ECO 7550 More Health Capital

The Demand for Health Capital

Cost of capital, in terms of foregone resources (for health capital, both time and money) is a supply concept.

Other needed tool is the concept of the marginal efficiency of investment, the MEI, a demand concept which relates the return to investment to the amount of resources invested.

Marginal Efficiency of Investment (MEI) and Rate of Return

• The MEI can be described in terms of the X-ray machine example.

• A clinic which does considerable business may wish to own more than one such X-ray machine. How many?

• The clinic management may logically consider them in sequence.

Size of I (in $)

Rat

e of

Ret

urn

(%)

• The first X-ray machine purchased (if they were to buy only one) would yield a return. Suppose that return each year was $100,000.

• We can also calculate the rate of return, which would be $100,000/$500,000 or 20% per year. They would buy this X-ray machine if it covered its opportunity cost of capital and the depreciation.

Size of I (in $)

Rat

e of

Ret

urn

(%)

Marginal Efficiency of Investment (MEI) and Rate of Return

• Management would choose to own the first X-ray machine as long as the rate of return, 20%, was greater than the– interest rate (the

opportunity cost of capital)

– plus the depreciation rate.

Size of I (in $)

Rat

e of

Ret

urn

(%)

Marginal Efficiency of Investment (MEI) and Rate of Return

Cost of capital = interest rate + depreciation rate

Marginal Efficiency of Investment

• If they considered owning two X-ray machines, they would discover that the rate of return to the second X-ray machine was probably less than the first.

• Suppose that a clinic buying only one X-ray machine would assign it to the highest priority uses, those with the highest rate of return. If they add a second X-ray machine, then logically it could only be assigned to lesser priority uses (and might be idle on occasion) lower rate of return than the first.

• Clinic would also purchase the second X-ray machine only if its rate of return was still higher than interest plus depreciation.

Decreasing MEI• Let the marginal efficiency

of investment curve, MEI, describe the pattern of rates of return, declining as the amount of investment (measured on the horizontal axis) increases.

• The cost of capital, that is, the interest rate plus the depreciation rate, is shown as the horizontal line labeled (r + ).

Size of I (in $)

Rat

e of

Ret

urn

(%)

Cost of capital = interest rate (r) + depreciation rate ()

Optimum amount of capital

• The optimum amount of capital demanded is thus Ko, which represents the amount of capital at which the marginal efficiency of investment just equals the cost of capital.

• Like the mgl efficiency of investment curve in this example, the MEI curve for investments in health would also be downward sloping.

Size of I (in $)

Rat

e of

Ret

urn

(%)

I*

MEI Curve

Expenditures

↓ I may NOT mean ↓ Expenditures

Cost of capital = interest rate (r) + depreciation rate ()

Diminishing Marginal Returns

• This occurs because the production function for healthy days (Figure 7.4) exhibits diminishing marginal returns.

Health Inputs

Hea

lthy

Day

s

365

Total Product

Equilibria• Cost of capital for health

would similarly reflect the interest rate plus the rate of depreciation in health.

• Person’s health, like any capital good, will also depreciate over time. Thus the optimal demand for health is likewise given at the intersection of the MEI curve and the cost of capital curve, (r + ).

Size of I (in $)

Rat

e of

Ret

urn

(%)

Cost of capital = interest rate (r) + depreciation rate ()

I*

MEI Curve

Increased depreciation rate

I**

Pure Investment and Pure Consumption Models

• Do we invest in health because it makes us feel good, or do we invest in health because it makes us more productive?

• If all we care about is the money we can earn, then all we care about is bread. We have vertical indifference curves. We want only the amount that will allow us to earn as much as we can.

Hea

lth

Bread

PP curve

Pure investmenteq’m

Pure Investment and Consumption Models

• If all we also care about health, we get more conventional indifference curves.

Hea

lth

Bread

PPP

• Less bread -- more health

Pure investmenteq’m

Comparative Statics – Age • Age What happens to

MEI?

• Why?

Size of I (in $)

Rat

e of

Ret

urn

(%)

Cost of capital = interest rate (r) + depreciation rate ()

I*

MEI Curve

Comparative Statics – Education • Higher Education What

happens to MEI?

• Why?

Size of I (in $)

Rat

e of

Ret

urn

(%)

Cost of capital = interest rate (r) + depreciation rate ()

I*

MEI Curve

Comparative Statics – Wage • Wage What happens to

MEI?

• Why?

• But what if investment has a large wage component?

• As drawn the impact is positive, but mathematically it is ambiguous.

Size of I (in $)

Rat

e of

Ret

urn

(%)

Cost of capital = interest rate (r) + depreciation rate ()

I*

MEI Curve

One More Example of MEI – Uncertainty

• What is impact of increased uncertainty.

• Some models say I ↑. Others say I ↓.

• Let’s look ex ante.• You’re uncertain about the

future.• You can invest in I, or in F

(non-health financial asset), which by assumption is less risky.

• What do we do this year.

Investment

Cost of Capital

MEI

I*

Uncertainty

• Depends!• An ↑ in I this year will

increase health capital next year. If this ↑ productivity, MEI shifts right Do it (i.e. Invest)!

Investment

MEI

I*

MEI'

• An ↑ in I will increase health capital next year. If this does NOT ↑ productivity, you move down MEI curve Don’t do it (i.e. Don’t Invest.)

Cost of Capital

• If on net, sum of the impacts is positive, uncertainty increases health investment.

• If on net, sum of impacts is negative, uncertainty decreases health investment.

One More Example – Uncertainty

So, what does Grossman tell us?

