managerial economics professor geoffrey heal 616 uris hall phone: (212) 854-6459 e-mail:...

71
Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: [email protected] (note - gmh “one” not “L”)

Upload: moses-ward

Post on 15-Jan-2016

217 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Managerial Economics

Professor Geoffrey Heal

616 Uris Hall

Phone: (212) 854-6459

e-mail: [email protected]

(note - gmh “one” not “L”)

Page 2: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Course Outline:

(I) Analyzing the structure of a market

Part A: Demand & Supply

Part B: Costs

(II) Pricing (most important part of course)

(III) e-con.com (application and review)

(IV) Foundations of Strategy

Page 3: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Aim: to understand key aspects of markets:

nature of demands for the products

closeness or otherwise of competitors

structure of costs

dependence of profits on the level of output

Analyzing the Structure of a Market

Page 4: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Material to be covered:

Analysis of demand

– demand curves, – price, income & cross elasticities of demand– use of demand parameters in forecasting

Structure of costs:– fixed & variable costs– break-even analysis– opportunity costs and sunk costs– learning curves & economies of scale.

Page 5: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

How much product should you produce and what price should you charge for it?

How can you best segment your market if there are different types of buyers with different demand characteristics (e.g., business travelers vs. vacation travelers, home PC buyers vs. corporate buyers)?

What are the types of pricing schemes available (e.g., bundling, promotional offers, loyalty bonuses, volume discounts)?

Pricing

Page 6: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

e-con.com

Applications of market analysis to electronic commerce

How does the internet affect demand, pricing, and other aspects of running a business.

e-commerce business strategies. Auctions and the internet.

Page 7: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Foundations of Strategy

Interacting with competitors Anticipating their reactions Forecasting the final outcome when

everyone has reacted.

Page 8: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Aim of Course

To teach you to use basic economic ideas in making business decisions.

Decisions about opening and closing businesses.

Decisions about pricing and other policies.

Page 9: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Level of Course

Emphasis on understanding concepts and where and how they can be used.

Don’t aim to make you an economist, but an intelligent consumer of economics.

Evaluate and understand works of consultants, staff. Ask the right questions.

Recognize BS when you see it!

Page 10: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)
Page 11: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Consumption & Price of Copper 1880-1998

Page 12: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)
Page 13: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Profit margin 1999/2001

Operating margin 1999/2001

MSFT

40.0%/30.5% 49.5%/46.3%

INTC 25%/17.7% 34.2%/20.7%

CPQ 1.5%/0.8% 2.4%/1.2%

DELL 7%/6.7% 9.9%/8%

Page 14: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Compare Internet companies

eBay AOL Yahoo Amazon.com

Page 15: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Demand and Supply

Page 16: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Demand Curve

Shows amount purchased as a function of price

Depends on:

- income

- tastes

- prices of competitive products

- prices of complementary products

Page 17: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Supply Curve

Amount offered for sale as a function of price

Depends on costs of production, which in turn depend on

- costs of inputs

- technology

Page 18: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The Market Mechanism

Quantity

D

S

The curves intersect atequilibrium, or market-

clearing, price. At P0 thequantity supplied is equalto the quantity demanded

at Q0 .

P0

Q0

Price($ per unit)

Page 19: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The Market Mechanism

Characteristics of the equilibrium or market clearing price: QD = QS

No shortage No excess supply No pressure on the price to change

Page 20: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Demand Curve -Income Rises

Page 21: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Demand Shifts

Page 22: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Supply shifts

Page 23: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

D & S shift

Page 24: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The Market Mechanism

D

S

Q1

Assume the price is P1 , then:1) Qs : Q1 > Qd : Q2 2) Excess supply is Q1:Q2.3) Producers lower price.4) Quantity supplied decreases

and quantity demanded increases.

5) Equilibrium at P2Q3

P1

Surplus

Q2 Quantity

Price($ per unit)

P2

Q3

Page 25: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The Market Mechanism

The market price is above equilibrium There is excess supply Producers lower prices Quantity demanded increases and quantity

supplied decreases The market continues to adjust until the

equilibrium price is reached.

A SurplusA Surplus

Page 26: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The Market Mechanism

D

S

Q1 Q2

P2

Shortage

Quantity

Price($ per unit)

Assume the price is P2 , then:1) Qd : Q2 > Qs : Q1

2) Shortage is Q1:Q2.3) Producers raise price.

4) Quantity supplied increases and quantity demanded decreases.

