Transcript Lecture 3

Prevention or cure?
•Factors that influence the demand for prevention
and the demand for cure
Today – reference: Hey &Patel
•Moral hazard in the insurance market
Covered by Kari Eika: Demand for insurance
01/02/2005
Tor Iversen
• Many decisions involve a trade-off between prevention and cure
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Undertake a cost now to avoid a greater cost at a later stage
Checking the car’s level of engine oil
Regularly check-ups at the dentist
A healthy diet
A proper amount of physical exercise
Attending health screening programmes
• How much resources (money and time) should one spend on
prevention?
• How much resources should one spend on cure if sick?
• Is the demand for prevention likely to be influenced by the price of
prevention?
• What about the demand for curative services?
• What about the effect of prevention? Does more effective prevention
imply an increase in the demand for prevention? What about the
demand for curative services?
• And what about the effect of curative services? Does a more
effective cure imply an increase in the demand for cure? What about
the impact on the demand for prevention?
• Interaction between prevention and cure – these interactions are
easier to analyze by the means of a model
The Hey and Patel model
Two states: healthy and sick
Discrete time:
p=the probability of being healthy in period t+1 given that you are healthy in period t
q=the probability of being healthy in period t+1 given that you are sick in period t
Period t+1
Period t
Healthy
Sick
Healthy
p
q
Only this period’s state affects next period’s probability
First order Markov process
Sick
1-p
1-q
p = p(x)
0<p(x)<1
p’(x)>0
p’’(x)<0
q = q(y)
0<q(y)<1
q’(y)>0
q’’(y)<0
x = quantity of preventive care purchased
y = quantity of curative care purchased
I = money income received per period (exogenous)
P = price per unit of preventive care
Q = price per unit of curative care
R = residual income
R = I – Px
R = I – Qy
if healthy
if sick
State dependent utility:
U(R) = V(R) if healthy
U(R) = W(R) if sick
V(R)>W(R) for all R
Risk aversion in both state:
V’(R)>0, V’’(R)<0 for all R
W’(R)>0, W’’(R)<0 for all R
It is assumed that the individual lives an infinite number of periods.
The expected life time utility evaluated from period T:

t T
r
 U ( Rt )
t T
r = a constant discount factor
(6)
Problem: Choose levels of x and y that maximize lifetime expected utility (6)
With an infinite perspective the future looks similar regardless of whether the future
is considered from period t or period t+1
v = the maximum of future expected utility if healthy initially
w = the maximum of future expected utility if sick initially
Then:
v = max{V ( I  Px)  r[ p( x)v (1  p( x)) w]}
x
and
w = max{W ( I  Qy)  r[q( y)v (1  q( y)) w]}
(7)
(8)
y
First order conditions for optimal x and y:
PV ' ( I  Px)  r (v  w) p ' ( x)
QW ' ( I  Qy)  r (v  w)q ' ( y)
Marginal loss of utility today
equals expected marginal gain in
future utility of one unit of
prevention/cure
Second order condition:
A  P 2V '' ( I  Px)  r (v  w) p '' ( x)  0
B  Q 2W '' ( I  Qy )  r (v  w)q '' ( y )  0
Hence, necessary and sufficient conditions for an interior maximum of v and w
requires that v-w=u is positive which is reasonable.
From (7) and (8) with the maximum values of x and y inserted:
v = V ( I  Px)  r[ p( x)v (1  p( x)) w]
w = W ( I  Qy )  r[q ( y )v (1  q ( y )) w]
Subtracting the second from the first and rearranging:
V ( I  Px)  W ( I  Qy )  u[1  rp( x)v  rq ( y )]
The optimal x, y and u is then determined by:
PV ' ( I  Px)  rup ' ( x)
QW ' ( I  Qy)  ruq ' ( y)
(15)
(16)
V ( I  Px)  W ( I  Qy )  u[1  rp( x)v  rq ( y )]
(17)
We would now like to study the effects on x and y of a change in:
• Income (I)
• Price of prevention (P)
• Price of curative care (Q)
• The probability of staying healthy tomorrow given healthy today (p)
• The probability of becoming healthy tomorrow given sick today (q)
Formally the comparative statics can be done by differentiating the first order
conditions (15)-(17) and using Cramer’s rule. Here we go straight to the
solutions.
The effect of an increase in income (I):
Three cases:
(1) V’>W’ (marginal utility of income is greater as healthy than as sick) :
u
 0,
I
x
 0,
I
y
0
I
(2)
V’=W’ (marginal utility of income as healthy equals marginal utility of
income as sick) :
u
x
y
 0,
 0,
0
I
I
I
(3)
V’<W’ (marginal utility of income is smaller as healthy than as sick) :
u
 0,
I
x
?,
I
y
?
I
The effect of an increase in price per unit of prevention (P):
u
 0,
P
x
 0,
P
y
0
P
Demand for prevention decreases because:
•
More costly to buy prevention today
•
More costly to stay healthy in future periods
In increase in P also decreases the demand for curative care. It becomes less
attractive to become healthy because the expenditures related to staying healthy increase
The effect of an increase in price per unit of curative care (Q):
u
 0,
Q
x
 0,
Q
y
?
Q
Note: Misprint in the article – wrong sign on the effect on x on p. 129 – also in Table 4, p.
133
An increase in Q initiates an increase in demand for prevention because it is now more
expensive to become sick
The effect of increase in Q on demand for curative care – two opposing effects:
•
More costly to buy cure today
•
More attractive to become healthy in future periods.
The effect of a change in the technology of prevention:
(1) A shift (a) in the function: p(x) + a
u
 0,
a
x
 0,
a
y
0
a
In increase in a gives an increase in both x and y because the lifetime
expected utility of being healthy compared to being sick has increased
(2) A shift in the marginal efficacy of prevention (g): gp(x)
u
 0,
g
x
 0,
g
y
0
g
An increase in g initiates an increase in both x and y because the
marginal effect of prevention increases
Similarly: The effect of a change in the technology of cure:
(1) A shift (b) in the function: q(y) + b
u
 0,
b
x
 0,
b
y
0
b
In increase in b gives a decline in both x and y because the utility of
being healthy compared to being sick has declined
(2) A shift in the marginal efficacy of cure (h): hq(y)
u
 0,
h
x
 0,
h
y
0
h
An increase in h initiates an increase in y because the
marginal effect of curative care increases. Both u and x decline.
Important to specify the type of technology change regarding the effect
on y
Critique of assumptions:
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p and q independent of an individual’s age
p and q independent of previous states
no state of death
income independent of health
no insurance that lowers Q
prevention and cure cannot both be demanded simultaneously
Examples of market imperfections in the market for
prevention
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External effects – infectious diseases
Information as a public good – information about preventive technologies (a
and g)
Preferences – as healthy you may have unrealistic perceptions of what it is
like to be ill
Market imperfection because of
– Publicly provided health insurance – insurance premium independent of your
preventive effort
– Or health insurance in general with asymmetric information regarding preventive
effort
Marginal benefit
Utility
Marginal cost
x’
x*
x
PV ' ( I  Px)  rup ' ( x)
Marginal cost = Marginal expected utility
If imperfection because of low Q, the chosen x is too low. By subsidizing
prevention, P declines, marginal cost of prevention declines and x increases
to x*, closer to the socially optimal level