Conditional Versus Unconditional Utility as Welfare Criterion: Two Examples

Abstract

This paper provides two illustrative examples on how a choice of social welfare criterion (conditional vs. unconditional utility) can generate different welfare implications. The first example is based on the standard linear-quadratic permanent income model, and the other example uses a simple two-country DSGE model under autarky and under complete markets. When the conditional welfare criterion is used—with the social discount factor set at the private discount factor—we obtain the well-known results that the government should not intervene when there are no market imperfections and that complete markets generate risk sharing gains over autarky. In contrast, using an unconditional welfare criterion—which effectively implies that the social discount factor is set to unity—can generate unconventional welfare results.

Appendix

1.1 Autarky

We assume that log productivity shock follows an i.i.d. process with mean zero and variance $\sigma _{a}^{2}$, and the lower case represents the log deviation from the deterministic steady state (i.e. $x=\log X-\log \bar{X}$ ). From the Euler equation (21) and the budget constraint (20), we can calculate the second-order solution as

$$\begin{aligned} k_{t+1}= & {} \epsilon _{kk}k_{t}+\epsilon _{ka}a_{t}+\frac{\epsilon _{kkk}}{2} \left( k_{t}\right) ^{2}+\epsilon _{kka}k_{t}a_{t}+\frac{\epsilon _{kaa}}{2} \left( a_{t}\right) ^{2}+\frac{\epsilon _{k\sigma \sigma }}{2}\sigma _{a}^{2}, \\ c_{t}= & {} \epsilon _{ck}k_{t+1}+\frac{\epsilon _{ckk}}{2}\left( k_{t+1}\right) ^{2}+\frac{\epsilon _{c\sigma \sigma }}{2}\sigma _{a}^{2}, \end{aligned}$$

where the first-order coefficients are

$$\begin{aligned} \epsilon _{kk}= & {} \frac{1}{2}\left( 1+\chi -\sqrt{(1+\chi )^{2}-4/\beta } \right) , \\ \epsilon _{ka}= & {} \frac{\beta \delta }{s}\epsilon _{kk}, \\ \epsilon _{ck}= & {} \frac{1-\beta \epsilon _{kk}}{(1-s)\epsilon _{ka}}, \end{aligned}$$

and

$$\begin{aligned} \Delta= & {} 1-\beta +\beta \delta , \\ s= & {} \frac{\alpha \beta \delta }{\Delta }, \\ \chi= & {} \frac{\alpha \gamma +\left( 1-\alpha \right) (1-s)\Delta ^{2}}{ \alpha \beta \gamma }. \end{aligned}$$

The second-order coefficients are

$$\begin{aligned} \epsilon _{kkk}= & {} \frac{\gamma \left( 1-\epsilon _{kk}^{2}\right) \left[ \alpha ^{2}+\frac{s\left( 1-\delta \right) }{\delta }-\left[ \left( 1-s\right) \epsilon _{ck}^{2}+\frac{s}{\delta }\right] \epsilon _{kk}^{2} \right] +\Delta \left( 1-\Delta \right) \left( 1-\alpha \right) ^{2}\left( 1-s\right) \epsilon _{kk}^{2}}{\gamma \left( 1-s\right) \epsilon _{ck}+\gamma \left( 1-\epsilon _{kk}^{2}\right) \frac{s}{\delta }}, \\ \epsilon _{ckk}= & {} \frac{-\Delta \left( 1-\Delta \right) \left( 1-\alpha \right) ^{2}\left[ \left( 1-s\right) \epsilon _{ck}+\frac{s}{\delta }\right] +\gamma \epsilon _{ck}\left[ \alpha ^{2}+\frac{s\left( 1-\delta \right) }{ \delta }-\left[ \left( 1-s\right) \epsilon _{ck}^{2}+\frac{s}{\delta }\right] \epsilon _{kk}^{2}\right] }{\gamma \left( 1-s\right) \epsilon _{ck}+\gamma \left( 1-\epsilon _{kk}^{2}\right) \frac{s}{\delta }}, \end{aligned}$$

