### Appendix 1: Derivation of value trade-off in MCDA model

The multiplicative MAUT assumes utility independence between health, h, and consumption. Then, there exists univariate utilities \( u^{c} \left( c \right), u^{h} \left( h \right). \) The MAUT objective function is:

$$ V\left( {c,h} \right) = k_{c} u^{c} \left( c \right) + k_{h} u^{h} \left( h \right) + Kk_{c} k_{h} u^{c} \left( c \right)u^{h} \left( h \right) $$

\( k_{c} \) and \( k_{h} \) are scaling constants which satisfy \( 0 \le k_{j} \le 1, j = c,h \), and *K* is an additional scaling constant which captures interaction between consumption, *c*, and health, *h*, satisfying

$$ K = \frac{{1 - k_{c} - k_{h} }}{{k_{c} k_{h} }} .$$

The value trade-off in terms of consumption for health can be derived as the slope of any indifference curve \( V\left( {c,h} \right)\text{ = }\overline{V} \), where \( \frac{\partial V}{\partial c}dc + \frac{\partial V}{\partial h}dh = 0{:}\)

$$ - \frac{dc}{dh} = \frac{\partial V/\partial h}{\partial V/\partial c} = \frac{{u^{{h{\prime }}} }}{{u^{{c{\prime }}} }}\frac{{\left[ {k_{h} + \left( {1 - k_{c} - k_{h} } \right)u^{c} \left( c \right)} \right]}}{{\left[ {k_{c} + \left( {1 - k_{c} - k_{h} } \right)u^{h} \left( h \right)} \right]}} \left[ {MAUT \, VTO} \right] .$$

Considering the MAUT properties, the range of the scalings constants, \( k_{c} , k_{h} , \) and the utilities, \( u^{c} \left( c \right) \), \( u^{h} \left( h \right) \) is the unitary interval. Assuming standard utlity properties, \( u^{c} \left( c \right) \) and \( u^{h} \left( h \right) \) are monotonically increasing functions \( \left( {u^{{c{\prime }}} \text{ > 0;}u^{{h{\prime }}} \text{ > }0} \right), \) with non-increasing marginal utility \( \left( {u^{{c{\prime \prime }}} \le 0;u^{{h{\prime \prime }}} \le 0} \right) \). Then, the MAUT value trade-off, MAUT VTO, has the following properties:

MAUT VTO is positive

$$ MAUT \, VTO = \frac{{u^{{h{\prime }}} }}{{u^{{c{\prime }}} }}\frac{{\left[ {k_{h} + \left( {1 - k_{c} - k_{h} } \right)u^{c} \left( c \right)} \right]}}{{\left[ {k_{c} + \left( {1 - k_{c} - k_{h} } \right)u^{h} \left( h \right)} \right]}} = \frac{{u^{{h{\prime }}} }}{{u^{{c{\prime }}} }}\frac{{\left[ {k_{h} \left( {1 - u^{c} \left( c \right)} \right) + \left( {1 - k_{c} } \right)u^{c} \left( c \right)} \right]}}{{\left[ {k_{c} \left( {1 - u^{h} \left( h \right)} \right) + \left( {1 - k_{h} } \right)u^{h} \left( h \right)} \right]}} > 0. $$

MAUT VTO is strictly positive since both the numerator and denominator are positive.

MAUT VTO is not defined for an enrollee with maximum health \( h^{*} \) and maximum consumption \( c^{*} \) \( \left( {u^{c} \left( {c^{*} } \right) = u^{h} \left( {h^{*} } \right) = 1} \right) \), and defined as “non-trader” or “extreme risk-averse” because she elicits \( k_{h} = k_{c} = 1 \) in the probabilistic scaling trade-off task. That is, the extreme risk-averse enrollee strictly prefers corner solutions, \( \left( {c^{*} ,0} \right), \left( {0,h^{*} } \right) \) to the lottery, in the sense that she is only indifferent to the “lottery” which results in the best outcome \( \left( {c^{*} ,h^{*} } \right) \) with certainty. In this case, both the numerator and the denominator are zero and the value trade-off cannot be elicited.

