Benefit-Cost Analysis FGS - Ch. 4 © Allen C. Goodman 2013.

Benefit-Cost Analysis

FGS - Ch. 4

© Allen C. Goodman 2013

What is the Right Amount?

• Economists usually rely on market solutions, but what if we don’t have markets?

• What kind of mechanism can we devise?

Some First Principles

What is the “right” amount of a good to provide for society?

Let’s look at consumers surplus and producers surplus.

0 Q 1

C o n s u m e r s ’ S u r p lu s

P r ic e

Q u a n ti ty

P 1

D e m a n d

C o n s u m e r E x p e n d i tu r e s

F ig u r e 4 - 1 C o n s u m e rs ’ S u rp lu s

P 1

More consumers surplus makes consumers happier!

AdditionalSurplus

Some First Principles

What is the “right” amount of a good to provide for society?

Let’s look at consumers surplus and producers surplus.

More producers surplus makes producers happier!

Producer

P ri ce

Q u a n tit y

P 1

F ig u re 4 -2 P ro d u c ers ’ S u rp lu s

Q 1 0

Producers’ Surplus

Cos t to Selle rs

S u p p ly

P 1

AdditionalSurplus

What’s the “right” quantity?

• We seek to maximize sum of CS + PS.

• At Q < Q1, ↑ Q ↑ both CS and PS.

• At Q > Q1, ↑ Q costs more (S) than it is worth (D).

D em a n d

C o n s u m e r s ’ S u r p lu s

P r o d u c e r C o s t t o S e l l e r s

P r ic e

Q u an tity

P 1

S u p p ly

Q 1

F igu re 4-3 E ff ic ien t Q u an tity

0

P r o d u c e r s ’ S u r p lu s

P 1

D em a n d

C o n s u m e r s ’ S u r p lu s

P r o d u c e r C o s t t o S e l l e r s

P r ic e

Q u an tity

P 1

S u p p ly

Q 1

F igu re 4-3 E ff ic ien t Q u an tity

0

P r o d u c e r s ’ S u r p lu s

P 1

What’s the “right” quantity?

• We seek to maximize sum of CS + PS.

• At Q < Q1, ↑ Q ↑ both CS and PS.

+

What’s the “right” quantity?

• We seek to maximize sum of CS + PS.

• At Q > Q1, ↑ Q costs more (S) than it is worth (D).

D em a n d

C o n s u m e r s ’ S u r p lu s

P r o d u c e r C o s t t o S e l l e r s

P r ic e

Q u an tity

P 1

S u p p ly

Q 1

F igu re 4-3 E ff ic ien t Q u an tity

0

P r o d u c e r s ’ S u r p lu s

P 1

SocietalCosts

What’s the “right” quantity?

• We seek to maximize sum of CS + PS.

• At Q > Q1, ↑ Q costs more (S) than it is worth (D).

D em a n d

C o n s u m e r s ’ S u r p lu s

P r o d u c e r C o s t t o S e l l e r s

P r ic e

Q u an tity

P 1

S u p p ly

Q 1

F igu re 4-3 E ff ic ien t Q u an tity

0

P r o d u c e r s ’ S u r p lu s

P 1

SocietalCosts

SocietalBenefits

-

Key Point

• Efficiency is ALL ABOUT Q!

• A monopolist is BAD because Q* < Q1.

D em a n d

C o n s u m e r s ’ S u r p lu s

P r o d u c e r C o s t t o S e l l e r s

P r ic e

Q u an tity

P 1

S u p p ly

Q 1

F igu re 4-3 E ff ic ien t Q u an tity

0

P r o d u c e r s ’ S u r p lu s

P 1

• BUT, a perfectly discriminating monopolist appropriates all of the CS.

• Eq’m quantity is EFFICIENT!

Q*

MR

Benefit-Cost Analysis

• In a sense, everything economists do is benefit-cost analysis.

• Competitive markets get us to the “right” amount.

• Why don’t we just depend on markets?

Benefit-Cost

This is of particular concern with the public health sector, in which you are considering various types of public interventions.

