Lecture 04 Production Theory

Transcription

1 ecture 04 Production Theory 1. How to define a Firm? roles, objectives, and more a) Specialization in production. b) Team production is more efficient than individual production in many aspects. ess transaction costs but more monitoring costs in firm. c) Internal control than external market transactions with much transaction cost. Opportunistic behaviors due to asset specificity. But, is internal control always reasonable? Not really. Why? Ex) Vehicle brand owner d) Profit-maximizers? e) Ownership and management are separated. 2. Production Technology: Basic Analysis a) Time periods SR with at least one fixed input / R with all variable inputs b) Production function (total product curve) = f (, ) (1) : is maximum amount of output a firm could get from a given combination of inputs. So, inefficient management could reduce output from what is technologically possible. If we invert the equation (1) to get = g( ), which tells us the minimum amount of labor required to produce a given amount of output, this function is labor requirement function. 3. TP, MP, and AP TP AP = MP = d d I R Up to I (inflection point), MP is increasing (increasing marginal returns to labor), and after that MP starts diminishing. MP, AP A Up to S, AP is increasing, so AP < MP. After S, AP is decreasing, AP > MP B C J S AP aw of Diminishing Marginal Returns: Principle that as the usage of one input increases, the quantities of other inputs being held fixed, a point will be reached beyond which the marginal product of the variable input will decrease. MP 11

2 4. Production Technology: Two Variable Inputs Production Surface (or Total Product Hill) A O B 5. Isoquants A curve that shows all of the combinations of labor and capital that can produce a given level of output. Ridge ines R S C Δ Δ D O 6. MRTS (Marginal rate of Technical Substitution) MRTS, = Δ Δ From = f (, ), we can get the following equation by totally differentiating the production function. Δ= MP Δ+ MP Δ = 0. Finally, we can get 12

3 MRTS Δ, = = Δ MP MP 2 and d > 0implies the aw of Diminishing MRTS. 2 d 7. Elasticity of Substitution Δ percentage change in capital labor ratio σ = = percentage change in MRTS, Δ MRTS MRTS / at A = slope of ray OA = 4 MRTS at A = 4 20 A / at B = slope of ray OB = 1 MRTS at B = 1 10 As the firm moves from A to B, the B elasticity of substitution equals (σ is relatively small) (σ is relatively big) ittle opportunity to substitute between inputs Big opportunity to substitute between inputs 13

4 8. Special Production Functions a) inear Production Function (Perfect Substitutes, σ = ) = c+ d, c and d are positive constants. Ex) natural gas or fuel oil in manufacturing process. company data storage b/w high-capacity and low-capacity computers. b) Fixed-proportions Production Function (Perfect Complements, σ = 0, eontief Function) = min( a, b), a and b are positive constants. Ex) fixed portions of oxygen and hydrogen atoms to make water molecules. One frame with two tires for bicycle. One chassis with four tires for a car. c) Cobb-Douglas Production Function α = A, A, α, are positive constants. Homogeneous Function of degree r y = f ( x, z) If we k-fold all the independent variables x and z, f kx kz k r r (, ) f ( x, z) = k y If r = 1, the function is also known as linear homogeneous function. Ex) Identify the following functions. 2 2 x a a) y = 3x + xz 2z b) y = + 5 c) y = x z 3z 1 a Returns to scale for a Cobb-Douglas Production Function et 1 and 1 denote the initial quantities of labor and capital, and let 1 denote the initial α output, so 1 = A1 1. Now let s increase all input quantities by the same proportional amount λ, where λ > 1, and let 2 denote the resulting volume of output: α α+ α α = A( λ ) ( λ ) = λ A = λ. From this, we can see that if: α+ a) α + >1, then λ > λ, and so 2 > λ 1(increasing returns to scale, IRS) α+ b) α + =1, then λ = λ, and so 2 = λ 1 (constant returns to scale, CRS) α+ c) α + <1, then λ < λ, and so < λ (decreasing returns to scale, DRS) 2 1 d) Constant Elasticity of Substitution (CES) Production Function σ is independent of MRTS, or input ratio ( / ) or even output. r ρ = A a + ( 1 a), where A> 0, 0< a< 1, ρ 1. 1 r is the degree of homogeneity. σ = 1 + ρ a) If ρ = 1 ( σ = ), = A[ a+ ( 1 a) ] r (isoquant is a straight-line). 14

5 b) If ρ = 0 ( σ = 1 ), we need a trick because we can t define 1. Taking log on both sides, we can get log= log A r log[ a + ( 1 a) ]. ρ But the second term of r.h.s. is indeterminate because of 0. The best way to solve this 0 problem is to use Hospital rule. f Generally, when we have lim f ( ρ) = 0, lim g( ρ) = 0, lim ( ρ ) f lim ( ρ = ) ρ 0 ρ 0 ρ 0 g( ρ) ρ 0 g ( ρ) et s define f ( ρ) = rlog[ a + ( 1 a) ] and g( ρ) = ρ. r lim f ( ρ) = lim ( a log + ( 1 a) log ) ρ 0 ρ 0 a + ( 1 a) = r a + a = r a 1 a ( log ( 1 )log ) log( ) 1 lim g ( ρ) = 1 ρ 0 f Finally, we know limlog log lim ( ρ ) 1 = A = log A+ rlog( a a ) ρ 0 ρ 0 g( ρ) lim = A( a a ) r (Cobb-Douglas Production Function) ρ 0 1 (You can check out why Cobb-Douglas has σ = 1) c) If ρ (σ = 0 ). eontief Production Function = min[ a, b] σ = 0 σ = 01. σ = 1 σ = σ = 5 9. Returns to Scale: revisited f ( k, k) = kf (, ) : CRS f ( k, k) > kf (, ) : IRS f ( k, k) < kf (, ) : DRS 15

6 CRS IRS DRS 16