Week 4
Part 1: Health Investment (The Grossman Model)
This model treats "Good Health" as a capital stock (like a machine) that produces "healthy time"
for you. You are born with some health, but it wears out over time.
1. The Core Equation (Law of Motion)
Your health stock in the current period (ht) depends on your previous health, how much it
deteriorated (depreciation), and how much you repaired it (investment).
Concept: Current Health = Previous Health - Depreciation + Investment
ℎ! = ℎ!"# − 𝛾ℎ!"# + 𝐼!
• Ht: Health stock at time t.
• 𝛾: The depreciation rate (how fast health wears out naturally).
• It: Investment in health (medical care, diet, exercise).
2. The Decision Rule (How much to invest?)
A rational person invests in health until the marginal benefit equals the marginal cost.
• Marginal Efficiency of Capital (MEC): This is the benefit
curve. It represents the extra utility (satisfaction/earnings)
you get from one more unit of health capital. It slopes
downwards because of diminishing marginal returns (going
from sick to healthy is huge; going from healthy to super-
healthy helps less).
• Cost of Capital: This is the cost line. It consists of the
opportunity cost of your money (interest rate r) and the
depreciation rate (gamma).
Optimization Condition: You choose the level of health (h*)
where: 𝑀𝐸𝐶 = 𝑟 + 𝛾
3. Exercise Applications
A. The Effect of Ageing (Exercise 1: Q1 & Q2)
• Scenario: As you get older, your body wears out faster.
• Model mechanism: The depreciation rate (gamma) increases with age.
• Graph result: The cost line (r + gamma) shifts UP.
• Outcome: You intersect the MEC curve at a lower point. Optimal health stock (h^*)
decreases.
• Rational Death: Eventually, gamma becomes so high that the cost to maintain minimal
health is too expensive. The model predicts a rational individual "chooses" death when
investment stops paying off.
o [Exercise 1 Q2: Rational Rita] Rita moves from point A (young) to point D (old)
because her costs increased.
,B. The Effect of Wages (Exercise 1: Q3)
• Scenario: High-wage workers earn more per "healthy hour."
• Model mechanism: Being healthy is more valuable because it allows you to earn more
money.
• Graph result: The MEC curve shifts to the RIGHT (upward).
• Outcome: High-wage earners (Hugh) choose a higher optimal health stock (hH) than
low-wage earners (Larry, hL).
o [Exercise 1 Q3b: Manual Labor] If Larry has a physically demanding job, his
depreciation (gamma) increases faster as he ages. His health will deteriorate
faster than Hugh's because his cost line shifts up more rapidly.
C. The Effect of Time Preference (Exercise 1: Q4)
• Scenario: Some people are impatient (live for today), while others are patient (plan for
the future).
• Model mechanism: Impatient people discount the future heavily. This is represented by
a higher interest/discount rate (r or sigma𝜎).
• Graph result: For impatient people (Edna), the effective cost (r + gamma) is higher. The
cost line shifts UP.
• Outcome: Impatient people invest less and choose lower health stocks than patient
people (Pam).
Part 2: The Rational Addiction Model (RAM)
This model argues that addiction can be a rational choice where individuals maximize utility
while anticipating future consequences.
1. The Utility Function
Your happiness today depends on current consumption (ct), other goods (zt), and your
"addiction stock" (St) built up from the past. 𝑈! = 𝑢(𝑐! , 𝑧! , 𝑆! )
2. Key Definitions of Addiction
For a good to be addictive in this model, it must have two traits:
1. Tolerance: Past consumption lowers your current happiness (you need the drug just to
feel normal). Mathematically: US < 0.
2. Reinforcement (Crucial): Past consumption increases the marginal utility (craving) of
current consumption. Mathematically: UcS > 0.
3. Steady State
The steady state is where your consumption exactly replaces the addiction stock that wears off.