5) Equilibrium at P3, Q3

Q3

P3

Page 27: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The Market Mechanism

The market price is below equilibrium: There is a shortage Producers raise prices Quantity demanded decreases and quantity

supplied increases The market continues to adjust until the new

equilibrium price is reached.

ShortageShortage

Page 28: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The Market Mechanism

Market Mechanism - Summary:

1) Supply and demand interact to determine the market-clearing price.

2) When not in equilibrium, the market will adjust to alleviate a shortage or surplus and return the market to equilibrium.

3) Markets must be competitive for the mechanism to be efficient.

Page 29: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)
Page 30: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)
Page 31: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Consumption & Price of Copper 1880-1998

Page 32: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The Long-Run Behaviorof Natural Resource Prices

Observations Consumption of copper has increased about a

hundred fold from 1880 through 1998 indicating a large increase in demand.

The real price for copper has remained relatively constant.

Page 33: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

S1998

D1998

D1900

S1900

S1950

D1950

Long-Run Path of

Price and Consumption

Changes In Market Equilibrium

Quantity

Price

Page 34: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Conclusion Decreases in the costs of production have

increased the supply by more than enough to offset the increase in demand.

Changes In Market Equilibrium

Page 35: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Changes In Market Equilibrium

Wage Inequality in the United States

Real after-tax income from 1977 to 1999:

– Rose 40+% for the top 20% of the income distribution

– Fell 10+% for the bottom 20%

Page 36: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Changes In Market Equilibrium

Question

Why did the income distribution become more unequal for 1977 to 1999?

Page 37: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Price elasticity of demand:

Measures responsiveness of demand to price.

Defined as E = (Q/Q)/(P/P) = (Q/P)*(P/Q)

Why is it defined in proportional terms?

- Unit free.

- Scale sensitive.

A negative number.

Page 38: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)
Page 39: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Q = 8 - 2P or P = 4 - 0.5Q

Elasticity = (Q/Q)/(P/P) = (Q/P)*(P/Q) = -2*(P/Q)

Page 40: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Elasticity and Pricing

If elasticity is between 0 and -1 then raising price will raise profits - it will raise revenues and lower costs.

If elasticity is lower than -1 then raising price will lower revenues and also costs, so the effect on profits is not clear.

Moral - never operate where the elasticity is between 0 and -1.

Page 41: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Q = 8 - 2P

or

P = 4 - 0.5Q

so as revenue R is price times quantity

R = 4Q - 0.5Q2

Relationship between demand, quantity and revenue:

Page 42: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

PED = -1 PED = 0

Revenue rises as price rises

Revenue falls as price rises

Page 43: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

This is a quadratic pointing up.

The slope is:

R Q

which is zero at Q = 4.

Slope is positive for Q<4 and vice versa.

Maximum revenue comes when Q = 4, therefore P = 2, and max revenue is 8

= 4 - Q

Page 44: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

PED when revenue is maximum

Revenue is max when Q = 4, P = 2. E = (Q/Q)/(P/P) = (Q/P)*(P/Q) So E = (Q/P)*(1/2) and Q/P = -2 so E = -2 * 1/2 = -1 when R is

at a maximum.

Page 45: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The responsiveness of demand for good A to change in price of good B:

QA/QA = QA * PB

PB/PB PB PA

Example:

responsiveness of demand for Dell computers to

prices of Gateway computers

Cross price elasticity of demand:

Page 46: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Supply Elasticity

The responsiveness of supply to price changes.

(S/S)/(P/P), proportional change in supply divided by proportional change in price.

Usually positive.

Page 47: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Elasticities of Supply and Demand

1981 Supply Curve for Wheat QS = 1,800 + 240P

1981 Demand Curve for Wheat QD = 3,550 - 266P

The Market for WheatThe Market for Wheat

Page 48: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Elasticities of Supply and Demand

Equilibrium: Q S = Q D

PP 266550,3240800,1

750,1506 P

bushelP /46.3

bushels million 630,2)46.3)(240(800,1 Q

The Market for WheatThe Market for Wheat

Page 49: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The Market for WheatThe Market for Wheat

Elasticities of Supply and Demand

ED=(P/Q) (QD/P) = (3.46/2630)(-266)= -0.35 ES=(P/Q) (QS/P) = (3.46/2630)(+240)= 0.32

Page 50: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

1981 1800 + 240P 3550 - 266P 1800+240P = 3550-266P506P = 1750

P1981 = $3.46/bushel

1998 1,944 + 207P 3,244 - 283P 1,944+207P = 3,244-283P P1998 = $2.65/bushel

Supply (Qs) Demand (QD) Equilibrium Price (Qs = QD)