If the interaction term \( \varvec{K} \ge 0, \) MAUT VTO is increasing with the level of consumption

$$ \frac{\partial }{\partial c}\left( {MAUT \, VTO} \right) = \frac{\partial }{\partial c}\frac{\left[ N \right]}{\left[ D \right]} = \frac{{u^{{c{\prime }}} u^{{h{\prime }}} \left( {1 - k_{c} - k_{h} } \right)}}{\left[ D \right]} - \frac{{u^{{c{\prime \prime }}} \left[ {k_{c} + \left( {1 - k_{c} - k_{h} } \right)u^{h} \left( h \right)} \right]\left[ N \right]}}{{\left[ {D^{2} } \right]}} \ge 0 $$

. where \( \left[ N \right] \) and \( \left[ D \right] \) denote the numerator and denominator of the *MAUT VTO* expression.

###
*Proof*

Given that \( \left[ N \right] > 0 \) and \( \left[ D \right] > 0 \) (except that \( \left[ N \right] = 0 \) and \( \left[ D \right] = 0 \) for the extreme case defined as “non-trader” in the maximum as above).

If \( K \ge 0 \Leftrightarrow \left( {k_{c} \text{ + }k{}_{h}} \right) \le 1, \) \( \frac{{u^{c'} u^{h'} \left( {1 - k_{c} - k_{h} } \right)}}{\left[ D \right]} \ge 0; \) Since \( u^{{c{\prime \prime }}} \le 0 \Rightarrow \frac{{u^{c''} \left[ {k_{c} + \left( {1 - k_{c} - k_{h} } \right)u^{h} \left( h \right)} \right]\left[ N \right]}}{{\left[ {D^{2} } \right]}} \le 0.\)

Therefore, \( If \, K \ge 0, \frac{\partial }{\partial c}\left( {MAUT \, VTO} \right) \ge 0 . \) We can consider two cases:

Risk neutrality: \( \varvec{u}^{\varvec{c}} \left( \varvec{c} \right) \) is linear, \( \varvec{u}^{{\varvec{c''}}} = 0. \) In this case, \( \frac{\partial }{{\partial \varvec{c}}}\left( {\varvec{MAUT}\,\varvec{VTO}} \right) \) has the same sign as the interaction term, that is, as \( \left( {1 - \varvec{k}_{\varvec{c}} - \varvec{k}_{\varvec{h}} } \right).\varvec{ } \) A positive interaction term is more intuitive since it implies that WTP is larger the larger the enrollee’s consumption level. If MAUT is additive, \( \left( {1 - \varvec{k}_{\varvec{c}} - \varvec{k}_{\varvec{h}} } \right) = 0,\varvec{ } \) and *MAUT VTO* does not change with the level of consumption.

Risk aversion: \( \varvec{u}^{\varvec{c}} \left( \varvec{c} \right) \) is concave, \( \varvec{u}^{{\varvec{c''}}} < 0. \) In this case, if the interaction term is non-negative, MAUT VTO is strictly increasing in *c*, including with additive MAUT \( \left( {\frac{\partial }{{\partial \varvec{c}}}\left( {\varvec{MAUT}\,\varvec{VTO}} \right) > 0} \right). \) If the interaction term is negative, the sign of \( \frac{\partial }{{\partial \varvec{c}}}\left( {\varvec{MAUT}\,\varvec{VTO}} \right) \) is ambiguous. Note that this is the standard definition of risk aversion through the utility function and it is not related to the case of a decision maker eliciting scaling constant equal to one, labelled above as “extreme risk-averse”, that results in a negative interaction term *K *= -*1*.

