A decomposition method to evaluate the ‘paradox of progress’, with evidence for Argentina

Abstract

The ‘paradox of progress’ is an empirical regularity that associates more education with larger income inequality. Two driving and competing factors behind this phenomenon are the convexity of the ‘Mincer equation’ (that links wages and education) and the heterogeneity in the returns to education, as captured by quantile regressions. We propose a joint least-squares and quantile regression statistical framework to derive a decomposition to evaluate the relative contribution of each explanation. We apply the proposed decomposition strategy to the case of Argentina 1992 to 2015.

Appendix

1.1 A.1 Convexity and heterogeneity lead to higher unconditional inequality

In this Appendix we establish two results. The first one shows that a location shift in a convex Mincer equation leads to more unconditional inequality. The second one proves that a location shift in a linear quantile regression model with increasing heterogeneity leads to more unconditional inequality. We make use of two results in See and Chen (2008).

Convexity leads to higher inequality

Let $Y = f(X+ \epsilon )$, where X is a random variable and f is a differentiable, increasing and convex function. Then

$$\begin{aligned} \frac{ d\; V(Y)}{ d\; \epsilon } \ge 0 \end{aligned}$$

Proof

Start from

$$\begin{aligned} V[f(X+\epsilon )]=E\{f^2(X+\epsilon )\}-E^2[f(X+\epsilon )]. \end{aligned}$$

Taking derivatives with respect to $\epsilon$, and using Lemma 2.2 in See and Chen (2008),

$$\begin{aligned} \frac{d}{d \epsilon } V[f(X+\epsilon )]=E\{2f(X+\epsilon )\frac{d}{d \epsilon }f(X+\epsilon )\}-2E[f(X+\epsilon )]E[\frac{d}{d \epsilon }f(X+\epsilon )]. \end{aligned}$$

Let $h(X):=f(X+\epsilon )$ and $g(X) := \frac{d}{d \epsilon }f(X+\epsilon )$, then

$$\begin{aligned} \frac{d}{d \epsilon } V[f(X+\epsilon )] = 2E\{h(X)g(X)\}-2E[h(X)]E[g(X)] \ge 0, \end{aligned}$$

by Lemma 2.1 in See and Chen (2008), since both h and g are increasing under the assumptions about f.

Heterogeneity leads to more inequality

Now assume $Y=f(X+\epsilon ,U)$, where U represents unobserved heterogeneity and X is a scalar random variable. Consider the following linear quantile regression model $f(X+\epsilon ,U)=a(X+\epsilon )+b(X+\epsilon )\;g(U)$, with $a(X+\epsilon )=a_0 + a_1 (X+ \epsilon )$, where $a_0$ and $a_1$ are scalars, b and g are positive and increasing functions, and $U|X \sim \text{ Uniform }(0,1)$ and independent of X. This is the representation of linear quantile regressions proposed in Koenker and Xiao (2006), as non-linear functions of uniform random variables. More concretely, if, in general $Y = X'\alpha (U)$, and $\alpha (.)$ is an increasing monotonic function, then $Q_{Y|X}(\tau ) = X'\alpha (Q_{Y|X}(\tau ))$ -the standard QR representation-, since quantiles are equivariant under monotonic transformations and U|X is uniform.

Assume $E[g(U)]=0$ and $V[g(U)]=\sigma _{g}^{2}$. Then

$$\begin{aligned} \frac{ d\; V(Y)}{ d\; \epsilon } \ge 0. \end{aligned}$$

Proof

Start from the ‘law of total variance’:

$$\begin{aligned} V(Y) = V[E(Y|X)] + E[V(Y|X)]. \end{aligned}$$

Note that $E(Y|X)=a_0 + a_1\;(X+\epsilon )$ and $V(Y|X)=b(X+\epsilon )^2\sigma _{g}^{2}$. Then,

$$\begin{aligned} V(Y) = a_{1}^{2}\;V(X) + E[b^2(X+\epsilon )]\sigma _{g}^{2}. \end{aligned}$$

Using Lemma 2.2 in See and Chen (2008),

$$\begin{aligned} \frac{d}{d \epsilon } V(Y) = E[2b(X)c(X)]\sigma _{g}^{2} =2E[b(X)c(X)]\sigma _{g}^{2}, \end{aligned}$$

where $c(X):=\frac{\partial }{\partial \epsilon }b(X+\epsilon )$. The result follows since b is positive and increasing.

