs; it demonstrates real-world relevance for choosing production scales and input usage.