Public information, education and welfare

Abstract

We study investments in education made by a continuum of individuals under imperfect information on returns. In the model, the wage accruing to single units of human capital is bargained in the labour market. In addition, this wage depends positively on a stochastic fundamental and, negatively, on the aggregate supply of human capital. At the time of investment, any agent observes a public and a private signal of the fundamental. We show that an increase in the precision of the public signal has an ambiguous impact on the expected social welfare. On the one hand, social welfare improves as schooling decisions become closer to those decisions that would be made under perfect information. On the other hand, social welfare deteriorates as externalities and coordination failures become more severe. The key result of our analysis is that an increase in the quality of public information turns out to be welfare improving, unless the distortions on the labour market are particularly strong.

Appendices

Appendix 1: Bargaining

Using the Lagrange multiplier $\lambda$, the Nash problem in Eq. (1) may be written as follows:

$$\begin{aligned} Max_{\lambda ,w,\left\{ H_{j}\right\} _{j=1\ldots m}}\,\left[ \sum \limits _{j}\left( Y_{j}-wH_{j}\right) \right] ^{\beta }\left( w\sum \limits _{j}H_{j}\right) ^{1-\beta }+\lambda \left[ H-\sum \limits _{j}H_{j}\right] , \ \ \ \lambda \ge 0 \end{aligned}$$

First order conditions with respect to $H_{j}$, $j=1\ldots m$ and w are

$$\begin{aligned} \begin{array}{c} \left[ \left( 1-v\right) \theta H_{j}^{-v}-w\right] \beta \left[ \frac{ w\sum \nolimits _{j}H_{j}}{\sum \nolimits _{j}\left( \theta H_{j}^{1-v}-wH_{j}\right) }\right] ^{1-\beta }+\left( 1-\beta \right) w\left[ \frac{\sum \nolimits _{j}\left( \theta H_{j}^{1-v}-wH_{j}\right) }{ w\sum \nolimits _{j}H_{j}}\right] ^{\beta }-\lambda =0 \\ \\ \frac{w\sum \nolimits _{j}H_{j}}{\sum \nolimits _{j}\left( \theta H_{j}^{1-v}-wH_{j}\right) }=\frac{1-\beta }{\beta } \end{array} \end{aligned}$$

Substitute the second into the first and rearrange:

$$\begin{aligned} \lambda =\left( 1-v\right) \theta H_{j}^{-v}\left( 1-\beta \right) \left[ \frac{\beta }{1-\beta }\right] ^{\beta } \end{aligned}$$

(31)

An implication of Eq. (31) is that $\lambda >0$: the constraint on the total amount of human capital is strictly binding. A second implication is that all firms employ the same amount of human capital:

$$\begin{aligned} H_{j}=H/m \end{aligned}$$

Substitute the latter in the f.o.c. for w and solve:

$$\begin{aligned} w=\left( 1-\beta \right) \theta m^{v}H^{-v} \end{aligned}$$

This expression coincides with the wage equation reported in the main text [Eq. (2)]

Appendix 2: Linear-quadratic approximation

1.1 2.1 Utility function

In this section we compute the linear-quadratic approximation of the objective function

$$\begin{aligned} {\mathcal {U}}(h_{i},H,\theta )=\beta m^{v}\theta H^{1-v}+\left( 1-\beta \right) m^{v}\theta H^{-v}h_{i}-\frac{h_{i}^{1+\phi }}{\delta \left( 1+\phi \right) } \end{aligned}$$

The approximation is computed around the non-stochastic equilibrium as defined by the following first order condition:

$$\begin{aligned} h_{i}^{\phi }=\delta \left( 1-\beta \right) m^{v}\theta H^{-v}, \end{aligned}$$

(32)

Solving this condition gives equilibrium investments, output and wages:

$$\begin{aligned} {\overline{h}}_{i}={\overline{H}}=\left[ \delta \left( 1-\beta \right) {\overline{\theta }}m^{v}\right] ^{\frac{1}{\phi +v}} \ \ \ \ \ \ {\overline{Y}}= {\overline{\theta }}m^{v}{\overline{H}}^{1-v} \ \ \ \ \ \ \ \ {\overline{w}} =\left( 1-\beta \right) {\overline{Y}}/{\overline{H}} \end{aligned}$$