If the interaction term \( \varvec{K} \ge 0, \) MAUT VTO is decreasing with the level of health

$$ \frac{\partial }{\partial h}\left( {MAUT \, VTO} \right) = \frac{\partial }{\partial h}\frac{\left[ N \right]}{\left[ D \right]} = \frac{{u^{h} {\prime \prime }\left[ {k_{h} + \left( {1 - k_{c} - k_{h} } \right)u^{c} \left( c \right)} \right]}}{\left[ D \right]} - \frac{{u^{c} {\prime }u^{h} {\prime }\left( {1 - k_{c} - k_{h} } \right)}}{{\left[ {D^{2} } \right]}} $$

.

where \( \left[ N \right] \) and \( \left[ D \right] \) denote the numerator and denominator of the *MAUT VTO* expression.

Therefore, \( If \, K \ge 0, \frac{\partial }{\partial h}\left( {MAUT \, VTO} \right) \le 0 . \) We can consider two cases:

Risk neutrality: \( \varvec{u}^{\varvec{h}} \left( \varvec{h} \right) \) is linear, \( \varvec{u}^{{\varvec{h''}}} = 0. \) In this case, \( \frac{\partial }{{\partial \varvec{h}}}\left( {\varvec{MAUT}\, \varvec{VTO}} \right) \) has the opposite sign to the interaction term, that is, negative if \( \left( {1 - \varvec{k}_{\varvec{c}} - \varvec{k}_{\varvec{h}} } \right) > 0. \) A positive interaction term is more intuitive since it implies that WTP is lower the larger the enrollee’s health level. If MAUT is additive, \( \left( {1 - \varvec{k}_{\varvec{c}} - \varvec{k}_{\varvec{h}} } \right) = 0,\varvec{ } \) and MAUT VTO does not change with the level of health.

Risk aversion: \( \varvec{u}^{\varvec{h}} \left( \varvec{h} \right) \) is concave, \( \varvec{u}^{{\varvec{h''}}} < 0. \) In this case, if the interaction term is non-negative, MAUT VTO is strictly decreasing in *h*, including with additive MAUT \( \left( {\frac{\partial }{{\partial \varvec{c}}}\left( {\varvec{MAUT}\,\varvec{VTO}} \right) > 0} \right). \) If the interaction term is negative, the sign of \( \frac{\partial }{{\partial \varvec{h}}}\left( {\varvec{MAUT} \varvec{VTO}} \right) \) is ambiguous. Similarly, this concept of risk aversion operates through utility, not scaling constants.

The proofs below demonstrate the effect of increases of income (↑ *y*), costs of the medical technology (↑ *p*), and severity of disease (↓ \( {\text{h}} \)) on the *MAUT VTO*, considering that these effects channel through changes in consumption or health and how *MAUT VTO* changes:

*MCDA Property 1: For a medical technology, the value trade*-*off between costs and health gain (QALY) decreases with the level of costs.*

###
*Proof*

*If p* ↑, c \( \downarrow \) \( \Rightarrow \) *MAUT VTO* \( \downarrow \) (increasing in *c*).

*MCDA Property 2: For a medical technology, the value trade*-*off between consumption and health gain (QALY) increases with the level of income.*

###
*Proof*

*If y* ↑, c \( \uparrow \). And assume \( k_{c} \) does not change because the consumption range \( \left[ {0,y^{w} } \right] \) does not change \( \Rightarrow \) *MAUT VTO* \( \uparrow \) (increasing in *c*).

*MCDA Property 3: For a medical technology, the value trade*-*off between consumption and health gain (QALY) increases with the severity of disease.*

###
*Proof*

Severity of disease can have two effects: (a) Reducing the level of health: *If h* \( \downarrow \varvec{ } \) *MAUT VTO* \( \uparrow \) (decreasing in *h*). (b) Increasing the range of health gain and this causes the health attribute becomes more predominant: \( \varvec{k }_{\varvec{h}} \) ↑. Taking derivatives of *MAUT VTO* w.r.t \( \varvec{k }_{\varvec{h}} \) ↑ gives:

$$ \frac{\partial }{{\partial k_{h} }}\left( {MAUT \, VTO} \right) = \frac{{{\text{u}}^{{{\text{h}}{\prime }}} \left[ {1 - {\text{u}}^{\text{c}} \left( {\text{c}} \right)} \right]}}{{\left[ {\text{D}} \right]}} + \frac{{{\text{u}}^{{{\text{c}}{\prime }}} {\text{u}}^{\text{h}} \left( {\text{h}} \right)\left[ {\text{N}} \right]}}{{\left[ {\text{D}} \right]^{2} }} > 0. $$

Given that \( 0 \le u^{c} \left( c \right) \le 1,\) the sign of the derivative is positive: *MAUT VTO* \( \uparrow \).