𝑐 = 𝛾𝑆
4. Exercise Applications
A. Who becomes addicted? (Exercise 2: Q1)
, • Time Preference: People with high time preference (impatient/Eager Edna) are more
likely to addict. They care less about the future harm (tolerance) and more about the
current craving (reinforcement).
• Depreciation of Addiction Stock: If the drug leaves your system quickly (High gamma,
Fast Eddy), you are more likely to addict. The future "punishment" (withdrawal) feels
smaller/shorter.
B. Price Effects (Exercise 2: Q2)
• Future Prices Matter: Because rational addicts plan ahead, they will reduce
consumption now if they know prices will go up later.
• Permanent vs. Temporary: A permanent price increase causes a larger drop in
consumption than a temporary one. The addict knows they cannot sustain the habit in
the long run, so they quit/cut down immediately.
Part 3: Empirical Testing (Data Analysis)
This section deals with the "Opium in Dutch East Indies" exercise, testing if the Rational
Addiction model works in real life.
1. The Regression Equation
To test for addiction, we estimate demand. Current consumption depends on past
consumption, price, and income.
ln(𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦! ) = 𝛼 + 𝛽# ln(𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦!"# ) + 𝛽$ ln(𝑃𝑟𝑖𝑐𝑒! ) + 𝛽% ln(𝐼𝑛𝑐𝑜𝑚𝑒! )
2. Testing for Addiction
• The Test: Look at beta_1 (the coefficient of lagged consumption Qt-1.
• Interpretation: If beta_1 > 0 and is statistically significant, it means past use causes
current use. This proves Reinforcement (Addiction).
o Exercise Result: In the opium data, beta_1 ≈ 0.74. This confirms strong addiction.
3. Short-Run vs. Long-Run Elasticity
• Short-Run Elasticity (SR): This is simply beta_2. It is the immediate reaction to a price
change.
• Long-Run Elasticity (LR): This takes into account the cumulative effect over time.
𝛽$
𝜖&' =
1 − 𝛽#
• Theory Prediction: For addictive goods, the Long-Run response should be larger (more
negative) than the Short-Run response.
o Exercise Result: SR elasticity was -0.36, but LR elasticity was -1.36. This confirms
the theory.
, 4. OLS vs. 2SLS (Exercise 2: Q3)
• Problem: Using simple OLS regression is biased because errors are correlated. It often
overestimates the degree of addiction.
• Solution: Use Two-Stage Least Squares (2SLS) with an Instrumental Variable (Lagged
Price).
• Result: When using 2SLS, the addiction coefficient (beta_1) dropped from 0.74 to 0.35.
This is a more accurate estimate.
Exercise 1
Question 1: Health investment over the life cycle
Consider a version of Grossman’s model of the demand for health in which health capital depreciates
by 5% in each year of life and investment in health through medical care is just sufficient to replace
health lost through deprecia>on each year.
a) What relationship between health and age would this model predict?
The model predicts that health stock (ht) remains constant over the individual's life, irrespective
of age.
• Reasoning: In this simplified version, the depreciation rate is constant (gamma = 0.05)
and investment (I) is exactly equal to the depreciation (gamma*ht-1).
b) To test this prediction, what data would you need and how would you use them?
• Data Needed: data on the health (e.g., EQ-VAS scores) and age of individuals.
• Method: Perform a regression of health on age, while controlling for other socio-
demographic determinants of health (like income, education, etc.).
• Test: If the model is correct, the coefficient on age should be zero (indicating no
relationship). In reality, as shown in the class slides, the coefficient is typically negative
and significant, meaning health declines with age.
c) What assumption is critical to Grossman capturing the lifecycle profile of health?
The critical assumption is that the depreciation rate (gamma) is not constant but increases
disproportionately with age.
• As an individual ages, the biological rate at which
their health wears out increases. This raises the
effective cost of holding health capital (r + gamma),
making it more expensive to maintain the same
level of health. Consequently, rational individuals
choose a lower optimal level of health capital each
year.