The derivatives of ${\mathcal {U}}(h_{i},H,\theta )$ computed in this equilibrium are the following:

$$\begin{aligned} \begin{array}{ccc} {\mathcal {U}}_{h}=0 &{} {\mathcal {U}}_{hh}=-\phi \left( 1-\beta \right) {\overline{Y}}{\overline{H}}^{-2} &{} {\mathcal {U}}_{hH}=-v\left( 1-\beta \right) {\overline{Y}} {\overline{H}}^{-2} \\ {\mathcal {U}}_{h\theta }=\left( 1-\beta \right) {\overline{Y}}{\overline{H}}^{-1} {\overline{\theta }}^{-1} &{} {\mathcal {U}}_{H}=\left( \beta -v\right) {\overline{Y}} {\overline{H}}^{-1} &{} {\mathcal {U}}_{HH}=v\left( 1+v-2\beta \right) {\overline{Y}} {\overline{H}}^{-2} \\ {\mathcal {U}}_{H\theta }=\left( \beta -v\right) {\overline{Y}}{\overline{H}}^{-1} {\overline{\theta }}^{-1} &{} {\mathcal {U}}_{\theta }={\overline{Y}}{\overline{\theta }}^{-1} &{} {\mathcal {U}}_{\theta \theta }=0 \end{array} \end{aligned}$$

The approximated objective function $U(h_{i},H,\theta )$ is

$$\begin{aligned} U(h_{i},H,\theta )= & {} {\mathcal {U}}\left( {\overline{H}},{\overline{H}},{\overline{\theta }}\right) +{\mathcal {U}}_{H}\left( H-{\overline{H}}\right) +{\mathcal {U}} _{\theta }\left( \theta -{\overline{\theta }}\right) \\&+\frac{{\mathcal {U}}_{hh}}{2}\left( h_{i}-{\overline{H}}\right) ^{2}+{\mathcal {U}} _{hH}\left( h_{i}-{\overline{H}}\right) \left( H-{\overline{H}}\right) +\frac{ {\mathcal {U}}_{HH}}{2}\left( H-{\overline{H}}\right) ^{2} \\&+{\mathcal {U}}_{h\theta }\left( h_{i}-{\overline{H}}\right) \left( \theta - {\overline{\theta }}\right) +{\mathcal {U}}_{H\theta }\left( H-{\overline{H}} \right) \left( \theta -{\overline{\theta }}\right) -C(\pi _{x_{i}}). \end{aligned}$$

Since the decisions concerning information acquisition take place at the earliest stage of the model, it is useful to compute the ex-ante expected utility. Notice that, using the definition for $h(x_{i},z)$ provided in (7), that for $H(\theta ,z)$ in (8), and (11), we have that

$$\begin{aligned} E\left[ \left( H-{\overline{H}}\right) ^{2}\right]&=\varrho ^{2}\left( \frac{ 1}{\pi _{\theta }}+\frac{\gamma ^{2}}{\pi _{z}}-\frac{2\gamma }{\pi _{z}} \right) , \\ E\left[ \left( h_{i}-{\overline{H}}\right) \left( H-{\overline{H}}\right) \right]&=E\left[ \left( h_{i}-H\right) \left( H-{\overline{H}}\right) +\left( H- {\overline{H}}\right) ^{2}\right] =E\left[ \left( H-{\overline{H}}\right) ^{2} \right] , \\ E\left[ \left( h_{i}-{\overline{H}}\right) ^{2}\right]&=E\left[ \left( h_{i}-H+H-{\overline{H}}\right) ^{2}\right] =\varrho ^{2}\frac{\left( 1-\gamma \right) ^{2}}{\pi _{x_{i}}}+\varrho ^{2}\left( \frac{1}{\pi _{\theta }}+ \frac{\gamma ^{2}}{\pi _{z}}-\frac{2\gamma }{\pi _{z}}\right) , \\ E\left[ \left( H-{\overline{H}}\right) \left( \theta -\overline{\theta } \right) \right]&=\varrho \left( \frac{1}{\pi _{\theta }}-\frac{\gamma }{ \pi _{z}}\right) , \\ E\left[ \left( h_{i}-{\overline{H}}\right) \left( \theta -{\overline{\theta }} \right) \right]&=E\left[ \left( h_{i}-H\right) \left( \theta -{\overline{\theta }}\right) +\left( H-{\overline{H}}\right) \left( \theta -{\overline{\theta }}\right) \right] =E\left[ \left( H-{\overline{H}}\right) \left( \theta - {\overline{\theta }}\right) \right] . \end{aligned}$$

Notice that the notation $\pi _{x_{i}}$ underscores the fact that this is the precision that individual i takes into account; notice also that there is no need to distinguish the ‘individual’ weight $\gamma _{i}$ from the ‘average’ $\gamma$ since the envelope theorem guarantees that the impact of $\pi _{x_{i}}$ passing through ‘individual’ weight $\gamma _{i}$ is nil.

