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. PBB +PMM + C ≤I .
Thus the Lagrangian for this optimization problem can be written as follows:
L = U(B, H(B,M; η), C) + λ(I - PbB - PmM - C).
Where I is income, PB and PM 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) LC =UC -λ=0
1d) Lλ=I-PBB-PMM-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 ≡ UhHm —m-(Ub + UhHb ) = 0.
HM B HB
PB