Production, Inputs, and Outputs

• Robots are used in modern industries
• GM started using robots to save on labor costs in the 1990s.
• Robots are used in semiconductor chip factories to keep them 1,000 times cleaner than hospital operating rooms
• Using robots for sophisticated tasks can be expensive.
• Firms must weigh the cost of investment in robots with the savings from production costs.

Key Concepts

• Inputs or factors of production include labor and equipment used to manufacture goods and services.
• Output refers to the number of goods and services produced by a firm.
• Production transforms inputs into outputs.
• Technology determines the maximum output achievable with a given amount of inputs.

Production Function

• A production function tells the maximum output attainable for any given quantity of inputs.
• Production Function: Q = f(L, K)
• Q represents output, K represents capital, and L represents labor.
• The production set represents technically feasible combinations of inputs and outputs.

Technical Efficiency and Inefficiency

• Technical efficiency describes points on the production function that maximize output for a given input level.
• Q = f(L, K).
• Technical inefficiency describes points below the production function where output is less than possible for a given input level.
• Q < f(L, K)

Labor Requirements Function

• A labor requirements function gives the amount of labor, L = g(Q).
• It is the inversion of the production function.
• Example: L = Q², so Q = √L.

Total Product

• A total product function is a single-input production function.
• It shows how total output depends on the level of one input.
• Increasing marginal returns to labor: An increase in labor increases total output at an increasing rate.
• Diminishing marginal returns to labor: An increase in labor increases total output at a decreasing rate.
• Diminishing total returns to labor: An increase in labor decreases total output.

Marginal Product

• Marginal product: the change in output resulting from a small change in one input, holding other inputs constant.
  • MPL = ∆Q/∆L
  • MPK = ∆Q/∆K
• Example production function: Q = K¹/²L¹/²
  • MPL = (1/2)L⁻¹/²K¹/²
  • MPK = (1/2)K⁻¹/²L¹/²

Average Product & Diminishing Returns

• The average product is the total output divided by the quantity of the input used in its production.

• Equation forms

  • APL = Q/L
  • APK = Q/K

• Example:

  • APL = [K¹/²L¹/²]/L = K¹/²L⁻¹/²
  • APK = [K¹/²L¹/²]/K = L¹/²K⁻¹/²

• The law of diminishing marginal returns states that marginal products eventually decline as the quantity used of a single input increases.

Total, Average, and Marginal Products

• The relationship between labor (L), output (Q), average product (APL) and marginal product (MPL)

Average and Marginal Product Functions

APL increases when MPL > APL and APL decreases when MPL < APL

Production Functions with Two Inputs

• Two input production function, where the MPL is the change in output over the change in labor, MPL = ∆Q/ ∆L
• MPK is the change in the change in output over the change in capital MPK = ∆Q/ ∆K
• APL the average product of Labor function is Q(L,K)/L
• APK the average product of capital function is Q(L,K)/K
• Q(L,K)= 10K^(1/2)L^(1/2); allows one to determine find MP₁, MPK, APL, APK
• Q(L,K)= 2K^(1/3)L^(1/6); allows one to determine find MP₁, MPK, APL, APK

Isoquants

• Isoquants are a reduction of a 3D plot to 2D.
• Isoquants traces out all the input mixes such as labor and capital that enable a firm to generate the same level of output.
• Isoquants are generally downward sloping, and convex to the origin
• Isoquants cannot intersect each other
• Higher Isoquant means higher level of production
• In the example, Q = K^(1/2)L^(1/2), with Q = 20.
  • the equation of an Isoquant is K = 400/L

Marginal Rate of Technical Substitution

• The marginal rate of technical substitution is the amount of input L, that is required to compensate for using less of K in order to produce the same level of output
• Expressed as MRTS(subscript L,K) = - Change in K / Change in L
• Marginal Products and MRTS are related where MPL(Change in L) + MPK(change in K) = 0
  • MPL/MPK = - Change in K / Change in L = MRTS(subscript L,K)
• Rate at which the quantity of Capital can be decreased with every unit increase in labor
• Or Rate at which quantity of capital can be increased for every unit decrease in labor
• MRTS(subscript L,K) = dK/dL = MPl/MPk, and that it is always negative if both marginal products are positive
• If diminishing returns occur, then the marginal rate of technical substitution also occurs
• The marginal rate of technical substitution for capital diminishes as the quantity of labor increases

Cobb-Douglas Function

• Example Cobb-Douglas function Q = 10K^(1/2)L^(1/2)
  • MPK= 5K (-1/2)L^(1/2)
  • MPL- 5K^(1/2)L^(-1/2)
  • Marginal Rate of Technical Subsitution equals K/L
    • At input level (L,K) = (10,20), MRTS equals 2
  • At input level (L,K) = (20,20), MRTS equals 1

Economic and Uneconomic Regions of Production

• Isoquants are backward bending when MPK is less than zero
• Isoquants are upward sloping when MPL is less than zero

Elasticity of Substition

• How easily a firm can substiute labor for capital or vis versa for the capital-labor ratio
• The function expresses how the ratio of capital labor changes in response to changes in Marginal Rate of Technical Substitution
• When elasticity of subtitution is close to zero there is little oppurtunity to subustitue inputs, and when is it is large then a firm can easily substitute inputs

Perfect Substitutes

• If the two inputs are perfect substitutes then the isoquants are downward sloping and linear
• The function here is Q= AK + BL; and MRTS is constant in this case

Perfect Complements

• When inputs are perfect complements, the isoquants are L shaped
• A common function here can be Q= A min(aK, bL) In this model elasticity of substitution is said to be zero, there is no change along isoquant

Returns to Scale

• A measure of how output increases when all inputs increase by a particular amount
• Equation = % Change in quantity of output/ % Change in quantity of inputs
• Increasing returns is denoted if φ > λ
• Decreaseing returns is denoted if φ < λ
• Constant Returns are denoted if φ = λ

Returns to Scale versus Marginal return

• In Returns to scale all inputs are increased simultaneously
• When measuring returns to scale one increases quanitity as a single input
• The maginial product of a single factor may diminsh while returns scale does not
• Returns to scale does not have to be same at different levels of output
• Increased Returns to Scale (IRTS) occurs when a 1% increase in all inputs lead to a more than 1% increase in output
• Decreased Returns to Scale (DRTS) occurs when a 1% increase in all inputs lead to a less than 1% decrease in output
• Constatnt Returns to Scale (CRTS) occurs when a 1% increase in all inputs lead to a precisely 1% increase in output