The variances and covariances obtained above allow to express the ex-ante utility as

$$\begin{aligned} E\left[ U(h_{i},H,\theta )\right]= & {} {\mathcal {U}}\left( {\overline{H}},{\overline{H}},{\overline{\theta }}\right) +\left( \frac{{\mathcal {U}}_{hh}}{2}+{\mathcal {U}} _{hH}+\frac{{\mathcal {U}}_{HH}}{2}\right) \frac{\varrho ^{2}}{\pi _{\theta }} +\left( {\mathcal {U}}_{h\theta }+{\mathcal {U}}_{H\theta }\right) \varrho \frac{1 }{\pi _{\theta }} \nonumber \\&+\frac{{\mathcal {U}}_{hh}}{2}\varrho ^{2}\frac{\left( 1-\gamma \right) ^{2}}{ \pi _{x_{i}}}+\left( \frac{{\mathcal {U}}_{hh}}{2}+{\mathcal {U}}_{hH}+\frac{ {\mathcal {U}}_{HH}}{2}\right) \varrho ^{2}\left( \frac{\gamma ^{2}}{\pi _{z}}- \frac{2\gamma }{\pi _{z}}\right) \nonumber \\&-\left( {\mathcal {U}}_{h\theta }+{\mathcal {U}}_{H\theta }\right) \varrho \frac{ \gamma }{\pi _{z}}-C(\pi _{x_{i}}). \end{aligned}$$

(33)

1.2 2.2 Information acquisition and expected welfare

In this section we derive the equation that determines the optimal precision of the private signal $\pi _{x_{i}}$ [Eq. (13) in the main text].

The equation corresponds to the first order condition of Problem (12), which we report below for immediate reference:

$$\begin{aligned} \underset{\pi _{x_{i}}}{\max }E\left[ U\left( h_{i},H,\theta |z,x_{i}\right) \right] -C(\pi _{x_{i}}) \end{aligned}$$

To solve this problem we first compute the expectation $E\left[ U\left( h_{i},H,\theta |z,x_{i}\right) \right]$ by substituting—where relevant—the values for the derivatives of $U(h_{i},H,\theta )$ obtained in Appendix 2.1:

$$\begin{aligned} E\left[ U(h_{i},H,\theta )\right]= & {} {\mathcal {U}}\left( {\overline{H}},{\overline{H}},{\overline{\theta }}\right) +\left( \frac{{\mathcal {U}}_{hh}}{2}+{\mathcal {U}} _{hH}+\frac{{\mathcal {U}}_{HH}}{2}\right) \frac{\varrho ^{2}}{\pi _{\theta }} +\left( {\mathcal {U}}_{h\theta }+{\mathcal {U}}_{H\theta }\right) \varrho \frac{1 }{\pi _{\theta }} \nonumber \\&-\frac{1}{2}\frac{{\overline{Y}}}{{\overline{H}}^{2}}\phi \left( 1-\beta \right) \varrho ^{2}\frac{\left( 1-\gamma \right) ^{2}}{\pi _{x_{i}}}-\frac{1}{2} \frac{{\overline{Y}}}{{\overline{H}}^{2}}\left( \left( \phi +v\right) \left( 1-v\right) +\phi \left( v-\beta \right) \right) \varrho ^{2}\left( \frac{ \gamma ^{2}}{\pi _{z}}-\frac{2\gamma }{\pi _{z}}\right) \nonumber \\&-\frac{{\overline{Y}}}{{\overline{H}}{\overline{\theta }}}\left( 1-v\right) \varrho \frac{\gamma }{\pi _{z}}-C(\pi _{x_{i}}). \end{aligned}$$

(34)