Prime example, and a very successful one, is fluoridation of water. It is something that most (although not all) will agree has been profoundly successful. Yet, it is unlikely to be considered on a nonpublic basis. Moreover, it may be subject to substantive economies of scale.

It is also useful to consider the aspects of the jargon, that often get confused.

Nelson and Swint, 1976

• Performed a prospective cost-benefit analysis of fluoridating a segment of the water supply for Houston, Texas,

• Explicitly introduced and evaluated the time pattern of the costs and benefits. Showed that neglect of the time structure of the costs and benefits would significantly bias the results.

• A benefit-cost ratio of 1.51 and a net present value (or “social profit”) of $1,102,970 were found. The results are biased downwards and should be considered a lower bound.

W Nelson, J M Swint Cost-benefit analysis of fluoridation in Houston, TexasJournal of public health dentistry. 01/02/1976; 36(2):88-95.

ISSN: 0022-4006

Terms• Efficiency Marginal Benefit = Marginal Cost. In principle, it

would pay to do all projects up to where marginal benefit = marginal cost. This is our standard economic analysis.

• Benefit-Cost A way of ranking alternative projects, that typically aren't brought forward by the market. We want to consider health care interventions, and I'll do some analytical stuff in a moment. In a sense, it tries to provide some market signals for goods for which markets do not exist.

• Cost-Effectiveness (Efficiency) This is often confused, particularly by non-economists. It does not require satisfying any type of efficiency calculation. Basically, it assumes that a chosen project that is beneficial. You then want to consider the cheapest way to produce it. DOES NOT imply efficiency.

TB, TC

Quantity0

TC

TB

W = TB(Q) – TC(Q)dW/dQ = TB'(Q) - TC'(Q) = 0MB = MC

TB, TC

Quantity0

TB

B/C > 1

TC

Efficient (MB=MC)

Cost Efficient – everywhere on this curve

W = TB(Q) – TC(Q)dW/dQ = TB'(Q) - TC'(Q) = 0MB = MC

BMB

Q

CMC

Q

Exercise

TC = a + bQ + cQ2, (b, c >0)

TB = d + eQ + fQ2, (e>0, f<0)

Calculate: Q*,

Q | B/C 1

Flu Vaccines

A good example with which to look at a health care problem that requires some sorts of public interventions is flu vaccinations. In this type of situation, community health becomes a public stock. If you are vaccinated, I am likely to be more healthy.

Consider a simple n person world. For each person, well-being depends on the consumption of numeraire good x and the production of health H, which comes from input (inoculation) i.

So each is optimizing:

Flu (2)

Each person’s well-being depends on the consumption of numeraire good x and the production of health H, which comes from input (inoculation) i. p is the price of an inoculation.

U1 = U1 [x1, H1 (I)] + 1 (y1 - x1 - pi1)

U2 = …

… …

Un = Un [x2, Hn (I)] + n (yn - xn - pin)

I = Σj ij

Flu (3)

Optimizing w.r.t. xj, ij, we get:

Uj1 - j = 0

Uj2Hj' - jp = 0, leading to:

Uj2Hj'/Uj

1 = p.

This indicates the market level for Person j.

BUT, is it optimal?

Inoculations

$

p

Uj2Hj'/Uj

1

ij*

External Benefits

Measuring Benefits

• A key feature of benefit-cost analysis is measurement of the benefits.

• Key in the measurement of the benefits is the estimation of the willingness-to-pay for them. This is the inverse demand curve.

• In contrast to situation where we are saying “here is the price; how much are you willing to buy?” we say instead, “here is an amount; how much would you be

willing to pay?”

Willingness to pay

• One of the major problems is that since we do not usually have market signals (which is why we are doing benefit cost analysis), we have to guess what the willingness to pay is. We could save thousands of lives by lowering the speed limit to 15 M.P.H. Why don't we?

• We have moved to automobiles that are much much cleaner than they were in the 1950s and 1960s. There is an interesting question as to how we measure the benefits of the cleaner cars, as opposed to the costs. Many studies argue that we have cars that are essentially cleaner than optimal, given the marginal benefits.

