Production

What is technology? Figure: We study this abstract production process. YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 3 / 32

2. Production Function: single-output When there is only one output and several inputs, we can use production function to represent the technology. Definition (Production Function) • Production function is a function showing the highest output that a firm can produce for every specified combination of inputs. y = f ( x ) . The green curve. YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 6 / 32

Transformation Function T : R n → R . Transformation Function T : R n → R represents the efficient production plans, such that T(-x,y)=0 iff (-x,y) is efficient. Remarks: Just as the production function picks out the maximum scalar output as a function of inputs, the transformation function picks out the maximal vectors of net outputs. • Given inputs x, what is the combination of efficient output vectors? –Add a figure here in class: production possibility frontier YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 9 / 32

Examples of Isoquant Example 1: Perfect substitute Production Function y = ax 1 + bx 2 . Figure: Isoquant of Perfect substitute Technology: y = x 1 + 3 x 2 . YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 12 / 32

Concave Production Function For simplicity, we further assume that production function is CONCAVE. Concave production function implies: decreasing marginal output ( ∂ f ( x ) ∂ x i ) in input x i . Concave production function is important to guarantee a unique solution of firm maximization problem. Figure: Concave production function YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 15 / 32

Decreasing/Increasing Marginal Product Decreasing Marginal Product: — Marginal product of input x i decreases when use more x i and fixed other inputs. Increasing Marginal Product: — Marginal product of input x i increases when use more x i and fixed other inputs. Constant Marginal Product: YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 18 / 32

Technical Rate of Substitution Question: suppose x 1 is decreased by one unit, how much of x 2 do we need to increase to keep the same output level? Definition (Technical Rate of Substitution) The Technical Rate of Substitution (TRS) is defined as TRS 1 , 2 = dx 2 dx 1 = − ∂ f ( x ) ∂ x 1 ∂ f ( x ) ∂ x 2 Proof: production function f ( x 1 , x 2 ) = y . Total differentiation and keeping y constant gives ∂ f ∂ x 1 dx 1 + ∂ f ∂ x 2 dx 2 = 0 Rearranging it yields TRS. YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 21 / 32

Diminishing Technical Rate of Substitution Figure: Isoquant and Technical Rate of Substitution MRS 1 , 2 = △ x 2 △ x 1 Q: (1)Diminishing TRS 1 , 2 in x 1 ? (2) Counterpart in consumer theory? YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 22 / 32

Technical Rate of Substitution and Slope of Isoquant Technical Rate of Substitution is the Slope of Isoquant. Figure: Technical Rate of Substitution and Slope of Isoquant YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 23 / 32

Example: compute MP and TRS Cobb-Douglas production function: f ( x 1 , x 2 ) = x α 1 x β 2 . Compute its technical rate of substitution TRS 1 , 2 . Solution: Compute the technical rate of substitution MP 1 = ∂ f ( x ) ∂ x 1 = α x α − 1 1 x β 2 , MP 2 = ∂ f ( x ) ∂ x 2 = β x α 1 x β − 1 2 TRS 1 , 2 = − ∂ f ( x ) ∂ x 1 ∂ f ( x ) ∂ x 2 = − α β ( x 2 x 1 ) YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 24 / 32

Returns to Scale YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 25 / 32

Returns to Scale Motivation: firm size. — Generally, firm size in auto mobile industry is much larger than that in bakery industry. Why? Question we ask: if we double the input x, how will it change the output y? 1 Output also doubled? 2 Output more than doubled? 3 Output less than doubled? The three answers correspond to three types of technology. YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 26 / 32

Returns to scale Definition 1 Constant returns to scale if f ( tx 1 , tx 2 , ..., tx n ) = tf ( x 1 , x 2 , ..., x n ) for all t ≥ 0. 2 Increasing returns to scale if f ( tx 1 , tx 2 , ..., tx n ) > tf ( x 1 , x 2 , ..., x n ) for all t > 1. 3 Decreasing returns to scale if f ( tx 1 , tx 2 , ..., tx n ) < tf ( x 1 , x 2 , ..., x n ) for all t > 1. Figure: Constant Return to sacle: single input YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 27 / 32

Decreasing Return to Scale Figure: Decreasing Return to sacle: single input YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 28 / 32

Increasing Return to Scale Figure: Increasing Return to sacle: single input YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 29 / 32

Return to Scale: Two Inputs Figure: Isoquant: Constant/Increasing Return to sacle (two inputs) YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 30 / 32

Returns to Scale: Cobb-Douglas Example Question: Cobb-Douglas production function: f ( x 1 , x 2 ) = x α 1 x β 2 . Does it show CRS, IRS, or DRS? Solution: Pick any t ≥ 1, f ( tx 1 , tx 2 ) = ( tx 1 ) α ( tx 2 ) β = t α + β x α 1 x β 2 = t α + β f ( x 1 , x 2 ) So, If ( α + β ) > 1, f ( tx 1 , tx 2 ) > tf ( x 1 , x 2 ) for all t > 1 , ⇒ IRS. If ( α + β ) < 1, f ( tx 1 , tx 2 ) < tf ( x 1 , x 2 ) for all t > 1 , ⇒ DRS. If ( α + β ) = 1, f ( tx 1 , tx 2 ) = tf ( x 1 , x 2 ) for all t ≥ 0, ⇒ CRS. ♣ Thinking: f ( x 1 , x 2 ) = x 3 / 4 1 x 1 / 2 2 + x 1 − 10. Increasing, decreasing, or constant return to scale? YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 31 / 32

Reading: Chapter 1 of Hal Varian: Microeconomic Analysis YF Fong (HKUST) Producer Theory: Technology and Its Representation September 17, 2023 32 / 32