Second, we differentiate $E\left[ U\left( h_{i},H,\theta \right) \right]$ with respect to $\pi _{x_{i}}$. In computing the derivative, we take account of the fact that, by the envelope theorem, the impact of $\pi _{x_{i}}$ passing through the change in parameters $h_{1}$ and $h_{2}$ is nil:

$$\begin{aligned} \frac{dE\left[ U(.)\right] }{d\pi _{x_{i}}}=\frac{1}{2}\frac{{\overline{Y}}}{ {\overline{H}}^{2}}\phi \left( 1-\beta \right) \varrho ^{2}\frac{\left( 1-\gamma \right) ^{2}}{\pi _{x_{i}}^{2}}\left( =\frac{1}{2}\phi \left( 1-\beta \right) \frac{{\overline{Y}}}{{\overline{H}}^{2}}\frac{h_{1}^{2}}{\pi _{x_{i}}^{2}}\right) \end{aligned}$$

Notice that this derivative coincides with the LHS of Eq. (13) in the main text. The RHS is straightforward.

Having computed the expected utility for the i-th individual—i.e. $E\left[ U\left( h_{i},H,\theta \right) \right]$—the expected welfare under the laissez faire can be immediately derived by aggregation. In fact, the welfare expression used in the main text corresponds to the aggregation of equation (34) after accounting for the fact that, in equilibrium, all agents acquire the same level of private information ($\pi _{x_{i}}=\pi _{x}$).

Appendix 3: Proofs

Proof for Proposition 2

Substitute Equations (29) and (26) in Equation (24):

$$\begin{aligned} \frac{dE\left[ W\right] }{d\pi _{z}}= & {} \frac{1}{2}\frac{{\overline{Y}}}{ {\overline{H}}^{2}}\left( \left( \phi +v\right) \left( 1-v\right) +\phi \left( v-\beta \right) \right) \varrho ^{2}\frac{\gamma ^{2}}{\pi _{z}^{2}}-\frac{ {\overline{Y}}}{{\overline{H}}^{2}}\phi \left( v-\beta \right) \varrho ^{2}\frac{ \gamma }{\pi _{z}^{2}}+ \nonumber \\&+\varrho ^{2}\frac{{\overline{Y}}}{{\overline{H}}^{2}}\frac{\pi _{z}+\pi _{x}}{ \pi _{z}}\frac{\phi \left( \phi +v\right) }{\left( \phi \pi _{z}+\left( \phi +v\right) \pi _{x}\right) }\left( v-\beta \right) \frac{d\gamma }{d\pi _{z}} \end{aligned}$$

(35)

Next, substitute in the latter the expression for $\frac{d\gamma }{d\pi _{z}}$ that holds when acquisition costs are linear—i.e. Equation (16)—and the expression for $\gamma$ contained in Equation (10):

$$\begin{aligned} \left. \frac{dE\left[ W\right] }{d\pi _{z}}\right| _{C^{^{\prime \prime }}=0}=\frac{1}{2}\frac{{\overline{Y}}}{{\overline{H}}^{2}}\left[ \left( \phi +v\right) \left( 1-v\right) +\left( 2v+\phi \right) \left( v-\beta \right) \right] \varrho ^{2}\frac{\gamma ^{2}}{\pi _{z}^{2}} \end{aligned}$$

(36)

From Equation (36) it is immediate to conclude that Restriction (30) is both necessary and sufficient for the derivative $\left. \frac{dE\left[ W\right] }{d\pi _{z}}\right| _{C^{^{\prime \prime }}=0}$ to be negative. $\square$

Proof for Proposition 3

When $\beta -v>0,$ the indirect effect can determine negative consequences on welfare. Eq. (25) and the result presented in Proposition 1 allow to conclude that the indirect effect is strongest when the information acquisition costs are linear. Hence, Restriction (30) is necessary (but not sufficient) to guarantee the result.

When instead $\ \beta -v<0$ the indirect effect has positive effects on welfare. Hence, we now consider the most favorable cost structure for the social value of public information to be negative, namely the one that nullifies the substitution effect between public and private information. From Proposition 1, this happens when $C^{^{\prime \prime }}\left( \pi _{x}\right) \longrightarrow \infty .$