1.2 A.2 Derivation of eq. (4)

Consider equation (1) and calculate the expectation of W conditional on $X=x$, then

$$\begin{aligned} E(W|x)=x' E[\alpha (U)|x]=x' E[\alpha (U)] := x'\beta . \end{aligned}$$

(8)

Note that $\beta$ has been defined as the expectation of the random vector $\alpha (U)$, where U|x is Uniform(0, 1). That is, the parameters $\beta$ of the conditional expectation are the average of the parameters of all the conditional quantiles.

On the other hand, consider (2) and compute the conditional variance

$$\begin{aligned} Var(W|x) =Var[x'\beta |x] + Var[x'\gamma (U)|x] = x' Var[\gamma (U)] x := x' \Omega x, \end{aligned}$$

(9)

where $\Omega$ has been defined as the matrix of variances of the vector $\gamma (U)$. Note that by construction the expectation $E[\gamma (U)]=E[\alpha (U)-\beta ]=0$ and therefore $Var[\gamma (U)]=E[\gamma (U)\gamma (U)' ]=\Omega$. That is, the matrix $\Omega$ is a notion of distance between the mean and the quantiles of the distribution W|X.

Combining and using the Law of Iterated Variances,

$$\begin{aligned} Var(W)=Var[X' \beta ]+E[X' \Omega X] \end{aligned}$$

Then, using standard properties of variance of the product of vectors and properties of the expectation for quadratic forms:

$$\begin{aligned} Var(W) = \beta ' Var(X)\beta + tr[\Omega Var(X)] + E(X)' \Omega E(X). \end{aligned}$$

1.3 A.3 Derivation of eq. (5)

Equation (5) is a particular case of the notion of (partial) functional derivative proposed by Firpo et al. (2009). In this literature it is usual to assume that the distribution of w|x is not affected by changes in the distribution of x. This assumption translated into our quantile model means that the parameters $\beta$ and $\Omega$ do not change as a consequence of a location shift in any of the regressors included in x. Intuitively, this assumption makes explicit the fact that it is a partial equilibrium analysis, in the sense that a small change in education (measured by h) does not change the returns to education. The functional derivative of the inequality I with respect to a horizontal translation in h is obtained by computing a differential limit of equation (3.4). For example, to derive the expression $\beta 'V\beta$ (first term of I) we solve the following limit:

$$\begin{aligned} \delta [\beta ' Var(x) \beta ]&= \lim _{\varepsilon \rightarrow 0} \frac{[\beta ' Var(x+\varepsilon )\beta ] - [\beta ' Var(x) \beta ]}{\varepsilon } \\&= \lim _{\varepsilon \rightarrow 0} \frac{\beta ' [Var(x+\varepsilon ) - Var(x)] \beta }{\varepsilon } \\&= \beta ' \left[ \lim _{\varepsilon \rightarrow 0} \frac{Var(x+\varepsilon ) - Var(x)}{\varepsilon }\right] \beta := \beta ' \delta (V) \beta \end{aligned}$$

Using the previous reasoning but applied to the rest of the terms of equation (4), it follows that:

$$\begin{aligned} \delta [tr(\Omega V)] = tr[\Omega \delta (V)]&,&\text{ and }&,&\delta (E' \Omega E) = 2E'\Omega \delta (E) \end{aligned}$$

Finally, adding these three components gives equation (5) as a result.

1.4 A.4 Derivation of eq. (6)

To obtain the expressions (6) it is convenient to analyze each of the elements in E and V. The matrix V contains all the variances and covariances of the variables included in the vector x, while E is a vector that contains the expectation x:

$$\begin{aligned} E :=\left[ \begin{matrix} E_0 \\ E_1 \\ E_2 \\ E_z \end{matrix} \right]&,&\text{ and }&,&V :=\left[ \begin{matrix} V_{00} &{} V_{01} &{} V_{02} &{} V_{0z}\\ V_{10} &{} V_{11} &{} V_{12} &{} V_{1z}\\ V_{20} &{} V_{21} &{} V_{22} &{} V_{2z}\\ V_{z0} &{} V_{z1} &{} V_{z2} &{} V_{zz} \end{matrix} \right] , \end{aligned}$$

where the element notation includes the following scalars,

$$\begin{aligned} E_k=E(h^k)&,&\text{ and }&,&V_{jk}=Cov(h^j,h^k ) \end{aligned}$$

for $k=0,1,2$ y $j=0,1,2$, together with the following $(Q\times 1)$ vectors

$$\begin{aligned} E_z=E(z)&,&\text{ and }&,&M_{kz}=Cov(h^k,z)=M_{zk}' \end{aligned}$$

for $k=0,1,2$, and the $(Q \times Q)$ matrix:

$$\begin{aligned} M_{zz}=V(z) \end{aligned}$$

Note that when $k=0$ the vector $M_{0z}$ is a null vector, because it is the covariance between $h^0=1$ with each of the regressors z.