$$\begin{aligned} \epsilon _{kka}= & {} \left[ \left( 1-s\right) \epsilon _{ck}+\frac{s}{\delta } \right] ^{-1}\left[ \alpha -\left[ \left( 1-s\right) \left( \epsilon _{ckk}+\epsilon _{ck}^{2}\right) +\frac{s}{\delta }\right] \epsilon _{kk}\epsilon _{ka}\right] , \\ \epsilon _{kaa}= & {} \left[ \left( 1-s\right) \epsilon _{ck}+\frac{s}{\delta } \right] ^{-1}\left[ 1-\left[ \left( 1-s\right) \left( \epsilon _{ckk}+\epsilon _{ck}^{2}\right) +\frac{s}{\delta }\right] \epsilon _{ka}^{2} \right] , \\ \epsilon _{k\sigma \sigma }= & {} -\epsilon _{kaa}+\left( \gamma \epsilon _{ck}\right) ^{-1}\left[ -\gamma \epsilon _{ckk}\epsilon _{ka}^{2}+\Delta \left( 1-\Delta \right) +\left( \Delta -\gamma \epsilon _{ck}\epsilon _{ka}\right) ^{2}\right] , \\ \epsilon _{c\sigma \sigma }= & {} -\left[ \epsilon _{ck}+\frac{s}{\delta \left( 1-s\right) }\right] \epsilon _{k\sigma \sigma }. \end{aligned}$$

Taking expectations of these two second-order solutions generate the mean and variance of $k_{t}$ and $c_{t}.$

$$\begin{aligned} Var\left[ k_{t+1}\right]= & {} \epsilon _{kk}^{2}Var\left[ k_{t}\right] +\epsilon _{ka}^{2}\sigma _{a}^{2}, \\ Var\left[ c_{t}\right]= & {} \epsilon _{ck}^{2}Var\left[ k_{t+1}\right] , \\ E\left[ k_{t+1}\right]= & {} \epsilon _{kk}E\left[ k_{t}\right] +\frac{\epsilon _{kkk}}{2}Var\left[ k_{t}\right] +\frac{\epsilon _{kaa}}{2}\sigma _{a}^{2}+ \frac{\epsilon _{k\sigma \sigma }}{2}\sigma _{a}^{2}, \\ E\left[ c_{t}\right]= & {} \epsilon _{ck}E\left[ k_{t+1}\right] +\frac{\epsilon _{ckk}}{2}Var\left[ k_{t+1}\right] +\frac{\epsilon _{c\sigma \sigma }}{2} \sigma _{a}^{2}. \end{aligned}$$

Unconditional mean and variance of consumption become

$$\begin{aligned}&\displaystyle \text {E}\left[ c_{t}^{U}\right] =\left[ \frac{\epsilon _{ck}}{1-\epsilon _{kk}}\left( \frac{\epsilon _{ka}^{2}}{1-\epsilon _{kk}^{2}}\epsilon _{kkk}+\epsilon _{kaa}+\epsilon _{k\sigma \sigma }\right) +\epsilon _{ckk} \frac{\epsilon _{ka}^{2}}{1-\epsilon _{kk}^{2}}+\epsilon _{c\sigma \sigma } \right] \frac{\sigma _{a}^{2}}{2}, \\&\displaystyle \text {Var}\left[ c_{t}^{U}\right] =\epsilon _{ck}^{2}\frac{\epsilon _{ka}^{2}\sigma _{a}^{2}}{1-\epsilon _{kk}^{2}}, \end{aligned}$$

where superscript U denotes unconditional. Unconditional welfare and the certainty equivalent consumption can be calculated as follows

$$\begin{aligned} E\left[ U_{t}^{U}\right]= & {} E\left[ c_{t}^{U}\right] +\frac{1-\gamma }{2}Var \left[ c_{t}^{U}\right] , \\ c_{t}^{U,CE}= & {} \left[ E\left[ U_{t}^{U}\right] \left( 1-\gamma \right) +1 \right] ^{\frac{1}{1-\gamma }}. \end{aligned}$$

1.2 Complete Markets Economy

Solution for the complete markets model implies that consumption should be equal across countries: $C_{1s}=C_{2s}$ for all $s\ge t$. Also, due to the independence assumption of the shocks, the social planner allocates the same amount of capital: $K_{1s}=K_{2s}$ for all $s\ge t+1$.

