Estimating capabilities with random scale models: women’s freedom of movement

Abstract

In Sen’s capability approach well-being is evaluated not only in terms of functionings (what they do and who they are) but also in terms of capabilities (what people are free to do and to be). It implies that individuals with the same observed functionings may have different well-being because their choice sets (i.e. capabilities) are different. We utilise a Random Scale Model to measure the latent capability of Italian women to move based on observations of their realized choices. We demonstrate that such models can offer a suitable framework for measuring how individuals are restricted in their capabilities. Our estimations show that the percentage of women predicted to be restricted in their freedom of movement (have restricted capability sets) is 23–25%. If all women were unconstrained, our model predicts that 15–17% of them would choose to do more activities.

Appendices

Appendix A: Expected utility is increasing in opportunities

In the following we discuss in more detail our assertion in Sect. 2 that expected utility is increasing in opportunities. It is based on a discussion of the indirect random scale. This is analogous to indirect utility, which gives the maximal attainable utility when faced with given choice set. It reflects both preferences and the choice set. Towards the end, we also discuss how one might analyse how welfare varies across households, though we do not do this in the present paper.

The conditional indirect random scale ${V}{C}(\varepsilon _1,\ldots ,\varepsilon _{H})$    will under our distributional assumptions be extreme value distributed. Let $\bar{V}( {C_s } )$ be the deterministic part (representative part) of the conditional indirect scale, conditional on choice set $C_{s}$ being available, defined as $\bar{V}\left( {C_s } \right) =\hbox {E max}_{k\in C_s} U_k$. Due to the distributional assumptions about $U_{k}$,it is well known that one obtains

$$\begin{aligned} {{\overline{V}}} \left( {C_s } \right) =\log \left( {\sum _{k\in C_s } {\exp \left( {v_k } \right) } } \right) ,\end{aligned}$$

(a1)

where it should be noted that the evaluation $\hbox {exp}\left( {\nu _k } \right)$ is the same across choice sets.

From equation (C.1) it follows that in our case, with ranked latent capability sets, we have $\bar{V} \left( {C_1 } \right)<\bar{V} \left( {C_2 } \right)<\cdots < \bar{V} \left( {C_H } \right)$. In other words, the conditional indirect scale is increasing in the size of the opportunity set.

As a measure of the well-being of individuals, it thereby has the desired property of valuing opportunities instead of only choices. In the following analysis of freedom of movement, we will not be using this measure, since we only consider a one-dimensional concept of freedom and thereby can directly say that it is better to have an unconstrained freedom of movement than a constrained one. If we were trying to evaluate different combinations of freedoms, then having a measure of the above type would be valuable. The unconditional representative indirect scale function is defined by

$$\begin{aligned} \hbox {E} \bar{V} \left( C \right) =\mathop \sum \limits _{s=1}^H \bar{V} \left( {C_s } \right) \cdot r\left( {C_s } \right) =\mathop \sum \limits _{s=1}^H r\left( {C_s } \right) \cdot \log \left( {\mathop \sum \limits _{k\in C_s } \hbox {exp}\left( {\nu _k } \right) } \right) . \end{aligned}$$

(a2)

Thus the conditional indirect scale function is the mean value of the chosen functioning restricted to a given capability set $C_{s}$, whereas the unconditional indirect scale function is the mean value of the conditional indirect scale where the mean is taken over the possible capability sets. By means of $\hbox {E }{{\overline{V}}} \left( C \right)$ one may analyse how welfare (in an ordinal sense) varies across households (identified by covariate values) for given selected capability sets. See Dagsvik (2013) for more details on this and for a discussion of how to develop a welfare function and a capability adjusted income distribution based on the indirect random scale function.

Appendix B: Identification

In the following we illustrate how identification can be achieved by introducing observed discrete covariates into the preference terms $\{v_j \}$ and the restriction probabilities. To see that the model can be identified in this case, we show that the unrestricted choice probabilities and the restriction probabilities can be expressed as functions of the observable probabilities, $Q_{j}$. By this we mean that, within subsamples of observationally identical households, all the probabilities $r\left( {C_k } \right)$ and $P_j \left( {C_k } \right)$, $j\in C_k$, $k = {1, 2,\ldots , L}$, can in principle be estimated by replacing the respective observable probabilities by their empirical counterparts, provided the subsamples are sufficiently large.

