Short-Run Production Function Analysis: Total, Marginal, and Average Product

Short-Run Production Function and Mathematical Derivations

• Production Function Equation:   $Q = TP(L) = 30L^2 - L^3$   Where $L$ represents the number of workers and $TP$ represents units of total output. Capital is assumed to be fixed in the short run.

• Derivation of Marginal Product of Labor ($MP_L$):

  • Derivative Formula:     $MP_L = \frac{dQ}{dL} = \frac{d}{dL}(30L^2 - L^3) = 2 \times 30L^{2-1} - 3L^2 = 60L - 3L^2$
  • Meaning on Total Product ($TP$) Graph: Measures the slope of a line tangent to the $TP$ curve at each given level of labor $L$

• Derivation of Average Product of Labor ($AP_L$):

  • Ratio Formula:     $AP_L = \frac{Q}{L} = \frac{30L^2 - L^3}{L} = 30L - L^2$
  • Meaning on Total Product ($TP$) Graph: Measures the slope of a ray drawn from the origin to a specific point on the $TP$ curve at each given level of labor $L$

Short-Run Production Schedule and Numerical Evaluation

• Evaluation Table across Labor Inputs:

  • At $L = 0$:

  • Total Product ($TP$): $0$

  • Interval Output Change ($\frac{\Delta Q}{\Delta L}$): $\text{N/A}$

  • Marginal Product of Labor ($MP_L$): $\text{N/A}$

  • Average Product of Labor ($AP_L$): $\text{N/A}$

  • At $L = 5$:

  • Total Product ($TP$): $30(5)^2 - (5)^3 = 30(25) - 125 = 625$

  • Interval Output Change ($\frac{\Delta Q}{\Delta L}$): $\frac{625 - 0}{5 - 0} = \frac{625}{5} = 125$

  • Marginal Product of Labor ($MP_L$): $60(5) - 3(5)^2 = 300 - 75 = 225$

  • Average Product of Labor ($AP_L$): $\frac{625}{5} = 125$ or $30(5) - (5)^2 = 150 - 25 = 125$

  • At $L = 10$:

  • Total Product ($TP$): $30(10)^2 - (10)^3 = 3000 - 1000 = 2000$

  • Interval Output Change ($\frac{\Delta Q}{\Delta L}$): $\frac{2000 - 625}{10 - 5} = \frac{1375}{5} = 275$

  • Marginal Product of Labor ($MP_L$): $60(10) - 3(10)^2 = 600 - 300 = 300$

  • Average Product of Labor ($AP_L$): $\frac{2000}{10} = 200$ or $30(10) - (10)^2 = 300 - 100 = 200$

  • At $L = 15$:

  • Total Product ($TP$): $30(15)^2 - (15)^3 = 6750 - 3375 = 3375$

  • Interval Output Change ($\frac{\Delta Q}{\Delta L}$): $\frac{3375 - 2000}{15 - 10} = \frac{1375}{5} = 275$

  • Marginal Product of Labor ($MP_L$): $60(15) - 3(15)^2 = 900 - 675 = 225$

  • Average Product of Labor ($AP_L$): $\frac{3375}{15} = 225$ or $30(15) - (15)^2 = 450 - 225 = 225$

  • At $L = 20$:

  • Total Product ($TP$): $30(20)^2 - (20)^3 = 12000 - 8000 = 4000$

  • Interval Output Change ($\frac{\Delta Q}{\Delta L}$): $\frac{4000 - 3375}{20 - 15} = \frac{625}{5} = 125$

  • Marginal Product of Labor ($MP_L$): $60(20) - 3(20)^2 = 1200 - 1200 = 0$

  • Average Product of Labor ($AP_L$): $\frac{4000}{20} = 200$ or $30(20) - (20)^2 = 600 - 400 = 200$

  • At $L = 25$:

  • Total Product ($TP$): $30(25)^2 - (25)^3 = 18750 - 15625 = 3125$

  • Interval Output Change ($\frac{\Delta Q}{\Delta L}$): $\frac{3125 - 4000}{25 - 20} = \frac{-875}{5} = -175$