### Appendix 2: Derivation of value trade-off in ACEA model

To demonstrate ACEA Key Properties, we analyse the changes implied for the value trade-off, which is defined as the marginal rate of substitution between consumption and health along the EU indifference curve \( E\left( u \right)\text{ = }\pi u\left( {c^{s} ,h^{s} } \right)\text{ + }\left( {1\text{ - }\pi } \right)u\left( {c^{w} ,h^{w} } \right)\text{ = }\overline{EU},\) so that it is invariant to changes in health and consumption caused by access to the medical treatment, that is:

$$ \frac{\partial E\left( u \right)}{{\partial c^{s} }}dc^{s} + \frac{\partial E\left( u \right)}{{\partial h^{s} }}dh^{s} + \frac{\partial E\left( u \right)}{{\partial c^{w} }}dc^{w} = \pi u_{c}^{s} dc^{s} + \pi u_{h}^{s} dh^{s} + \left( {1 - \pi } \right)u_{c}^{w} dc^{w} = 0 $$

$$ MRS\left( {c^{s} ,h^{s} } \right) = \frac{{\partial E\left( u \right)/\partial h^{s} }}{{\partial E\left( u \right)/\partial c^{s} }} = - \frac{{dc^{s} }}{{dh^{s} }} = \frac{{u_{h}^{s} }}{{u_{c}^{s} }} + \frac{1 - \pi }{\pi }\frac{{u_{c}^{w} }}{{u_{c}^{s} }}\frac{{dc^{w} }}{{dh^{s} }} .$$

This *MRS* can be evaluated for a given value trade-off between consumption and health in the well state expressed as a proportion (γ) of the value trade-off between consumption and health in the sick state: \( \frac{{dc^{w} }}{{dh^{s} }} = \gamma \frac{{dc^{s} }}{{dh^{s} }} \). Then, the value trade-off of interest is defined as as money for health gain in the sick status, or how much consumption an enrollee would expect to give up for QALY gains obtained from the medical treatment and remain indifferent to not having the treatment. ACEA *ex ante* value trade-off is:

$$ - \frac{{dc^{s} }}{{dh^{s} }} = \frac{{\pi u_{h}^{s} }}{{\pi u_{c}^{s} + \gamma \left( {1 - \pi } \right)u_{c}^{w} }} \left[ {ACEA \, VTO} \right] .$$

Considering the medical treatment is funded through insurance coverage, consumption in the sick state is increased by the net payment of the insurance coverage which is \( \left( {1 - \pi } \right)p \) with complete coverage. Therefore \( c^{s} = y^{s} - \pi p \), so that medical spending decreases consumption in the sick state by:

$$ \frac{{dc^{s} }}{dp} = - \pi. $$

Therefore, the value trade-off between the cost of medical treatment and health gain defines the maximum increase in the cost of technology that the payer would expect to reimburse for QALY gains. ACEA payer’s value trade-off:

$$ \frac{dp}{{dh^{s} }} = \frac{{u_{h}^{s} }}{{\pi u_{c}^{s} + \gamma \left( {1 - \pi } \right)u_{c}^{w} }} \left[ {ACEA \, PVTO} \right]. $$

Preferences between consumption and QALY are represented by a monotonically increasing utility function, \( u\left( {c,h} \right) \), with decreasing marginal utilities: \( u_{c} > 0; u_{h} > 0; u_{cc} < 0; u_{hh} < 0. \) In principle, we allow positive or negative interactions between consumption and QALYs, that is, *c* and *h* are Hicksian q-complements if \( u_{ch} > 0 \), and q-substitutes if \( u_{ch} < 0. \) If \( u\left( {c,h} \right) \) is additively separable, \( u_{ch} = 0. \) Then, ACEA VTO has the following properties (the expressions for *ACEA PVTO* are similar and have the same sign).

