Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 1 of 25 Marginal Analysis Purposes: 1.To present a basic application of constrained optimization 2. Apply to Production Function to get criteria and patterns of optimal system design

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 2 of 25 Marginal Analysis: Outline 1.Definition and Assumptions 2.Optimality criteria  Analysis  Interpretation  Application 3.Key concepts  Expansion path  Cost function  Economies of scale 4.Summary

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 3 of 25 Marginal Analysis: Concept l Basic form of optimization of design l Combines: Production function - Technical efficiency Input cost function, c(X) Economic efficiency

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 4 of 25 Marginal Analysis: Assumptions l Feasible region is convex (over relevant portion) This is key. Why?  To guarantee no other optimum missed l No constraints on resources  To define a general solution l Models are “analytic” (continuously differentiable)  Finds optimum by looking at margins -- derivatives

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 5 of 25 Optimality Conditions for Design, by Marginal Analysis The Problem: Min C(Y’) = c(X) cost of inputs for any level of output, Y’ s.t. g(X) = Y’ production function The Lagrangean: L = c(X) - [g(X) - Y’] vector of resources Fixed level of output

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 6 of 25 Optimality Conditions for Design: Results l Key Result:  c(X) /  X i =  g(X) /  X i l Optimality Conditions: MP i / MC i = MP j / MC j = 1 / MC j / MP j = MC i / MP i = = Shadow Price on Product l A Each X i contributes “same bang for buck” balanced design marginal costmarginal product

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 7 of 25 Optimality Conditions: Graphical Interpretation of Costs (A) Input Cost Function X2X2 X1X1 P2P2 P1P1 B/P 2 B/P 1 Linear Case: B = Budget c(X) =  p i X i  B General case: Budget is non-linear (as in curved line)

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 8 of 25 Optimality Conditions: Graphical Interpretation of Results (B) Conditions X2X2 X1X1 Slope = MRS ij = - MP 1 / MP 2 = - MC 1 / MC 2 Isoquant $

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 9 of 25 Application of Optimality Conditions -- Conventional Case Problem: Y = a 0 X 1 a1 X 2 a2 c(X) =  p i X i Note: Linearity of Input Cost Function - typically assumed by economists - in general, not valid prices rise with demand wholesale, volume discounts Solution:[a 1 / X 1 *] Y / p 1 = [a 2 / X 2 *] Y / p 2 ==> [a 1 / X 1 *] p 1 = [a 2 / X 2 *] / p 2 ( * denotes an optimum value)

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 10 of 25 Optimality Conditions: Example Assume Y = 2X X (increasing RTS) c(X) = 5X 1 + 3X 2 Apply Conditions: = MP i / MC i [a 1 / X 1 *] / p 1 = [a 2 / X 2 *] / p 2 [0.48 /5] / X 1 * = [ 0.72/ 3] / X 2 * 9.6 / X 1 * = 24 / X 2 * This can be solved for a general relationship between resources => Expansion Path

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 11 of 25 Expansion Path l Locus of all optimal designs X* l Not a property of technical system alone l Depends on local prices l Optimal designs do not, in general, maintain constant ratios between optimal X i * Compare: crew of 20,000 ton ship crew of 200,000 ton ship larger ship does not need 10 times as many sailors as smaller ship

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 12 of 25 Expansion Path: Non-Linear Prices l Assume:Y = 2X X (increasing RTS) c(X) = X 1 + X (increasing costs) l Optimality Conditions: (0.48 / X 1 ) Y / 1 = (0.72 / X 2 ) Y / (1.5X ) => X 1 * = (X 2 *) 1.5 l Graphically: X2X2 Y X1X1 Expansion Path

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 13 of 25 Cost Functions: Concept l Not same as input cost function It represents the optimal cost of Y Not the cost of any set of X l C(Y) = C(X*) = f (Y) l It is obtained by inserting optimal values of resources (defined by expansion path) into input cost and production functions to give “best cost for any output”