QALYs

• Health community has resisted putting a $ value on health benefits. There are a lot of equity considerations:– Should the lives of poor people, elderly, be valued

differently than the lives of others?– Lots of this moves from economics to ethics.

• Health community has embraced the idea of Quality Adjusted Life Years, or QALYs. Idea is to adjust incremental years by the quality of life.

Example

• Someone faces an intervention (rather than dying) that can increase the expected time of death from age 70 to age 90.

• For the first 10 years, life will be fine. For the next 10, not so good.

• Each of the first 10 year increment is equivalent to 1 QALY. Each of the next 10 is equivalent to 0.5 QALY.

• So, the effectiveness of the intervention is:– 10 years * (1 QALY/year) + 10 years* (0.5 QALY/year) = 15

QALYs.

• This is your denominator.• Then, calculate cost/QALY.

Several Non-Trivial Issues

• What about children? How do we evaluate their QALYs?

• Who evaluates their QALYs?

• Do you add adult + children's QALYS?

• How are QALYs developed?

Ed and Harry

• Start at point M. Assume that Ed can gain health at a lower incremental cost than Harry. Hence, a given level of expenditures will give more incremental QALYs for Ed than for Harry.

• That’s why (Emax - E1) > (Hmax - H1).

Geometricpresentation below

Ed and Harry

10

10

30

20

Ed

Harry

M

What do we find?

• Conventional production-possibility frontier.• Equal outcomes @ 45o line.• Maximum production is tangent to a line w/ slope = -1.0.

( ) ( ) ( ) E E H H E HQ R Q R R R R

0

E

E E

Q

R R

0

H

H H

Q

R R

E H

E H

Q Q

R R

E H

E E

Q Q

R R

1

E

H

Q

Q

dRE=-dRHdRE=-dRH

MPs are equal!

Harry and Ed

• What if we think that Harry and Ed should have the same QALYs? Draw 45 degree line.

10

10

30

20

Ed

Harry

• What if we think that Harry and Ed should get the same inputs?

45o

• Why?

SH = SE 8

Slope = -1

Next Time

• Cost-Benefit readings from JHE

• Readings from Elgar– Reading 35– Reading 37– Reading 42

• Applied CBA.

Supplemental Material

• Remainder of Slides

Ed and Harry• At age 10, Harry and Ed

both have certain levels of health, 10 each.

• Assume that Ed (easy) can gain health at a lower incremental cost than Harry (hard). Hence, a given level of expenditures will give Ed 20 incremental points but would give Harry only 10.

• Suppose half of the people are like Ed and half are like Harry.

10

10

30

20

Ed

Harry

Geometric Treatment

Harry and Ed

• What if we think that Harry and Ed should have the same QALYs? Draw 45 degree line.

10

10

30

20

Ed

Harry

• What if we think that Harry and Ed should get the same inputs?

45o

• Why?

SH = SE 8

What’s the most cost-effective place?

• Thought experiment. Most cost effective place is where we get the highest mean score. Why?

10

10

30

20

Ed

Harry

45o

• We can draw a line with a slope of –1. This line gives us places with equal totals. Start with S = SE + SH = 10.

SE+SH=10

SE+SH=20

SE+SH= max

Mean = (0+10)/2 = 5

Mean = (8+8)/2 = 8

Mean = (20+0)/2 = 10

Highest mean!

What do we want?

10

10

30

20

Ed

Harry

45o

SE+SH= max

A

B

C

DE

Std. Dev.

Mean.

B'

A'

C'

D'E'

What do we want?

Std. Dev.

Mean.

B'

A'

• Utility Functions– Leveler – Will only

accept lower mean along with lower SD.

– Why?

• Utility Functions– Elitist – Will accept

lower mean with higher SD.

– Why?

C'E'

D'

L1

L2

L3

What do we want?

Std. Dev.

Mean.

B'

A'

• Utility Functions– Leveler – Will only

accept lower mean along with lower SD.

– Why?

• Utility Functions– Elitist – Will accept

lower mean with higher SD.

– Why?

C'

D'

E3

E1

E2

What do we want?

Std. Dev.

Mean.