  • Marginal Product of Labor ($MP_L$): $60(25) - 3(25)^2 = 1500 - 1875 = -375$

  • Average Product of Labor ($AP_L$): $\frac{3125}{25} = 125$ or $30(25) - (25)^2 = 750 - 625 = 125$

Critical Points on Production Curves

• Point A: Maximum Marginal Product of Labor ($MP_L$)

  • Labor Level ($L$): $L = 10$
  • Optimization Derivation:     $\frac{d}{dL}(MP_L) = \frac{d}{dL}(60L - 3L^2) = 60 - 6L = 0 \implies L = 10$
  • Values at Point A:
  • $MP_L = 300$
  • $TP = 2000$
  • $AP_L = 200$
  • Curvature of $TP$: Point A is the inflection point on the $TP$ curve. For $L < 10$, $TP$ is convex (increasing at an increasing rate). For $L > 10$, $TP$ becomes concave (increasing at a decreasing rate).

• Point B: Maximum Average Product of Labor ($AP_L$)

  • Labor Level ($L$): $L = 15$
  • Optimization Derivation:     $\frac{d}{dL}(AP_L) = \frac{d}{dL}(30L - L^2) = 30 - 2L = 0 \implies L = 15$
  • Values at Point B:
  • $AP_L = 225$
  • $MP_L = 225$
  • $TP = 3375$
  • Interrelationship: At the maximum of $AP_L$, $MP_L = AP_L = 225$

• Point C: Maximum Total Product ($TP$)

  • Labor Level ($L$): $L = 20$
  • Optimization Derivation:     $\frac{d TP}{dL} = MP_L = 60L - 3L^2 = 0 \implies 3L(20 - L) = 0 \implies L = 20$
  • Values at Point C:
  • $TP = 4000$
  • $MP_L = 0$
  • $AP_L = 200$
  • Significance: Maximum output achievable is $4000$ units, where marginal productivity drops to zero. Hiring beyond $L = 20$ leads to negative marginal productivity and reduced total output.

Graphical Analysis and Key Relationships

• Visual Layout and Alignment:

  • Top Panel Graph: Total Product $TP = 30L^2 - L^3$ (units of output) plotted against Labor $L$. Contains Point A $(10, 2000)$, Point B $(15, 3375)$, and Point C $(20, 4000)$.
  • Bottom Panel Graph: Marginal Product $MP_L$ and Average Product $AP_L$ plotted against Labor $L$. Contains Point A $(10, 300)$, Point B $(15, 225)$, and Point C $(20, 0)$.
  • Alignment Lines: Vertical dashed lines pass through Points A, B, and C across both panels to align labor levels across curves.

• Effect of $MP_L$ Dynamics on Total Product ($TP$):

  • When $MP_L$ is positive and rising ($0 < L < 10$): $TP$ increases at an increasing rate (convex shape).
  • When $MP_L$ is positive and falling ($10 < L < 20$): $TP$ increases at a decreasing rate (concave shape).
  • When $MP_L$ is zero ($L = 20$): $TP$ reaches its maximum peak value ($4000$).
  • When $MP_L$ is negative ($L > 20$): $TP$ decreases.

• Ray from the Origin to Point B and Tangency Conditions:

  • Slope of Ray: Any ray drawn from the origin to a point on the $TP$ curve has a slope equal to $\frac{TP}{L}$, which defines $AP_L$
  • Maxima Proof: The ray drawn from the origin tangent to the $TP$ curve at Point B $(15, 3375)$ is the steepest line from the origin that touches $TP$, demonstrating that $AP_L$ reaches its maximum at B.
  • Equivalence Proof: Because this steepest ray is tangent to $TP$ at B, its slope equals both $\frac{TP}{L}$ ($AP_L$) and the derivative $\frac{d TP}{dL}$ ($MP_L$), proving $MP_L = AP_L$ at Point B.

• Explanation of Intersection at Maximum $AP_L$:

  • When $MP_L > AP_L$: The marginal worker produces more than the current average, pulling $AP_L$ upward.
  • When $MP_L < AP_L$: The marginal worker produces less than the current average, pulling $AP_L$ downward.
  • Intersection Point: Because $AP_L$ rises when $MP_L$ is above it and falls when $MP_L$ is below it, $MP_L$ must cross $AP_L$ exactly at its peak where $AP_L$ reaches its maximum value.