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

Maximizing utility subject to the income constraint and to the health production function can be
written in standard form as
max U(B,H(B,M;n),C )

s.t. P_BB +P_MM + C ≤I .

Thus the Lagrangian for this optimization problem can be written as follows:

L = U(B, H(B,M; η), C) + λ(I - P_bB - P_mM - C).

Where I is income, P_B and P_M are the prices for behaviors and medication, and for simplicity,
the price of all other goods, C, is defined as the numeraire. The first order conditions are
1a) L = U + U H -λP = 0
B B HB B
1b) LM =UHHM-λPM =0
1c) L_C =U_C -λ=0
1d) L_λ=I-P_BB-P_MM-C=0.

Solving 1a and 1b for λ and equating (or for UC through 1c) exhausts the budget constraint 1d
and yields:

U +U H P
ɔʌ B____H B _ B

⁷ UH ~P'
HM   M

That is, the marginal rate of substitution between behaviors and pharmaceuticals that offset the
health cost of behaviors is equal to the price ratio. The marginal utility of behaviors is a net
concept as it includes the direct benefits as well as the health cost.

Equation (2) can be rewritten as

P

F ≡ U_hH_m —^m-(U_b + U_hH_b ) = 0.
HM    B HB

P_B

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