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Marginal Analysis

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Purposes:

1.To present a basic application of constrained optimization

2. Apply to Production Function to get criteria and patterns of optimal system design

1.Definition and Assumptions

2.Optimality criteria

- Analysis
- Interpretation
- Application
3.Key concepts

- Expansion path
- Cost function
- Economies of scale
4.Summary

- Basic form of optimization of design
- Combines:
Production function - Technical efficiency

Input cost function, c(X)

Economic efficiency

- Feasible region is convex (over relevant portion) This is key. Why?
- To guarantee no other optimum missed

- No constraints on resources
- To define a general solution

- Models are “analytic” (continuously differentiable)
- Finds optimum by looking at margins -- derivatives

Fixed level of output

vector

of resources

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’]

balanced design

- Key Result:
c(X) / Xi = g(X) / Xi

- Optimality Conditions:
MPi / MCi = MPj / MCj = 1 /

MCj / MPj = MCi / MPi = = Shadow Price on Product

- A
Each Xi contributes “same bang for buck”

marginal cost

marginal product

X2

X1

(A) Input Cost Function

Linear Case: B = Budget

c(X) = piXi B

B/P2

P2

General case:

Budget is non-linear

(as in curved line)

P1

B/P1

(B) Conditions

Slope = MRSij = - MP1 / MP2

= - MC1 / MC2

X2

Isoquant

$

X1

Note: Linearity of Input Cost Function

- typically assumed by economists

- in general, not valid

prices rise with demand

wholesale, volume discounts

Problem: Y = a0X1a1 X2a2

c(X) = piXi

Solution:[a1 / X1*] Y / p1 = [a2 / X2*] Y / p2 ==> [a1 / X1*] p1 = [a2 / X2*] / p2

( * denotes an optimum value)

Assume Y = 2X10.48 X20.72 (increasing RTS)

c(X) = 5X1 + 3X2

Apply Conditions: = MPi / MCi

[a1 / X1*] / p1 = [a2 / X2*] / p2

[0.48 /5] / X1* = [ 0.72/ 3] / X2*

9.6 / X1* = 24 / X2*

This can be solved for a general relationship between resources => Expansion Path

- Locus of all optimal designs X*
- Not a property of technical system alone
- Depends on local prices
- Optimal designs do not, in general, maintain constant ratios between optimal Xi*
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

X2

Y

Expansion Path

X1

- Assume:Y = 2X10.48 X20.72 (increasing RTS)
c(X) = X1 + X21.5 (increasing costs)

- Optimality Conditions:
(0.48 / X1) Y / 1 = (0.72 / X2) Y / (1.5X20.5)

=> X1* = (X2*)1.5

- Graphically:

- Not same as input cost function
It represents the optimal cost of Y

Not the cost of any set of X

- C(Y) = C(X*) = f (Y)
- 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”

Engineer’s view

Cost-Effectiveness

Economist’s view

- Graphically:
- Great practical use:
How much Y for budget?

Y for B?

Cost effectiveness, B / Y

C(Y)

Y

Feasible

Feasible

Y

C(Y)

- Cobb-Douglas Prod. Fcn: Y = a0Xiai
- Linear input cost function: c(X) = piXi
- Result
- C(Y) = A(piai/r)Y1/r where r = ai

=> Solution for ‘ai’

=> Estimate of prod. fcn. Y = a0Xiai

- Assume Again: Y = 2X10.48 X20.72
c(X) = X1 + X21.5

- Expansion Path:X1* = (X2*)1.5
- Thus: Y = 2(X2*)1.44
c(X*) = 2(X2*)1.5

=> X2* = (Y/2)0.7

c(Y) = c(X*) = (2-0.05)Y1.05

- System Design for Satellites at various altitudes and configurations
- Source: O. de Weck and MIT co-workers
- Data generated by a technical model that costs out wide variety of possible designs
- Establishes a Cost Function
- Note that we cannot in general expand along cost frontier. Technology limits what we can do: Only certain pathways are available

- Each star in the Trade Space corresponds to a vector:
- r: altitude of the satellites
- e: minimum acceptable elevation angle

- C: constellation type
- Pt: maximal transmitter power
- DA: Antenna diameter
- Dfc: bandwidth

- ISL: intersatellite links
- Tsat: Satellites lifetime

- r: altitude of the satellites
- Some are fixed:
- C: polar
- Dfc: 40 kHz

- C: polar

- r= 400 km
- = 35 deg
Nsats=1215

Lifecycle cost [B$]

Constants:

Pt=200 W

DA=1.5 m

ISL= Yes

Tsat= 5

- r= 400 km
- = 20 deg
Nsats=416

- r= 2000 km
- = 5 deg
Nsats=24

- r= 800 km
- = 5 deg
Nsats=24

- r= 400 km
- = 5 deg
Nsats=112

System Lifetime capacity [min]

- r= 400 km
- = 35 deg
N= 1215

Lifecycle cost [B$]

Constants:

Pt=200 W

DA=1.5 m

ISL= Yes

Tsat= 5

e = 35 deg

e = 20 deg

- r= 2000 km
- = 5 deg
N=24

System Lifetime capacity [min]

- A possible characteristic of cost function
- Concept similar to returns to scale, except
- ratio of ‘Xi’ not constant
- refers to costs (economies) not “returns”
- Not universal (as RTS) but depends on local costs

- Economies of scale exist if costs increase slower than product
Total cost = C(Y) = Y < 1.0

- If Cobb-Douglas, linear input costs, Increasing RTS (r = ai > 1.0
=> Economies of scale

Optimality Conditions: [a1/X1*] / p1 = [a2/X2*] / p2

Thus: Inputs Cost is a function of X1*

Also: Output is a function of [X1* ] r

So: X1* is a function of Y1/r

Finally: C(Y) = function of [ Y1/r ]

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*) = (2-0.05)Y1.05

Contrarily, if unit prices decrease with volume (quite common) we can have Economies of Scale, without Returns to Scale

- Assumptions --
-- convex feasible region

-- Unconstrained

- Optimality Criteria
- MC/MP same for all inputs

- Expansion path -- Locus of Optimal Design
- Cost function -- cost along Expansion Path
- Economies of scale (vs Returns to Scale)
-- Exist if Cost/Unit decreases with volume