How resources are allocated over time.How resources are allocated in any given period.Grossman focuses on the first.Ultimately the math is complex but it comes to the

equation:

)(])(

)(ˆ)([

)(

)(

)(

)(

]0[

)()(

qq

qr

H

t

t

Ye

tU

s

s

trs

[1] [2] [3] [4]

Marginal Benefits Marginal Costs=

Sick time

What does it mean?

[3] Increased health must reduce sick time (-). If not, I = 0.

)(])(

)(ˆ)([

)(

)(

)(

)(

]0[

)()(

qq

qr

H

t

t

Ye

tU

s

s

trs

[1] [2] [3] [4]

Marginal Benefits Marginal Costs=

[1] Valuation of health as a consumption good. Numerator (-) refers to increased utility that health buys. Denominator (+) tells about the increased income from financial assets (nonwage income), and what you can buy with it.[2] Increased labor income (-)pure investment effect

[4] Cost of capital * amount of capital.

Edgeworth Boxes and Constant Returns

An a% in goods and leisurean a% in health and home good

Health

Leisure

Expansion path for health

Expansion path for home good

Think about CRTS? What does the length of the ray mean?

Think about CRTS? What does the length of the ray mean?

Edgeworth Boxes and Increases in 1 Factor

An a% increase in goods an in goods-intensive output (here, health), but a in home good. Why?

Rybczynski - A little calculusLet:agI and atI denote the goods and leisure per unit of Health Investment, I

agc and atc denote the goods and leisure per unit of Home Good, C

These coefficients will vary with the relative factor prices {Leisure - wage rate; Home good - out-of-pocket price}, but since a given commodity price ratio (e.g. Health Investment/Home Good) uniquely determines a factor price ratio, these coefficients will be constants at the given commodity price-ratio (why?).

Denoting the total amounts of goods and leisure available as G and T respectively:  agII + agcC = G

  atII + atcC = T

Solving these equations for I/T and C/T yields: 

I/T = [atc (G/T) - agc] / [agIatc -atIagc] 

C/T = [agI - atI(G/T)] / [agIatc -atIagc] 

We can then solve for: 

I/C = [atc (G/T) - agc]/[agI - atI(G/T)]

This is the ratio of commodity outputs as a function of the goods/time ratio. 

Differentiating (I/C) with respect to (G/T) yields: 

d (I/C) / d (G/T) = (agIatc - atIagc) / (agI - atI (G/T))2

 

Then: 

d (I/C) / d (G/T) 0, as (agI/atI) (agc/atc).  

d (I/C) / d (G/T) 0, as (agI/atI) (agc/atc).  

agI/atI = (goods/leisure ratio)I 

agc/atc = (goods/leisure ratio)C.

x

xFactor the Denominators

Income Effects

As drawn,I is more mkt.-intensive.

An in Gleads to relativelylarge in I.

Bread

Invest.

Time

$

Obesity – An Application of Human Capital

• A leading risk factor for heart disease, hypertension (high blood pressure), certain cancers, and type-2 diabetes.

• According to reports from the CDC in 2012, over one third of U.S. adults (more than 72 million) people and 17% of U.S. children are obese. From 1980 through 2008, obesity rates for adults doubled and rates for children tripled.

• Obesity describes health capital:– may make the body less productive, – more susceptible to disease, and – possibly cause it to depreciate more quickly.

BMIHealth analysts usually measure obesity in terms of Body Mass Index, or BMI, with the formula

2

Weight in kilogramsBMI

(height in meters)

.

Category BMI range

Severely underweight less than 16

Underweight 16 to 18.5

Normal 18.5 to 25

Overweight 25 to 30

Obese Class I 30 to 35

Obese Class II 35 to 40

Obese Class III 40 and above

BMIBMI Calculator

2000

Obesity Trends* Among U.S. AdultsBRFSS, 1990, 2000, 2010

(*BMI 30, or about 30 lbs. overweight for 5’4” person)

2010

1990

No Data <10% 10%–14% 15%–19% 20%–24% 25%–29% ≥30%

http://www.cdc.gov/nchs/data/databriefs/db82.pdf

Yaniv, Rosin, and Tobol

• Calories are expended in both in physical activity and when the body is at rest. The rest component, known as Basal Metabolic Rate (BMR), is the largest source of energy expenditure, reflecting blood circulation, respiration and daily maintenance of body temperature.

• Differing BMRs among individuals indicate why one person can “eat like a horse” and gain little weight, while another may gain weight with far less intake of food.

Obesity – Economic Theory

• Weight gain as the outcome of rational choice that reflects a willingness to trade off some future health for the present pleasures of less restrained eating and lower physical activity. “Diets” reverse this.

Model

• Overweight individuals can determine consumption of junk-food meals, F, and healthy meals, H. They may also choose their level of exercise, x. The model defines the weight gain during a period, or obesity, S, as:

S = δF + εH − μx − BMR

YRT develop the model showing that taxes on junk food (reducing consumption), or subsidies to healthy food (increasing its consumption) could have important impacts on formation of health capital.

Why has obesity increased.

Cutler, Glaeser, and Shapiro (2003)

Changes in the time costs of food production

• Vacuum packing, improved preservatives.

• Mass preparation – French fries are a pain to make at home– Quick and easy at the restaurants– Food professionals and economies of scale

Time Costs by Group

Cutler, Glaeser, and Shapiro (2003)

104.4

ReferencesCutler, David M., Edward L. Glaeser and Jesse M. Shapiro, “Why Have Americans Become More Obese?” Journal of Economic Perspectives 17 (3): 93–118

Yaniv, Gideon, Odelia Rosin, and Yossef Tobel, “Junk-food, Home Cooking, Physical Activity and Obesity: The Effect of the Fat Tax and the Thin Subsidy,” Journal of Public Economics 93 (2009): 823–830