To see that introducing discrete covariates can identify our model, consider a two state model. From Eq. (1) we have

$$\begin{aligned} P_1 \left( {C_2 } \right)= & {} \frac{\exp \left( {X_i \beta } \right) }{1+\exp \left( {X_i \beta } \right) }, \end{aligned}$$

(b1)

$$\begin{aligned} P_2 \left( {C_2 } \right)= & {} \frac{1}{1+\exp \left( {X_i \beta } \right) }. \end{aligned}$$

(b2)

From this, together with Eqs. (3) and (5) we get

$$\begin{aligned} Q_1= & {} r\left( {C_1 } \right) +P_1 \left( {C_2 } \right) \cdot r\left( {C_2 } \right) =\frac{\exp \left( {X_i \beta } \right) +\exp \left( {Z_i \gamma } \right) +\exp \left( {X_i \beta +Z_i \gamma } \right) }{1+\exp \left( {X_i \beta } \right) +\exp \left( {Z_i \gamma } \right) +\exp \left( {X_i \beta +Z_i \gamma } \right) }\nonumber \\ \end{aligned}$$

(b3)

$$\begin{aligned} Q_2= & {} P_2 \left( {C_2 } \right) \cdot r\left( {C_2 } \right) =\frac{1}{1+\exp \left( {X_i \beta } \right) +\exp \left( {Z_i \gamma } \right) +\exp \left( {X_i \beta +Z_i \gamma } \right) }. \end{aligned}$$

(b4)

Rewriting Eqs. (b3, b4) as odds-ratios we get

$$\begin{aligned} \frac{Q_1 }{Q_2 }=\exp \left( {X_i \beta } \right) +\exp \left( {Z_i \gamma } \right) +\exp \left( {X_i \beta +Z_i \gamma } \right) . \end{aligned}$$

(b5)

Assume there is one dichotomous explanatory variable in each of the vectors so that $X_{i}=\{{1,x}_{i}\}$ and $Z_{i}=\{{1,z}_{i}\}$, with $x_i \in \left\{ {0,1} \right\}$ and $z_i \in \left\{ {0,1} \right\}$. This means that we can view women as belonging to one of four groups composed of the four different possible combinations of $x_{i }$ and $z_{i}$ (note that as the number of variables increases linearly, the number of possible combinations increases geometrically). We therefore get the following four equations for the four different subgroups among those who might be restricted in their choices:

$$\begin{aligned} \frac{Q_1 }{Q_2 }= & {} \exp \left( {\beta _0 } \right) +\exp \left( {\gamma _0 } \right) +\exp \left( {\beta _0 +\gamma _0 } \right) , \qquad \qquad \qquad \quad if \quad x_i =0,\quad z_i =0\nonumber \\ \end{aligned}$$

(b6)

$$\begin{aligned} \frac{Q_1 }{Q_2 }= & {} \exp \left( {\beta _0 +\beta _1 } \right) +\exp \left( {\gamma _0 } \right) +\exp \left( {\beta _0 +\beta _1 +\gamma _0 } \right) , \quad if \quad x_i =1,\quad z_i =0 \nonumber \\\end{aligned}$$

(b7)

$$\begin{aligned} \frac{Q_1 }{Q_2 }= & {} \exp \left( {\beta _0 } \right) +\exp \left( {\gamma _0 +\gamma _1 } \right) +\exp \left( {\beta _0 +\gamma _0 +\gamma _1 } \right) , \quad if \quad x_i =0,\quad z_i =1 \nonumber \\ \end{aligned}$$

(b8)

$$\begin{aligned} \frac{Q_1 }{Q_2 }= & {} \exp \left( {\beta _0 +\beta _1 } \right) +\exp \left( {\gamma _0 +\gamma _1 } \right) +\exp \left( {\beta _0 +\beta _1 +\gamma _0 +\gamma _1 } \right) , \nonumber \\&\quad if\quad x_i =1,\quad z_i =1 \end{aligned}$$

(b9)

where the parameter vectors are given as $\beta =\left\{ {\beta _0 ,\beta { }_1} \right\} ^{\prime }$ and $\gamma =\left\{ {\gamma _0 ,\gamma { }_1} \right\} ^{\prime }.$ This is four equations in four parameters, so there is now a possibility of the model being identified. Since these equations are non-linear, one cannot generally use a simple counting rule to generally establish identifiability, but the above indicates that a fairly small set of discrete explanatory variable should in practice lead to identification without requiring assumptions about who might be at risk of being restricted.

In general, the above model is only identified if we exogenously decide that a subgroup is never restricted, but in our case we have enough discrete explanatory variables to identify the model in the manner described above without needing to specify an unrestricted subgroup.