Following the same steps that led to Eq. (36) we obtain

$$\begin{aligned} \left. \frac{dE\left[ W\right] }{d\pi _{z}}\right| _{C^{^{\prime \prime }}\left( \pi _{x}\right) \longrightarrow \infty }= & {} \frac{1}{2}\frac{{\overline{Y}}}{{\overline{H}}^{2}}\left( \left( \phi +v\right) \left( 1-v\right) +\phi \left( v-\beta \right) \right) \varrho ^{2}\frac{\gamma ^{2}}{\pi _{z}^{2}}- \frac{{\overline{Y}}}{{\overline{H}}^{2}}\phi \left( v-\beta \right) \varrho ^{2} \frac{\gamma }{\pi _{z}^{2}} \nonumber \\&+\frac{{\overline{Y}}}{{\overline{H}}^{2}}\frac{\pi _{z}+\pi _{x}}{\pi _{z}} \frac{\phi \left( \phi +v\right) }{\left( \phi \pi _{z}+\left( \phi +v\right) \pi _{x}\right) }\left( v-\beta \right) \frac{\left( \phi +v\right) \pi _{x}}{\phi }\varrho ^{2}\frac{\gamma ^{2}}{\pi _{z}^{2}}, \end{aligned}$$

(37)

which readily becomes

$$\begin{aligned}&\left. \frac{dE\left[ W\right] }{d\pi _{z}}\right| _{C^{^{\prime \prime }}\left( \pi _{x}\right) \longrightarrow \infty }=\frac{1}{2}\frac{\overline{ Y}}{{\overline{H}}^{2}}\varrho ^{2}\frac{\gamma ^{2}}{\pi _{z}^{2}} \\&\quad \times \left( \left( \phi +v\right) \left( 1-v\right) +\phi \left( v-\beta \right) -2\frac{\phi \left( v-\beta \right) }{\gamma }+2\frac{\pi _{z}+\pi _{x}}{\pi _{z}}\frac{\left( v-\beta \right) \left( \phi +v\right) ^{2}\pi _{x}}{\phi \pi _{z}+\left( \phi +v\right) \pi _{x}}\right) . \end{aligned}$$

Exploiting the definition for $\gamma$ where appropriate we obtain

$$\begin{aligned} \left. \frac{dE\left[ W\right] }{d\pi _{z}}\right| _{C^{^{\prime \prime }}\left( \pi _{x}\right) \longrightarrow \infty }=\frac{1}{2}\frac{{\overline{Y}}}{{\overline{H}}^{2}}\varrho ^{2}\frac{\gamma ^{2}}{\pi _{z}^{2}}\left( \left( \phi +v\right) \left( 1-v\right) +\left( \beta -v\right) F\left( \pi _{x},\pi _{z}\right) \right) , \end{aligned}$$

where

$$\begin{aligned} F\left( \pi _{x},\pi _{z}\right) =\frac{\phi ^{2}\pi _{z}+\left( \phi +v\right) \left( \phi -2v\right) \pi _{x}}{\phi \pi _{z}+\left( \phi +v\right) \pi _{x}}\mathbf {.} \end{aligned}$$

Since $\beta <v$ the fact that

$$\begin{aligned} \frac{\partial F\left( \pi _{x},\pi _{z}\right) }{\partial \pi _{x}}<0 \end{aligned}$$

implies that the most favorable case for $\frac{dE\left[ W\right] }{d\pi _{z} }<0$ occurs when $\pi _{x}=0,$ and hence when $F\left( \pi _{x},\pi _{z}\right) =\phi$. Accordingly, for given $\left( \pi _{x},\pi _{z}\right) ,$

$$\begin{aligned} \left. \frac{dE\left[ W\right] }{d\pi _{z}}\right| _{C^{^{\prime \prime }}\left( \pi _{x}\right) \longrightarrow \infty }\ge \frac{1}{2}\frac{ {\overline{Y}}}{{\overline{H}}^{2}}\varrho ^{2}\frac{\gamma \left( \pi _{x},\pi _{z}\right) ^{2}}{\pi _{z}^{2}}\left( \left( \phi +v\right) \left( 1-v\right) +\left( \beta -v\right) \phi \right) . \end{aligned}$$

This guarantees that a necessary (but not sufficient) condition for $\frac{dE \left[ W\right] }{d\pi _{z}}<0$ is

$$\begin{aligned} \beta -v<-\frac{\left( \phi +v\right) \left( 1-v\right) }{\phi }, \end{aligned}$$

or

$$\begin{aligned} \left| \beta -v\right| >\overline{{\overline{d}}} \equiv \frac{\left( \phi +v\right) \left( 1-v\right) }{\phi }\mathbf {.} \end{aligned}$$

Because $\overline{{\overline{d}}}>$${\overline{d}},$ this completes the proof. $\square$