Estimating the Impact of Medication on Diabetics' Diet and Lifestyle Choices

As the function F is identically zero, the implicit function rule can be used to explore relations
among variables in the perceived health production function and in the utility function.

P

UHHMη⁺HMUHHHη^-P^M (UBHHη⁺UHHBη⁺HBUHHHη)

∂B   Fη                      P_B

^{∂η   FB}   U H +H U  -^PM (U +2U H +U H +U (H )²)

H MB M HB        BB BH B H BB HH B

P_B

Our goal is to sign the derivative — for a typical individual, not all mathematically possible
∂η

utility and production functions. We make conventional assumptions that

U _i > 0 and

U_ii,H_ii <0fori=B,H,C.

That is, the marginal utilities are positive, and utility and production functions are concave. The
marginal product of medicine is positive, but behaviors that bring enjoyment are assumed
unhealthful and information about health is assumed to be bad news, reducing perceived health
status.

H _j > 0 for j = M and
H _j < 0 for j = B,η.

Conventional utility and production function assumptions, however, are not sufficient to sign the
derivative. In addition, cross partials must meet a test of reasonableness or plausibility. We
assume that U_BH = U_HB ≥ 0. That is, bad behavior is more rewarding when in better health. The
equality allows for the possibility that the rewards from bad behavior are independent of health
status. We assume H_Mη ≥ 0, H_Bη ≤ 0, H_MB = H_BM = 0. Medicine becomes more important
(or of unchanged importance) to health when health news is bad, behaviors compromise health
more (or compromise health equally) when health news is bad, and the efficacy of medicine is
independent of poor diet and lifestyle choices. The latter condition is equivalent to insulin’s

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