For continuous variables identification is readily established. Let

$$\begin{aligned} R\left( {X,Z} \right) =Q_1 \left( {X,Z} \right) /Q_2 \left( {X,Z} \right) . \end{aligned}$$

(b10)

Assume that R(X, Z) is not constant, as a function of X for given Z and as a function of Z for given X. From (b5) we have that

$$\begin{aligned} R\left( {X,Z} \right) =\frac{Q_1 }{Q_2 }=\exp \left( {X_i \beta } \right) +\exp \left( {Z_i \gamma } \right) +\exp \left( {X_i \beta +Z_i \gamma } \right) . \end{aligned}$$

(b11)

Assume that the function R(X, Z) is known for all vectors (X, Z) belonging to some set A. If (b11) holds for all vectors in A we shall show that the vectors of coefficients $\beta$ and $\gamma$ are uniquely determined. From (b11) it follows that

$$\begin{aligned} \frac{\partial R(X,Z)}{\partial X_k }= & {} \exp (X\beta )\beta _k (1+\exp (Z\gamma )), \nonumber \\ \frac{\partial ^{2}R(X,Z)}{\partial X_k^2 }= & {} \exp (X\beta )\beta _k^2 (1+\exp (Z\gamma )) \end{aligned}$$

(b12)

and

$$\begin{aligned} \frac{\partial R(X,Z)}{\partial Z_k }= & {} \exp (Z\gamma )\gamma _k (1+\exp (X\beta )) \quad \quad \nonumber \\ \frac{\partial ^{2}R(X,Z)}{\partial Z_k^2 }= & {} \exp (Z\gamma )\gamma _k^2 (1+\exp (X\beta )). \end{aligned}$$

(b13)

From (b12) and (b13) it follows that

$$\begin{aligned} \frac{\partial ^{2}R(X,Z)}{\partial X_k^2 }\Bigg /\frac{\partial R(X,Z)}{\partial X_k }=\beta _k \end{aligned}$$

(b14)

and

$$\begin{aligned} \frac{\partial ^{2}R(X,Z)}{\partial Z_k^2 }\Bigg /\frac{\partial R(X,Z)}{\partial Z_k }=\gamma _k. \end{aligned}$$

(b15)

Since the partial derivatives of R(X, Z) are known, the relations above demonstrate that $\beta _k$ and $\gamma _k$ are identified for $k>$ 0. It remains to show that the constant terms $\beta _0$ and $\gamma _0$ are identified. Note that we can write (b11) as

$$\begin{aligned} R(X,Z)+1= & {} (1+\exp (X\beta ))(1+\exp (Z\gamma ))\nonumber \\= & {} (1+g_1 (X)\exp \beta _0 )(1+g_2 (Z)\exp \gamma _0 ) \end{aligned}$$

(b16)

where $g_1$ and $g_2$ are known functions, due to the identification results above. Let ${X}'$ be in A and be different from X. Hence, we obtain that

$$\begin{aligned} \frac{R(X,Z)+1}{R({X}',Z)+1}=\frac{1+g_1 (X)\exp \beta _0 }{1+g_1 ({X}')\exp \beta _0 } \end{aligned}$$

(b17)

which implies that

$$\begin{aligned} \left( {g_1 ({X}')\cdot \frac{R(X,Z)+1}{R({X}',Z)+1}-g_1 (X)} \right) \exp \beta _0 =1-\frac{R(X,Z)+1}{R({X}',Z)+1}. \end{aligned}$$

(b18)

Similarly, it follows that

$$\begin{aligned} \left( {g_2 ({Z}')\cdot \frac{R(X,Z)+1}{R(X,{Z}')+1}-g_2 (Z)} \right) \exp \gamma _0 =1-\frac{R(X,Z)+1}{R(X,{Z}')+1} \end{aligned}$$

(b19)

where ${Z}'$ is in A and is different from Z. Since by assumption $\partial R(X,Z)/\partial X\ne 0$ and $\partial R(X,Z)/\partial Z\ne 0$ the last two equations show that $\beta _0$ and $\gamma _0$ are identified (the equations only consist of known fuctions in the variables X and Z). Note that establishing that identification is possible in theory does not necessarily mean that it is always achieved in practice.

Appendix C: Sample selection, cumulative distribution of activities, health and psychological violence variables

Tables 9, 10, 11 and 12 show different statistics related to the data.

Table 9 Sample selection

Table 10 Cumulative distribution of the number of activities a woman participates in $^\mathrm{a}$

Table 11 Women who are healthy; not having health problems$^\mathrm{a}$, 17,350 women

Table 12 Women who have been subjected to psychological violence by partner, 17,350 women