Chapter 6. The Production Function. Production Jargon. Production

Transcription

1 Chapter 6 Production The Production Function A production function tells us the maximum output a firm can produce (in a given period) given available inputs. It is the economist s way of describing technology or engineering relationships. Q= f (, ) Production Jargon Factors of production: Inputs used in production (for example, and ). Production set: The set of points (combinations of inputs and outputs) that are feasible according to the production function. Technically inefficient: feasible production points that yield less than the maximum possible for given amounts of inputs. Technically efficient: feasible production points that yield the maximum possible for given amounts of inputs. 1

2 Total Product If we plot Q as a function of one input, say, this is a total product curve. Q = f (, ) Marginal Product ΔQ MP = Δ = The marginal product of labor (at a point on the total product curve) can be interpreted as the slope of the total product function. Average Product Q f(, ) AP = = The average product of labor (at a point on the total product curve) can be interpreted as the slope of a ray from the origin to a point on the total product function. 2

3 Average and Marginal Products Average and marginal products are related in the following ways: When the average product is increasing in labor, marginal product is greater than average product. When the average product is decreasing in labor, marginal product is less than average product. When the average product is neither increasing nor decreasing in labor (it is at a maximum) then marginal product is equal to average product. Marginal Returns to an Input Along the total product curve, as is increased, we may have increasing or decreasing marginal returns to labor (as marginal product is increasing or decreasing when rises). Production with 2 Inputs Varying Marginal products can be defined for both capital and labor (and more) inputs: ΔQ MP = Δ ΔQ MP = Δ = = 3

4 Isoquants Isoquant: A curve that shows all of the combinations of labor and capital that can produce a given level of output. Isoquants are normally negatively sloped (as indifference curves were). Marginal Rate of Technical Substitution The (absolute value of the) slope of an isoquant is called the marginal rate of technical substitution: Δ Δ Q = Q = MRTS, More on the Slope of an Isoquant For a movement along an isoquant, the following must hold: MP Δ + MP Δ = 0 MP Δ = MP Δ Δ MP = Δ MP 4

5 Elasticity of Substitution The elasticity of substitution is defined as: σ = % Change % Change MRTS, Special Cases: Elasticity of Substitution Consider the special cases of -shaped (fixed proportions) isoquants and straight-line (perfect substitutes) isoquants. In the first case, a change in the capital labor ratio can have large impact of the MRTS, and the elasticity of substitution is small. In the second case, a change in the capital labor ratio does not change MRTS, and the elasticity of substitution is large. CES Production Function The function below (do not bother to memorize) is the constant elasticity of substitution production function Special cases include Cobb-Douglas, fixed proportions, linear (perfect substitutes) production functions. σ σ 1 σ 1 σ 1 σ σ Q= a + b 5

6 Special Cases Q= A b 1 b H Q = min, O 2 Q = a + b Returns to Scale Suppose all inputs in production increase by a factor λ. If output goes up by more than in proportion to λ, the production function has increasing returns to scale. If output goes up by less than in proportion to λ, the production function has decreasing returns to scale. If output goes up exactly in proportion to λ, the production function has constant returns to scale (CRS). The Cobb-Douglas production function is CRS if α + β = 1. The End 6