1 / 25

# Marginal Analysis - PowerPoint PPT Presentation

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. Marginal Analysis: Outline. 1. Definition and Assumptions 2. Optimality criteria Analysis Interpretation

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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

vector

of resources

Optimality Conditionsfor 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’]

Optimality Conditions for Design: Results

• 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

Optimality Conditions: Graphical Interpretation of Costs

(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

Optimality Conditions: Graphical Interpretation of Results

(B) Conditions

Slope = MRSij = - MP1 / MP2

= - MC1 / MC2

X2

Isoquant

\$

X1

- typically assumed by economists

- in general, not valid

prices rise with demand

wholesale, volume discounts

Application of Optimality Conditions -- Conventional Case

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

Optimality Conditions: Example

• 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

Expansion Path: Non-Linear Prices

• 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”

Cost-Effectiveness

Economist’s view

Cost Functions: Graphical 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 = a0Xiai

• Linear input cost function: c(X) = piXi

• Result

• C(Y) = A(piai/r)Y1/r where r = ai

• Easy to estimate statistically

=> Solution for ‘ai’

=> Estimate of prod. fcn. Y = a0Xiai

• 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

• Some are fixed:

• C: polar

- Dfc: 40 kHz

• 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

• 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

• 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