Changes in the Market: 1981-1998The Market for WheatThe Market for Wheat

Page 51: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Marginal Revenue

Increase in revenue from one extra sale Rate of change of revenue with respect to

sales Typically less than price as demand curve

slopes down Depends on PED

Page 52: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Marginal Revenue & PED

MR = P{1 + 1/PED} Remember PED < 0 so MR < P. The larger PED as a number the nearer MR is to P If PED = - 1, then MR = 0. (Top of revenue curve) ………………………………………… Derivation - dR/dQ = d{P(Q).Q}/dQ = P + Q*dP/dQ = P{1 + (Q/P)*dP/dQ}

Page 53: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Responsiveness of demand to changes in income

IED = (Q/Q)/I/I) = (Q/I)*(I/Q)

Use to define necessities and luxuries

Income Elasticity of Demand:

Page 54: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Necessities - IED < 1

Luxuries - IED > 1

Cyclical vs. defensive sectors

Cyclical - high IED - foreign travel, consumer durables

Defensive - low IED - food, utilities

Page 55: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)
Page 56: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)
Page 57: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Critical in understanding oil market, energy markets, metal markets

Responding to a price movement takes time - possibly many years

Long-run elasticity measures total responseShort-run elasticity measures immediate response

Short-run vs. long-run elasticities

Page 58: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Short-run dropin demand

Long-run drop in demand

Po

P1

Short-rundemand

Long-run demand

Page 59: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Price -0.11 -0.22 -0.32 -0.49 -0.82 -1.17

Income 0.07 0.13 0.20 0.32 0.54 0.78

Years Following Price or Income Change

Elasticity 1 2 3 4 5 6

The Demand for GasolineThe Demand for Gasoline

Short-Run Versus Long-Run Elasticities

Page 60: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Price -1.20 -0.93 -0.75 -0.55 -0.42 -0.40

Income 3.00 2.33 1.88 1.38 1.02 1.00

Years Following Price or Income Change

Elasticity 1 2 3 4 5 6

The Demand for AutomobilesThe Demand for Automobiles

Short-Run Versus Long-Run Elasticities

Page 61: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Data Explains:

1) Why the price of oil did not continue to rise above $30/barrel even though it rose very rapidly in the early 1970s.

2) Why automobile sales are so sensitive to the business cycle.

Short-Run Versus Long-Run Elasticities

The Demand forGasoline and Automobiles

The Demand forGasoline and Automobiles

Page 62: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

The World Oil Market

In 1995: P* = $18/barrel World demand and total supply = 23 bb/yr (= 63

mbd) OPEC supply = 10 bb/yr (= 27 mbd) Non-OPEC supply = 13 bb/yr (= 35 mbd) US consumption about 17 mbd = 5.5 bb/yr

Page 63: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Price of Crude Oil

Page 64: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)
Page 65: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

D

Quantity(billions barrels/yr)

Price($ per barrel)

5

18

ST

0 5 15 20 25 30 3510

10

15

20

25

30

35

40

45

23

Impact of Saudi Production CutSC

Short-RunEffect

S’T

Page 66: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

D

Quantity(billions barrels/yr)

Price($ per barrel)

5

ST

0 5 15 20 25 30 3510

10

15

20

25

30

35

40

45

23

18

Impact of Saudi Production Cut

SC

Due to the elasticityof the long-run

supply and demand curves, the long-run

effect of a cutin production is

much less.

S’T Long-run Effect

Page 67: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

AMAX Case

Page 68: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Price (1980 $)

0

1

2

3

4

5

6

7

8

9

10

1975 1976 1977 1978 1979 1980

Year

Price (1980 $)

Page 69: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

Moly Consumption & Production

0

50

100

150

200

250

1975 1976 1977 1978 1979

Year

Consumption

Production

Page 70: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)

MC

0

2

4

6

8

10

12

140

25

50

75

10

0

12

5

15

0

17

5

20

0

22

5

25

0

27

5

30

0

32

5

35

0

Output

Ma

rgin

al C

os

t

MC

Marginal Costs

Page 71: Managerial Economics Professor Geoffrey Heal 616 Uris Hall Phone: (212) 854-6459 e-mail: gmh1@columbia.edu (note - gmh “one” not “L”)