2. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 1 of 34 Production Functions (PF) Outline 1.Definition 2.Technical Efficiency 3.Mathematical Representation 4.Characteristics

3. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 2 of 34 Production Function - Basic Model for Modeling Engineering Systems l Definition: — Represents technically efficient transform of physical resources X = (X 1 …X n ) into product or outputs Y (may be good or bad) l Example: — Use of aircraft, pilots, fuel (the X factors) to carry cargo, passengers and create pollution (the Y) l Typical focus on 1-dimensional output

4. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 3 of 34 Technical Efficiency l A Process is Technically Efficient if it provides Maximum product from a given set of resources X = X 1,... X n l Graph: Max Output Resource Feasible Region Note

5. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 4 of 34 Mathematical Representation -- General l Two Possibilities l Deductive -- Economic — Standard economic analysis — Fit data to convenient equation — Advantage - ease of use — Disadvantage - poor accuracy l Inductive -- Engineering — Create system model from knowledge of details — Advantage - accuracy — Disadvantage - careful technical analysis needed contrastcontrast

6. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 5 of 34 Mathematical Representation -- Deductive Standard Cobb-Douglas Production Function Y = a 0  X i a i = a 0 X 1 a 1... X n a n [  means multiplication] — Interpretation: ‘a i ’ are physically significant — Easy estimation by linear least squares log Y = a 0 +  a i log X i l Translog PF -- more recent, less common — log Y = a 0 +  a i log X i +  a ij log X i log X j — Allows for interactive effects — More subtle, more realistic l Economist models (no technical knowledge)

7. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 6 of 34 PF Example l One of the advantage of the “economist” models is that they make calculations easy. This is good for examples, even if not as realistic as Technical Cost Models (next) l Thus: Output = 2 M 0.4 N 0.8 l Let’s see what this looks like...

8. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 7 of 34 PF Example -- Calculation

9. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 8 of 34 PF Example -- Graphs

10. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 9 of 34 PF Example -- Graphs

11. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 10 of 34 Mathematical Representation -- Inductive l “Engineering models” of PF l Analytic expressions — Rarely applicable: manufacturing is inherently discontinuous — Exceptions: process exists in force field, for example transport in fluid, river l Detailed simulation, Technical Cost Model — Generally applicable — Requires research, data, effort — Wave of future -- not yet standard practice

12. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 11 of 34 Cooling Time, Part Weight, Cycle Time Correlation (MIT MSL, Dr.Field)

13. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 12 of 34 PF: Characteristics l Isoquants l Marginal Products l Marginal Rates of Substitution l Returns to Scale l Possible Convexity of Feasible Region

14. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 13 of 34 Characteristic: Isoquants l Isoquant is the Locus (contour) of equal product on production function l Graph: Y Xi Xj Isoquant Projection Production Function Surface

15. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 14 of 34 Important Implication of Isoquants l Many designs are technically efficient — All points on isoquant are technically efficient — no technical basis for choice among them — Example: * little land, much steel => tall building * more land, less steel => low building l Best System Design depends on Economics l Values are decisive

16. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 15 of 34 Isoquant Example -- Calculation

17. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 16 of 34 Isoquant Example -- Graph

18. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 17 of 34 Characteristic: Marginal Products l Marginal Product is the change in output as only one resource changes MP i =  Y/  X i l Graph: MP i XiXi

19. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 18 of 34 Diminishing Marginal Products l Math: Y = a 0 X 1 a 1... X i a i...X n a n  Y/  X i = (a i /X i )Y = f (X i a i -1 ) Diminishing Marginal Product if a i < 1.0 l “Law” of Diminishing Marginal Products — Commonly observed -- but not necessary — “Critical Mass” phenomenon => increasing marginal products

20. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 19 of 34 MP Example -- Calculations

21. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 20 of 34 MP Example -- Graph

22. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 21 of 34 Characteristic: Marginal Rate of Substitution l Marginal Rate of Substitution is the Rate at which one resource must substitute for another so that product is constant l Graph: XiXi XjXj XiXi XjXj Isoquant

23. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 22 of 34 Marginal Rate of Substitution (cont’d) Math: since  X I MP I +  X J MP J = 0 (no change in product) then MRS IJ =  X J /  X I = - MP I / MP J = - [(a I / X I ) Y] / [(a J / X J ) Y] = - (a I /a J ) (X J /X I ) l MRS is “slope” of isoquant — It is negative — Loss in 1 dimension made up by gain in other

24. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 23 of 34 MRS Example l For our example PF: Output = 2 M 0.4 N 0.8 l a M = 0.4 ; a N = 0.8 l At a specific point, say M = 5, N = 12.65 l MRS = - (0.4 / 0.8) (12.65 / 5) = - 1.265 l At that point, it takes ~ 5/4 times as much M as N to get the same change in output

25. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 24 of 34 Characteristic: Returns to Scale l Returns to Scale is the Ratio of rate of change in Y to rate of change in ALL X (each X i changes by same factor) l Graph: — Directions in which the rate of change in output is measured for MP and RTS XjXj XiXi MP j MP i RTS – along a ray from origin

26. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 25 of 34 Returns to Scale (cont’d) Math: Y’= a 0  X i a i Y’’= a    (sX i ) a i = Y’(s)  a all inputs increase by s RTS = (Y”/Y’)/s = s (  a i - 1) Y”/Y’ = % increase in Y if Y”/Y’ > s => Increasing RTS Increasing returns to scale (IRTS) if  a i > 1.0

27. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 26 of 34 Increasing RTS Example l The PF is: Output = 2 M 0.4 N 0.8 — Thus  a i = 0.4 + 0.8 = 1.2 > 1.0 — So the PF has Increasing Returns to Scale — Compare outputs for (5,10), (10,20), (20,40)

28. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 27 of 34 Importance of Increasing RTS l Increasing RTS means that bigger units are more productive than small ones l IRTS => concentration of production into larger units l Examples: — Generation of Electric power — Chemical, pharmaceutical processes

29. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 28 of 34 Practical Occurrence of IRTS l Frequent! l Generally where * Product = f (volume) and * Resources = f (surface) l Example: * ships, aircraft, rockets * pipelines, cables * chemical plants * etc.

30. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 29 of 34 Characteristic: Convexity of Feasible Region l A region is convex if it has no “reentrant” corners l Graph: CONVEX NOT CONVEX

31. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 30 of 34 Informal Test for Convexity of Feasible Region (cont’d) Math:If A, B are two vectors to any 2 points in region Convex if all T = KA + (1-K)B0  K  1 entirely in region   Origin

32. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 31 of 34 Convexity of Feasible Region for Production Function l Feasible region of Production function is convex if no reentrant corners l Convexity => Easier Optimization — by linear programming (discussed later) Y X Y X Convex Non- Convex

33. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 32 of 34 Test for Convexity of Feasible Region of Production Function Test for Convexity: Given A,B on PF If T = KA + (1-K)B0  K  1 Convex if all T in region For Cobb-Douglas, the test is if: all a i  1.0 and  a i  1.0 Y X Y X A B T A T B

34. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 33 of 34 Convexity Test Example l Example PF has Diminishing MP, so in the MP direction it looks like left side l But: it has IRTS, like bottom of right side l Feasible Region is not convex

35. Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Production Function Slide 34 of 34 Summary l Production models are the way to describe technically efficient systems l Important characteristics — Isoquants, Marginal products, Marginal rates of Substitution, Returns to scale, possible convexity l Two ways to represent — Economist formulas — Technical models (generally more accurate)