Part 1: Health Investment (The Grossman Model)
This model treats "Good Health" as a capital stock (like a machine) that produces "healthy time"
for you. You are born with some health, but it wears out over time.
1. The Core Equation (Law of Motion)
Your health stock in the current period (ht) depends on your previous health, how much it
deteriorated (depreciation), and how much you repaired it (investment).
Concept: Current Health = Previous Health - Depreciation + Investment
ℎ! = ℎ!"# − 𝛾ℎ!"# + 𝐼!
• Ht: Health stock at time t.
• 𝛾: The depreciation rate (how fast health wears out naturally).
• It: Investment in health (medical care, diet, exercise).
2. The Decision Rule (How much to invest?)
A rational person invests in health until the marginal benefit equals the marginal cost.
• Marginal Efficiency of Capital (MEC): This is the benefit
curve. It represents the extra utility (satisfaction/earnings)
you get from one more unit of health capital. It slopes
downwards because of diminishing marginal returns (going
from sick to healthy is huge; going from healthy to super-
healthy helps less).
• Cost of Capital: This is the cost line. It consists of the
opportunity cost of your money (interest rate r) and the
depreciation rate (gamma).
Optimization Condition: You choose the level of health (h*)
where: 𝑀𝐸𝐶 = 𝑟 + 𝛾
3. Exercise Applications
A. The Effect of Ageing (Exercise 1: Q1 & Q2)
• Scenario: As you get older, your body wears out faster.
• Model mechanism: The depreciation rate (gamma) increases with age.
• Graph result: The cost line (r + gamma) shifts UP.
• Outcome: You intersect the MEC curve at a lower point. Optimal health stock (h^*)
decreases.
• Rational Death: Eventually, gamma becomes so high that the cost to maintain minimal
health is too expensive. The model predicts a rational individual "chooses" death when
investment stops paying off.
o [Exercise 1 Q2: Rational Rita] Rita moves from point A (young) to point D (old)
because her costs increased.
,B. The Effect of Wages (Exercise 1: Q3)
• Scenario: High-wage workers earn more per "healthy hour."
• Model mechanism: Being healthy is more valuable because it allows you to earn more
money.
• Graph result: The MEC curve shifts to the RIGHT (upward).
• Outcome: High-wage earners (Hugh) choose a higher optimal health stock (hH) than
low-wage earners (Larry, hL).
o [Exercise 1 Q3b: Manual Labor] If Larry has a physically demanding job, his
depreciation (gamma) increases faster as he ages. His health will deteriorate
faster than Hugh's because his cost line shifts up more rapidly.
C. The Effect of Time Preference (Exercise 1: Q4)
• Scenario: Some people are impatient (live for today), while others are patient (plan for
the future).
• Model mechanism: Impatient people discount the future heavily. This is represented by
a higher interest/discount rate (r or sigma𝜎).
• Graph result: For impatient people (Edna), the effective cost (r + gamma) is higher. The
cost line shifts UP.
• Outcome: Impatient people invest less and choose lower health stocks than patient
people (Pam).
Part 2: The Rational Addiction Model (RAM)
This model argues that addiction can be a rational choice where individuals maximize utility
while anticipating future consequences.
1. The Utility Function
Your happiness today depends on current consumption (ct), other goods (zt), and your
"addiction stock" (St) built up from the past. 𝑈! = 𝑢(𝑐! , 𝑧! , 𝑆! )
2. Key Definitions of Addiction
For a good to be addictive in this model, it must have two traits:
1. Tolerance: Past consumption lowers your current happiness (you need the drug just to
feel normal). Mathematically: US < 0.
2. Reinforcement (Crucial): Past consumption increases the marginal utility (craving) of
current consumption. Mathematically: UcS > 0.
3. Steady State
The steady state is where your consumption exactly replaces the addiction stock that wears off.
