The Production Process

Production Analysis Production Function Q = f(K,L) Describes available technology and feasible means of converting inputs into maximum level of output, assuming efficient utilization of inputs: ensure firm operates on production function (incentives for workers to put max effort) use cost minimizing input mix Short-Run vs. Long-Run Decisions Fixed vs. Variable Inputs

Total Product Cobb-Douglas Production Function Example: Q = f(K,L) = K.5 L.5 K is fixed at 16 units. Short run production function: Q = (16).5 L.5 = 4 L.5 Production when 100 units of labor are used? Q = 4 (100).5 = 4(10) = 40 units

Marginal Product of Labor Continuous case: MP L = dQ/dL Discrete case: arc MP L =  Q/  L Measures the output produced by the last worker. Slope of the production function

Average Product of Labor AP L = Q/L Measures the output of an “average” worker. Slope of the line from origin onto the production function

Law of Diminishing Returns (MPs) TP increases at an increasing rate (MP > 0 and  ) until inflection, continues to increase at a diminishing rate (MP > 0 but  ) until max and then decreases (MP < 0). Three significant points are: Max MP L (TP inflects) Max AP L = MP L MP L = 0 (Max TP) A line from the origin is tangent to Total Product curve at the maximum average product. Increasing MP Diminishing MP Negative MP

LTP = Q = 10K 1/2 L 1/2

Isoquant The combinations of inputs (K, L) that yield the producer the same level of output. The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output. Slope or Marginal Rate of Technical Substitution can be derived using total differential of Q=f(K,L) set equal to zero (no change in Q along an isoquant)

Cobb-Douglas Production Function Q = K a L b Inputs are not perfectly substitutable (slope changes along the isoquant) Diminishing MRTS: slope becomes flatter Most production processes have isoquants of this shape Output requires both inputs Q1Q1 Q2Q2 Q3Q3 K -  K 1 || -  K 2  L 1 <  L 2 L Increasing Output

Linear Production Function Q = aK + bL Capital and labor are perfect substitutes (slope of isoquant is constant) y = ax + b K = Q/a - (b/a)L Output can be produced using only one input Q3Q3 Q2Q2 Q1Q1 Increasing Output L K

Leontief Production Function Q = min{aK, bL} Capital and labor are perfect complements and cannot be substituted (no MRTS no slope) Capital and labor are used in fixed-proportions Both inputs needed to produce output Q3Q3 Q2Q2 Q1Q1 K Increasing Output

Isocost The combinations of inputs that cost the same amount of money C = K*P K + L*P L For given input prices, isocosts farther from the origin are associated with higher costs. Changes in input prices change the slope (Market Rate of Substitution) of the isocost line K = C/P K - (P L /P K )L K L C1C1 C0C0 L K New Isocost for a decrease in the wage (labor price). New Isocost for an increase in the budget (total cost).

Long Run Cost Minimization Q L K -P L /P K < -MP L /MP K MP K /P K < MP L /P L -P L /P K > -MP L /MP K MP K /P K > MP L /P L Point of Cost Minimization -P L /P K = -MP L /MP K MP K /P K = MP L /P L Min cost where isocost is tangent to isoquant (slopes are the same) Expressed differently: MP (benefit) per dollar spent (cost) must be equal for all inputs

Returns to Scale Return (MP): How TP changes when one input increases RTS: How TP changes when all inputs increase by the same multiple > 0 Q = f(K, L) Q = 50K ½ L ½ Q = 100, L + 100K Q = 0.01K 3 + 4K 2 L + L 2 K L 3