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 14 of 25 Cost Functions: Graphical View l Graphically: l Great practical use: How much Y for budget?  Y for  B? Cost effectiveness,  B /  Y C(Y) Y Feasible Y C(Y) Economist’s view Engineer’s view Cost-Effectiveness

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 15 of 25 Cost Function Calculation: Linear Costs Cobb-Douglas Prod. Fcn: Y = a 0  X i a i Linear input cost function: c(X) =  p i X i l Result C(Y) = A(  p i a i / r )Y 1/r where r =  a i l Easy to estimate statistically => Solution for ‘a i ’ => Estimate of prod. fcn. Y = a 0  X i a i

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 16 of 25 Cost Function Calculation: Example l Assume Again: Y = 2X X c(X) = X 1 + X l Expansion Path:X 1 * = (X 2 *) 1.5 l Thus: Y = 2(X 2 *) 1.44 c(X*) = 2(X 2 *) 1.5 => X 2 * = (Y/2) 0.7 c(Y) = c(X*) = ( )Y 1.05

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 17 of 25 Real Example: Communications Satellites l System Design for Satellites at various altitudes and configurations l Source: O. de Weck and MIT co-workers l Data generated by a technical model that costs out wide variety of possible designs l Establishes a Cost Function l Note that we cannot in general expand along cost frontier. Technology limits what we can do: Only certain pathways are available

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 18 of 25 Key Parameters l Each star in the Trade Space corresponds to a vector:  r: altitude of the satellites  : minimum acceptable elevation angle  C: constellation type  Pt: maximal transmitter power  DA: Antenna diameter  f c : bandwidth  ISL: intersatellite links  T sat : Satellites lifetime l Some are fixed:  C: polar  f c : 40 kHz

Trade Space

Path example Constants: Pt=200 W DA=1.5 m ISL= Yes T sat = 5 r= 2000 km  = 5 deg N sats =24 Lifecycle cost [B$] System Lifetime capacity [min] r= 800 km  = 5 deg N sats =24 r= 400 km  = 5 deg N sats =112 r= 400 km  = 20 deg N sats =416 r= 400 km  = 35 deg N sats =1215

Tree example Constants: Pt=200 W DA=1.5 m ISL= Yes T sat = 5 r= 2000 km  = 5 deg N=24 e = 20 deg e = 35 deg r= 400 km  = 35 deg N= 1215 Lifecycle cost [B$] System Lifetime capacity [min]

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 22 of 25 Economies of Scale: Concept l A possible characteristic of cost function l Concept similar to returns to scale, except  ratio of ‘X i ’ not constant  refers to costs (economies) not “returns”  Not universal (as RTS) but depends on local costs l Economies of scale exist if costs increase slower than product Total cost = C(Y) = Y   < 1.0

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 23 of 25 Economies of Scale: Specific Cases If Cobb-Douglas, linear input costs, Increasing RTS (r =  a i > 1.0 => Economies of scale Optimality Conditions: [a 1 /X 1 *] / p 1 = [a 2 /X 2 *] / p 2 Thus: Inputs Cost is a function of X 1 * Also: Output is a function of [X 1 * ] r So: X 1 * is a function of Y 1/r Finally: C(Y) = function of [ Y 1/r ]

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 24 of 25 Economies of Scale: General Case Not necessarily true in general that Returns to scale => Economies of Scale Increasing costs may counteract advantages of returns to scale See example!! c(Y) = c(X*) = ( )Y 1.05 Contrarily, if unit prices decrease with volume (quite common) we can have Economies of Scale, without Returns to Scale

Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Marginal AnalysisSlide 25 of 25 Marginal Analysis: Summary l Assumptions convex feasible region -- Unconstrained l Optimality Criteria  MC/MP same for all inputs l Expansion path -- Locus of Optimal Design l Cost function -- cost along Expansion Path l Economies of scale (vs Returns to Scale) -- Exist if Cost/Unit decreases with volume