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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 – – – 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 Only this period’s state affects next period’s probability First order Markov process

p = p(x) 00 p’’(x)<0 q = q(y) 00 q’’(y)<0 x = quantity of preventive care purchased y = quantity of curative care purchased I = money income received 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: (6) r = a constant discount factor

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: (7) (8) First order conditions for optimal x and y: Marginal loss of utility today equals expected marginal gain in future utility of one unit of prevention/cure

Second order condition: 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: Subtracting the second from the first and rearranging:

The optimal x, y and u is then determined by: (15) (16) (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) : (2) V’=W’ (marginal utility of income as healthy equals marginal utility of income as sick) : (3) V’

The effect of an increase in price per unit of prevention (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): 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 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) 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 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) 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: • • • 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 • • 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 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