Technology. Production functions Short run and long run Examples of technology Marginal product Technical rate of substitution Returns to scale

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1 Technology Producion funcions Shor run and long run Examples of echnology Marginal produc Technical rae of subsiuion Reurns o scale

2 Analogies wih Consumer Theory Consumers Firms Maximize uiliy Maximize profis Consrain: budge line Consrain: producion funcion

3 Facors of Producion Inpus: labor, land, capial, raw maerials Physical capial: inpus ha are hemselves produced goods, such as racors, buildings, compuers, ec. Financial capial: money used o sar up or run a business (no an inpu o producion)

4 Producion Funcion Funcion ha describes he maximum amoun of oupu ha can be produced for a given level of inpus Inpus and oupus are measured in flow unis. E.g., wih labor, L, and capial, K, as inpus: ( K L) Y = F,

5 Technology, Flexibiliy, and Reurns o Scale Inpu Flexibiliy How flexible is a firm s echnology? To obain a paricular oupu is i possible o subsiue one inpu for anoher? A wha rae? Changing Scale of Operaions If a firm doubles all inpus, wha happens o oupu? Reurns o scale.

6 Shor Run and Long Run Shor run: some facors of producion are fixed a predeermined levels. Example: in shor run, a firm canno easily vary he size of a plan. I can use machines more inensively. E.g.: ( ) K L Y = F,

7 Shor Run and Long Run Long run: all facors of producion can be varied. No fixed facors. How long is he long run? I depends on he specific ype of producion. For auomobile manufacurer i can ake years o change size of plan. For ravel agency monhs.

8 Fixed Proporions: y = min( x1, x ) 2 x x1 Isoquan

9 Perfec Subsiues: y = ax + x 1 2 x2 a x1

10 Cobb-Douglas x2 Producion funcion: f ( x x Ax a x b 1, 2) = x1 Isoquans: 1 y b x2 = 1 ( ) Ax a 1 b

11 The Marginal Produc Consider a firm using inpus ( x1, x2) Q: by how much is oupu going o increase if he firm uses a bi more of inpu 1, while keeping inpu 2 fixed? A: Marginal produc of facor 1: f ( x x 1, x2) 1

12 The Marginal Produc Typical assumpion in economics: he marginal produc of a facor decreases as we use more and more of ha facor E.g.: nurses producing radiographies using given machine

13 The Marginal Produc: Cobb- Douglas Consider Cobb-Douglas producion funcion in K and L: Y = AK a L b Marginal produc of capial: Marginal produc of labor: Y a 1 K Y L = = aak b bak a L L b 1

14 Technical Rae of Subsiuion x2 X TRS is he slope of an isoquan a a given poin X Measures he rae a which he firm has o subsiue one inpu for anoher o keep oupu consan x1

15 Compuing he TRS Q: Wha is he slope of an isoquan (TRS)? A: TRS( x 1, x 2 ) = f f ( x 1, x ( x 1, 1 x x 2 2 ) ) x 2

16 Compuing he TRS: Cobb- Douglas Cobb-Douglas case: Y = AK a L b TRS( K L, ) ak a 1 L b = a b 1 bk L = a b L K

17 Reurns o Scale Q: Wha happens o oupu when you double he amoun of all he inpus? A1: If oupu doubles, he echnology exhibis consan reurns o scale. E.g.: ravel agency ha uses office space and ravel agens as inpus.

18 Increasing Reurns o Scale A2: If oupu more han doubles, he echnology exhibis increasing reurns o scale. E.g.: oil pipeline. Double diameer and quadruple crosssecion of he pipe.

19 Increasing Reurns o Scale If here are increasing reurns, i is economically advanageous for a large firm o produce, raher han many small firms. Issue of monopoly.

20 Reurns o Scale A3: If oupu less han doubles, he echnology exhibis decreasing reurns o scale. Applies o firms wih large-scale operaions where i is more difficul o coordinae asks and mainain communicaion beween managers and workers.

21 Cobb-Douglas and Reurns o Scale Cobb-Douglas: f ( x x Ax a x b 1, 2) = 1 2 Scale inpus by a facor >1: ( ) a b a b a b f ( x1, x2) = A x1 x2 + = Ax1 x2

22 Cobb-Douglas and Reurns o Scale Scale inpus by a facor : >1 ( ) a b a b a b f ( x1, x2) = A x1 x2 + = Ax1 x2 a + b >1 If, hen : a+ b > a b f ( x1, x2) > Ax1 x2

23 Cobb-Douglas and Reurns o Scale Scale inpus by a facor : >1 ( ) a b a b a b f ( x1, x2) = A x1 x2 + = Ax1 x2 a + b = 1 If, hen : a+ b = a b f ( x1, x2) = Ax1 x2

24 Cobb-Douglas and Reurns o Scale Scale inpus by a facor : >1 ( ) a b a b a b f ( x1, x2) = A x1 x2 + = Ax1 x2 a + b < 1 If, hen : a+ b < a b f ( x1, x2) < Ax1 x2