B'

A'

• So, it’s not altogether clear that we always want to raise the mean.

• The levelers here, want to push up the lower end, and this lowers the SD.

• Means fewer special programs.

C'

D'

E3

E1

E2

Old Stuff

Fuchs on Cost Containment and Cost-Benefit

You can ultimately contain costs in one of three ways:

1. Increase production efficiency. Old systems don't necessarily reward inefficient production. Most agree that there was more to do with what was delivered rather than how it was delivered.

2. Reduce input prices. In the short run, you can try to squeeze some of the inputs, like nurses' wages, physicians’ fees, or drug industry profits. In the long run they can go elsewhere.

3. Deliver fewer services.

What are impacts of cost containment?

H = aQ - bQ2.

If we want to maximize health, irrespective of costs, we maximize this, and we get:

dH/dQ = a - 2bQ = 0,

or Q* = a/2b.

Suppose a = 30, b = 1.

Q* = 15.

Fuchs on Cost Containment and Cost-Benefit

Health and Marginal Benefits

-50

0

50

100

150

200

250

0 5 10 15 20 25

Health Care

Hea

lth

Ben

efit

s

H

MBQ* = a/2b

We recognize that this differs from the optimum if we recognize the (constant) costs c, so that we are optimizing: L = Benefits - Costs

L = aQ - bQ2 - cQ.Here, we get:Q** = (a - c)/2b= Q* - (c/2b).Suppose a = 30, b = 1, c = 3Q* = 13.5.

Fuchs on Cost Containment and Cost-Benefit

Suppose, instead, we’re at QM, for the mean.

Marginal Benefits

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25

Health Care

Mar

gina

l Ben

efit

s, M

argi

nal C

osts

MB

MC

compare to optimum w/o costs

Reducing costsSuppose that the mean for the

population is QM. So the mean health is:

HM = aQM - bQM2. Then if we

reduce outputs by z:

HM' = a(QM - z) - b(QM - z)2.We may wish to discover whether

mean health improves or decreases with z.

Remember that Q* = a/2b.When we expand this expression, we

get:

H = HM' - HM = -2zb [Q* - (QM - z/2)].

Marginal Benefits

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25

Health Care

Mar

gina

l Ben

efit

s, M

argi

nal C

osts

MB

MC

DistributionsIf you think of this as a population distribution, all that you're doing is shifting the distribution. If you're moving toward social optimum, you have a similar situation. You're either moving more people toward the optimum (making society better off) or more people away (making society worse off).

What about if you mandate equal percentage decrease, rather than equal amount decrease. The algebra is trickier, and it is worthwhile to go to the marginal product - marginal cost diagram. Equal percentage decreases imply unequal absolute decreases, because those with larger amounts have larger decreases.

Distributions

Suppose that a = 12, b = 0.05, and c = 2.

Then Q* = a/2b = 120.

Q** = (a - c)/2b= Q* - (c/2b) = 120 - 20 = 100.

Suppose we have QM = 110, over 10 people.

Quantity120

2

100 110

Social optimum is 100, or 10/person

Distributions

Suppose you have Q = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} mean = 11

Several different mandates (in reducing Q by 10 overall)

• Reduce everyone by 1. First 5 are worse off; next 5 are better off. We now have {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

• Reduce everyone by 1/11.

We now have {1.8, 3.6, 5.5, 7.3, 9.1, 10.8, 12.7, 14.5, 16.3, 18.2}

EXCEL Slide (C_B_99)

Distribution• If social welfare is related to mean (+) and to variance (-), then with

option 1, we’re going to be better off, although some will be far worse off.• With option 2, effects on health and on social welfare depend on the size

of z, the mean of the distribution QM, and the variance 2. Starting with Q > Qopt, the larger the variance, the smaller can be QM consistent with a favorable effect on health or social welfare.

• Why? You're pulling those who are using the most services much closer, and, at worse, those who are using less (and possibly less than optimal) services less farther away. One could conceivably improve mean health, or mean welfare, even if you started below mean health or mean welfare. One might even do better, by reducing those at the right hand tail by even more, and those on the left hand tail by somewhat less.