Using the information that the solution is symmetric between the two countries, we compute the second-order solution as follows:

$$\begin{aligned} k_{1,t+1}= & {} 2\eta _{kk}k_{1t}+\eta _{ka}\left( a_{1t}+a_{2t}\right) +\frac{ \eta _{k\sigma \sigma }}{2}\sigma _{a}^{2} \\&+\frac{1}{2}\left[ \begin{array}{c} k_{1t} \\ a_{1t} \\ a_{2t} \end{array} \right] ^{\prime }\left[ \begin{array}{ccc} 4\eta _{kkk} &{} 2\eta _{kka} &{} 2\eta _{kka} \\ 2\eta _{kka} &{} \eta _{kaa} &{} \eta _{kaa^{*}} \\ 2\eta _{kka} &{} \eta _{kaa^{*}} &{} \eta _{kaa} \end{array} \right] \left[ \begin{array}{c} k_{1t} \\ a_{1t} \\ a_{2t} \end{array} \right] , \\ c_{t}= & {} 2\eta _{ck}k_{1,t+1}+2\eta _{ckk}k_{1,t+1}^{2}+\frac{\eta _{c\sigma \sigma }}{2}\sigma _{a}^{2}, \end{aligned}$$

where the coefficients $\eta ^{\prime }$s are

$$\begin{aligned} \eta _{ckk}= & {} \frac{\epsilon _{ckk}}{4},\quad \eta _{kkk}=\frac{\epsilon _{kkk}}{4 },\quad \eta _{kka}=\frac{\epsilon _{kka}}{4}, \\ \eta _{kaa}= & {} \frac{1}{4}\left( \epsilon _{kaa}+\epsilon _{ka}\right) ,\quad \eta _{kaa^{*}}=\frac{1}{4}\left( \epsilon _{kaa}-\epsilon _{ka}\right) , \\ \eta _{k\sigma \sigma }= & {} \frac{1}{2}\left( \epsilon _{k\sigma \sigma }+ \frac{\Delta -\gamma \epsilon _{ck}\epsilon _{ka}}{\gamma \epsilon _{ck}} \right) , \\ \eta _{c\sigma \sigma }= & {} -\left( \epsilon _{ck}+\frac{s}{\left( 1-s\right) \delta }\right) \left( \epsilon _{k\sigma \sigma }+\frac{\Delta -\gamma \epsilon _{ck}\epsilon _{ka}}{\gamma \epsilon _{ck}}\right) . \end{aligned}$$

Mean and variance of $k_{t}$ and $c_{t}$ follow:

$$\begin{aligned} Var\left[ k_{t+1}\right]= & {} 4\eta _{kk}^{2}Var\left[ k_{t}\right] +2\eta _{ka}^{2}\sigma _{a}^{2}, \\ Var\left[ c_{t}\right]= & {} 4\eta _{ck}^{2}Var\left[ k_{t+1}\right] , \\ E\left[ k_{t+1}\right]= & {} 2\eta _{kk}E\left[ k_{t}\right] +2\eta _{kkk}Var \left[ k_{t}\right] +\eta _{kaa}\sigma _{a}^{2}+\frac{\eta _{k\sigma \sigma } }{2}\sigma _{a}^{2}, \\ E\left[ c_{t}\right]= & {} 2\eta _{ck}E\left[ k_{t+1}\right] +2\eta _{ckk}Var \left[ k_{t+1}\right] +\frac{\eta _{c\sigma \sigma }}{2}\sigma _{a}^{2}. \end{aligned}$$

Unconditional mean and variance of consumption become

$$\begin{aligned}&\displaystyle \text {E}\left[ c_{t}^{U}\right] =\left[ \frac{\epsilon _{ck}}{1-\epsilon _{kk}}\left( \frac{2\epsilon _{ka}^{2}}{1-\epsilon _{kk}^{2}}\eta _{kkk}+2\eta _{kaa}+\eta _{k\sigma \sigma }\right) +\eta _{ckk}\frac{ 2\epsilon _{ka}^{2}}{1-\epsilon _{kk}^{2}}+\eta _{c\sigma \sigma }\right] \frac{\sigma _{a}^{2}}{2}, \\&\displaystyle \text {Var}\left[ c_{t}^{U}\right] =4\eta _{ck}^{2}\frac{2\eta _{ka}^{2}\sigma _{a}^{2}}{1-4\eta _{kk}^{2}}. \end{aligned}$$