ACEA VTO is positive. The proof is immediate since marginal utilities are strictly positive: \( u_{c} > 0; u_{h} > 0. \) Then, numerator and denominator are positive. If \( \varvec{u}_{{\varvec{ch}}} \ge 0 \), ACEA VTO is increasing with the level of consumption:

$$ \frac{\partial }{\partial c}\left( {ACEA VTO} \right) = \frac{\partial }{\partial c}\frac{\left[ N \right]}{\left[ D \right]} = \frac{{\pi u_{ch}^{s} }}{\left[ D \right]} - \frac{{\left( {\pi u_{cc}^{s} + \gamma \left( {1 - \pi } \right)u_{cc}^{w} } \right)\left[ N \right]}}{{\left[ {D^{2} } \right]}} > 0. $$

where \( \left[ N \right] \) and \( \left[ D \right] \) denote the numerator and denominator of the *ACEA VTO* expression.

###
*Proof*

The first term is non-negative \( \left( {if \, u_{ch}^{s} \ge 0} \right). \) The second term substracts a strictly negative term. If \( \varvec{u}_{{\varvec{ch}}} \ge 0, \) ACEA VTO is decreasing with the level of health:

$$ \frac{\partial }{\partial h}\left( {ACEA VTO} \right) = \frac{\partial }{\partial h}\frac{\left[ N \right]}{\left[ D \right]} = \frac{{\pi u_{hh}^{s} }}{\left[ D \right]} - \frac{{\left( {\pi u_{ch}^{s} + \gamma \left( {1 - \pi } \right)u_{ch}^{w} } \right)\left[ N \right]}}{{\left[ {D^{2} } \right]}} < 0. $$

where \( \left[ N \right] \) and \( \left[ D \right] \) denote the numerator and denominator of the *ACEA VTO* expression.

###
*Proof*

The first term is strictly negative \( \left( {u_{hh}^{s} < 0} \right). \) The second term substracts a non-negative term if \( u_{ch}^{s} \ge 0, \) and \( u_{ch}^{w} \ge 0. \)

The proofs below demonstrate the effect of increases of income (↑ *y*), the cost of the medical technology (↑ *p*), and severity of disease (↓ \( {\text{h}} \)) on the *ACEA VTO*, considering that these effects channel through changes in consumption or health and analyze how *ACEA VTO* (and *ACEA PVTO*) changes:

*ACEA Property 1: For a medical technology, the value trade-off between consumption and health gain (QALY), and the value trade-off between costs and health gain (QALY) decreases with the level of costs.*

###
*Proof*

Consider the consumption in the sick state for an insured individual \( c^{s} = y^{s} - \pi p, where \,\pi \, p \) is the insurance premium. This means that the net transfer to the individual in the sick state is \( p - \pi p = \left( {1 - \pi } \right)p. \)

*If p* ↑, \( c^{s} \downarrow \) \( \Rightarrow ACEA \) *VTO* \( \downarrow \) and \( ACEA \) *PVTO* \( \downarrow \) (increasing in *c*).

*ACEA Property 2: For a medical technology, the value trade-off between consumption and health (QALY), and the value trade-off between costs and health gain (QALY) increases with income.*

###
*Proof*

*If* \( y^{s} \uparrow \) *and/or* \( y^{w}\uparrow , c^{s}\uparrow, c^{w}\uparrow \Rightarrow ACEA \) *VTO* \( \uparrow \) and \( ACEA \) *PVTO* \( \uparrow \) (increasing in *c*).

*ACEA Property 3: For a medical technology, the value trade-off between consumption and health gain (QALY), and the value trade-off between costs and health gain (QALY) increase with the severity of disease.*

###
*Proof*

*If* \( h^{s} \downarrow \) \( \Rightarrow ACEA \) *VTO* \( \uparrow \) and \( ACEA \) *PVTO* \( \uparrow \) (decreasing in *h*).