𝑐 = 𝛾𝑆
4. Exercise Applications
A. Who becomes addicted? (Exercise 2: Q1)
, • Time Preference: People with high time preference (impatient/Eager Edna) are more
likely to addict. They care less about the future harm (tolerance) and more about the
current craving (reinforcement).
• Depreciation of Addiction Stock: If the drug leaves your system quickly (High gamma,
Fast Eddy), you are more likely to addict. The future "punishment" (withdrawal) feels
smaller/shorter.
B. Price Effects (Exercise 2: Q2)
• Future Prices Matter: Because rational addicts plan ahead, they will reduce
consumption now if they know prices will go up later.
• Permanent vs. Temporary: A permanent price increase causes a larger drop in
consumption than a temporary one. The addict knows they cannot sustain the habit in
the long run, so they quit/cut down immediately.
Part 3: Empirical Testing (Data Analysis)
This section deals with the "Opium in Dutch East Indies" exercise, testing if the Rational
Addiction model works in real life.
1. The Regression Equation
To test for addiction, we estimate demand. Current consumption depends on past
consumption, price, and income.
ln(𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦! ) = 𝛼 + 𝛽# ln(𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦!"# ) + 𝛽$ ln(𝑃𝑟𝑖𝑐𝑒! ) + 𝛽% ln(𝐼𝑛𝑐𝑜𝑚𝑒! )
2. Testing for Addiction
• The Test: Look at beta_1 (the coefficient of lagged consumption Qt-1.
• Interpretation: If beta_1 > 0 and is statistically significant, it means past use causes
current use. This proves Reinforcement (Addiction).
o Exercise Result: In the opium data, beta_1 ≈ 0.74. This confirms strong addiction.
3. Short-Run vs. Long-Run Elasticity
• Short-Run Elasticity (SR): This is simply beta_2. It is the immediate reaction to a price
change.
• Long-Run Elasticity (LR): This takes into account the cumulative effect over time.
𝛽$
𝜖&' =
1 − 𝛽#
• Theory Prediction: For addictive goods, the Long-Run response should be larger (more
negative) than the Short-Run response.
o Exercise Result: SR elasticity was -0.36, but LR elasticity was -1.36. This confirms
the theory.
, 4. OLS vs. 2SLS (Exercise 2: Q3)
• Problem: Using simple OLS regression is biased because errors are correlated. It often
overestimates the degree of addiction.
• Solution: Use Two-Stage Least Squares (2SLS) with an Instrumental Variable (Lagged
Price).
• Result: When using 2SLS, the addiction coefficient (beta_1) dropped from 0.74 to 0.35.
This is a more accurate estimate.
Exercise 1
Question 1: Health investment over the life cycle
Consider a version of Grossman’s model of the demand for health in which health capital depreciates
by 5% in each year of life and investment in health through medical care is just sufficient to replace
health lost through deprecia>on each year.
a) What relationship between health and age would this model predict?
The model predicts that health stock (ht) remains constant over the individual's life, irrespective
of age.
• Reasoning: In this simplified version, the depreciation rate is constant (gamma = 0.05)
and investment (I) is exactly equal to the depreciation (gamma*ht-1).
b) To test this prediction, what data would you need and how would you use them?
• Data Needed: data on the health (e.g., EQ-VAS scores) and age of individuals.
• Method: Perform a regression of health on age, while controlling for other socio-
demographic determinants of health (like income, education, etc.).
• Test: If the model is correct, the coefficient on age should be zero (indicating no
relationship). In reality, as shown in the class slides, the coefficient is typically negative
and significant, meaning health declines with age.
c) What assumption is critical to Grossman capturing the lifecycle profile of health?
The critical assumption is that the depreciation rate (gamma) is not constant but increases
disproportionately with age.
• As an individual ages, the biological rate at which
their health wears out increases. This raises the
effective cost of holding health capital (r + gamma),
making it more expensive to maintain the same
level of health. Consequently, rational individuals
choose a lower optimal level of health capital each
year.
