Aggregate Production Function. Production. Mark Huggett. Georgetown University. January 11, 2018

Transcription

1 Production Mark Huggett Georgetown University January 11, 2018

2 Aggregate Production Function 1. Many growth theories assume an aggregate production function. 2. Thus, there is a technological relationship between GDP, denoted Y t, and aggregate quantities of inputs of capital K t and labor L t all measured at time t. Y t = A t F (K t, L t ) 3. The variable A t denotes technology at time t.

3 Aggregate Production Function 1. A simple theoretical foundation: N firms (i = 1, 2,..., N) all have the same (constant returns) production technology Yt i = A t F (Kt, i L i t) 2. All firms minimize the cost of production and face the same input prices for capital and labor. 3. Claim: the entire economy behaves as if there is a single firm with technology Y t = A t F (K t, L t ) facing the same input prices where aggregate inputs and output are K t = K 1 t + K 2 t + K N t and L t = L 1 t + L 2 t + L N t Y t = Yt 1 + Yt 2 + Yt N

4 Properties of Production Functions Constant Returns to Scale A production function has constant returns to scale if whenever all factor inputs are scaled by a common factor λ > 0, then output is also scaled by the same factor λ. This can be restated mathematically as the property Y = F (K, L) λy = F (λk, λl) holds for all λ > 0 Example: when λ = 2 a doubling of all inputs leads to a doubling of output or when λ = 1/2 a halving of all inputs leads to a halving of output.

5 Properties of Production Functions Constant Returns to Scale A geometric description of constant returns to scale is sometimes useful. An isoquant of a production function Y = F (K, L) displays all the different combinations of inputs (K, L) that produce the same level of output. If F (K, L) is constant returns then all possible isoquants can be determined from the unit isoquant (the isoquant associated with output Y = 1). Specifically, the isoquant for any level of output Y > 0 is determined by expanding or contracting each point on the unit isoquant (radially) from the origin by the factor multiple Y > 0.

6 Properties of Production Functions Constant Returns to Scale An economic implication of constant returns If two countries have the same constant returns technology, then there is no economic advantage to size. Under constant returns a countries labor productivity (i.e. the ratio Y/L) is determined only by its capital-labor ratio K/L. Y = F (K, L) Y/L = F (K/L, L/L) = F (K/L, 1)

7 Properties of Production Functions Diminishing Marginal Products the marginal product of a factor of production is the extra output caused by increasing the factor by one (small) unit, other things equal. Mathematically, marginal products are partial derivatives. We will assume that all marginal products are diminishing (i.e. decreasing) as the relevant factor input increases, other things equal. Notation: F K (K, L) denotes the marginal product of capital at the input levels (K, L) F L (K, L) denotes the marginal product of labor at the input levels (K, L)

8 Properties of Production Functions If F (K, L) has constant returns and well-defined derivatives, then the following holds for all (K, L) values: ( ) Y = F (K, L) = F L (K, L)L F K (K, L)K Interpretation: This is Euler s Theorem for homogeneous functions (i.e. functions displaying constant returns). A competitive firm makes zero economic profit. The firm produces Y units of output. It pays labor F L (K, L)L and pays capital F K (K, L)K. Nothing is left after paying all factors with constant returns.

9 Implications of Profit Maximization P rofit = F (K, L) W L RK If a firm with technology F (K, L) maximzes profit taking input prices (W, R) as given, then W = F L (K, L) and R = F K (K, L) P rofit = F (K, L) W L RK = F (K, L) F L (K, L)L F K (K, L)K = 0 when F has constant returns.

10 Cobb-Douglas Production Function This production function displays (i) diminishing marginal products, (ii) constant returns and (iii) constant factor shares. Y = F (K, L) = AK β L 1 β diminishing marginal products F L (K, L) = (1 β)ak β L β and F K (K, L) = βak β 1 L 1 β constant returns: note that the exponents add up to 1! labor s share = F L(K,L)L F (K,L) capital s share = F K(K,L)K F (K,L) = (1 β)akβ L β L F (K,L) = (1 β) = βakβ 1 L 1 β K F (K,L) = β

11 Aggregate Production Function Why did the US college-wage premium increase despite increases in ratio of total college labor (L C ) to high school labor (L H )? Figure 1 College Graduate and High School Graduate Wage Premiums: 1915 to College graduate wage premium High school graduate wage premium

12 Why did the US college-wage premium increase despite increases in ratio of total college labor (L C ) to high school labor (L H )? Table 1 Changes in the College Wage Premium and the Supply and Demand for College Educated Workers: 1915 to 2005 (100 Annual Log Changes) Relative Demand (σsu = 1.4) Relative Demand (σsu = 1.64) Relative Demand (σsu = 1.84) Relative Wage Relative Supply Sources: The underlying data are presented in Appendix Table A8.1 and are derived from the 1915 Iowa State Census, 1940 to 2000 Census IPUMS, and 1980 to 2005 CPS MORG samples.

13 An Unsuccessful Explanation Y = A t F (K t, L H t, L C t ) = A t K β t (L H t ) α H (L C t ) α C Firm minimizes cost of production taking (Wt H, Wt C ) as given implies W C t W H t log W C t W H t = AtF C(K t,l H t,lc t ) A tf H (K t,l H t LC t ) = α CA tk β t (LH t )α H,(L C t )α C 1 α H A tk β t (LH t )α H 1 (L C t )α C = log α C α H Prediction: log W C t W H t + log LH t L C t = log α C α H log LC t L H t decreases when log LC t L H t increases. Clearly, something is wrong. Theory is missing a force capable of increasing the skill premium.

14 An More Successful Explanation: See Goldin and Katz (2007) Y = A t F (L H t, L C t, λ t ) = A t [λ t (L C t ) ρ + (1 λ t )(L H t ) ρ ] 1/ρ Firm minimizes cost of production taking (Wt H as given implies W C t W H t = AtF C(L H t,lc t,λt) A tf H (L H t,lc t,λt) = ( ) log W C t W H t λt(lc t )ρ 1 (1 λ t)(l H t )ρ 1 = log λ t (1 λ t) + (ρ 1) log LC t L H t, W C t ) Estimating the relationship (*) yields (ρ 1) = 0.61 Interpretation: US college wage premium is a race between the depressing effects of increased supply LC t L H t and the increasing effects of technological change λ t which complements skilled labor.

15 Technological Change Economists believe that technological change is the key reason why people in developed economies live vastly better lives than people 100 years ago. Embodied technological change: technological improvements are often embodied in new capital (e.g. a new car, new cell phone, new computer, new light bulb, new airplane, new tractor, new seeds,...). Disembodied technological change: you can take advantage of a technological change (largely) by using existing inputs (e.g. Adam Smith s pin factory, assembly line, double entry book keeping, just-in-time inventory, faster computer algorithms, new medical procedures,...)

16 Technological Change Embodied technological change is highly relevant but difficult to model. Need to keep track of many vintages of capital. Many influential growth theories (e.g Solow growth model) employ disembodied technical change Neutral (disembodied) technical change: Y t = A t F (K t, L t ) or Y t = F (K t A t, L t A t ) Labor augmenting (disembodied) technical change: Y t = F (K t, L t A t ) Capital augmenting (disembodied) technical change: Y t = F (K t A t, L t ) Goldin and Katz employ neutral and skill-augmenting technological change