The terms $\delta (E)$ and $\delta (V)$ are the functional derivatives of each of the elements of E and V, respectively. Consider a location shift $\varepsilon$ that only affects the distribution of years of education h. Then we can calculate the functional derivatives of each element in E and V as follows:

• (i) First-order moments of x First, analyze the effect on $E_k$:

  $$\begin{aligned} \delta (E_k) = \lim _{\varepsilon \rightarrow 0} \frac{E[(h+\varepsilon )^k] - E(h^k) }{\varepsilon } = E\left[ \lim _{\varepsilon \rightarrow 0} \frac{(h\varepsilon )^k - h^k }{\varepsilon }\right] = E(kh^{k-1}) = kE_{k-1} \end{aligned}$$

  (10)

  In addition, the vector $E_z=E(z)$ does not change with a horizontal translation in $\varepsilon$, therefore it is evident that $\delta (E_z)=0$. Substituting all this in $\delta (E)$ gives the first part of (6).

• (ii) Variance and covariance between $h^j$ and $h^k$ The origin of this block of matrix V is the inclusion of h and $h^2$ as part of the covariates x. Note that $V_{jk}=E_{j+k}-E_jE_k$, then using the result (10) follows that:

  $$\begin{aligned} \delta (V_{jk})&= \delta (E_{j+k}-E_j E_k )\nonumber \\&=\delta (E_{j+k})-\delta (E_k )E_j - \delta (E_j ) E_k\nonumber \\&= (j+k)E_{j+k-1} - kE_{k-1}E_{j} - jE_{j-1}E_{k}\nonumber \\&= kV_{j(k-1)} + jV_{(j-1)k} \end{aligned}$$

  (11)

  para $j=0,1,2$ y $k=0,1,2$.

• (iii) Covariance between $h^k$ and the regressors z Again, the inclusion of h and $h^2$ together with the rest of the covariates z results in this block of the matrix V. First, note that if $k=0$, the vector $M_{z0}$ is not affected by a location shift in h variable, therefore $\delta (M_{0z})$ is a vector of zeros of dimension Q:

  $$\begin{aligned} \delta (M_{0z})&= 0_{1 \times Q} \end{aligned}$$

  (12)

  For $k> 1$, each element is analyzed separately. Let $z_q$ be a covariable in z. The element q of the vector $M_{kz}$ is $Cov(h^k, z_q) = E(h^k z_q)-E (h^k)E(z_q)$, therefore:

  $$\begin{aligned} \delta [Cov(h^k,z_q )]&= \delta [E(h^k z_q )-E(h^k )E(z_q )] \\&= \delta [E(h^k z_q )]-\delta [E(h^k )]E(z_q ) \\&= kE(h^{k-1} z_q ) - kE(h^{k-1})E(z_q ) \\&= kCov(h^{k-1},z_q) \end{aligned}$$

  for $k=1,2$ and $q=1,2,..., Q$, where result (10) has been used together with the fact that $E(z_q)$ does not change due to a translation of h. Moreover,

  $$\begin{aligned} \delta [E(h^k z_q)]= E\left[ \lim _{\varepsilon \rightarrow 0} \frac{(h+\varepsilon )^k - h^k}{\varepsilon } \cdot z_q \right] =kE(h^{k-1}z_q) \end{aligned}$$

  Then, the vector $M_{kz}$ changes as follows:

  $$\begin{aligned} \delta [M_{kz}] = kM_{(k-1)z} \end{aligned}$$

  (13)

  for $k=1,2$. Note that when $k=1$, the change in $M_{1z}$ is a null vector of dimension Q, because $M_{0z}$ is a vector of zeros.

• (iv) Variances and covariances of z These moments do not depend on the h distribution, therefore $\delta (M_{zz})$ is a null matrix of dimension $Q \times Q$:

  $$\begin{aligned} \delta [M_{zz}] = 0_{Q \times Q} \end{aligned}$$

  (14)

  Substituting the results (11) - (14) in $\delta (V)$ gives as a result the second part of equation (6).

1.5 A.5 Robustness analysis: industry sector dummies to the Mincer equation

Table 3 Partial relationship between wage (log) and educational level. Argentina 1992 - 2015

Table 4 Marginal effect of education on inequality

Fig. 3

Convexity and Heterogeneity of the returns to education (controlling for industry sector). Source: own estimates based on the EPH. Note: the other covariates are evaluated at their sample means