Online Appendix Revisiting the Effect of Household Size on Consumption Over the Life-Cycle Not intended for publication

This appendix contains four sections supplementing the analysis in the main paper. First, we provide a further theoretical result which discusses differences in the level of per-adult equivalent consumption in the Demographics and Single Agent model. Second, we show and discuss insurance coefficients in the spirit of Blundell, Pistaferri, and Preston (2008) and Kaplan and Violante (2010) for our benchmark calibration. Third, we report the quantitative results for our benchmark model extended with a pensions system. Fourth, based on the empirical estimates in Attanasio, Banks, Meghir, and Weber (1999) we back out the implied utility weights δ in the Demographics model. 1 Consumption Levels Figure 1a in the paper compares the Single Agent model with the Demographics model with δ = φ. One can directly see that the (life-time) level of per-adult equivalent consumption differs between the two models. The following section provides the theoretical explanation for this. Proposition 1. Life-time per-adult equivalent consumption in the Demographics model and Single Agent model coincide if and only if κ2 = φ2 1−(φ2−1)1 Y2 ( 1− C1 Y1 ) . This result is straightforward to show. Life-time per-adult equivalent consumption in the Demographics model cD is given by cD = C1 + C2 φ2 = C1 + =C2 { }} { Y1 + Y2 − C1 φ2 = φ2 − 1 φ2 C1 + Y1 + Y2 φ2 (1) while in the Single Agent model life-time per-adult equivalent consumption cS equals life-time per-adult equivalent income: cS = c S 1 + c S 2 = Y1 + Y2 κ2 . (2) Equating Equations (1) and (2) yields the critical value of κ2 stated in Proposition 1 which also has a very intuitive interpretation. One can directly see from Equations (1) and (2) that when the household in the Demographics model optimally chooses to neither save nor borrow (C1 = Y1), the life-time per-adult equivalent consumption levels in both setups are the same if and only if κ2 = φ2. Now consider the case when the household in the Demographics model optimally chooses to be a borrower (C1 > Y1). Assume for a moment that κ2 is still equal to φ2 instead of the critical value stated in Proposition 1. In the Demographics model only period two consumption — which in this example is smaller than period two income — is deflated by the equivalence scale in the calculation of life-time per-adult equivalent consumption. This directly implies a higher life-time per-adult equivalent consumption level in the Demographics model compared to the Single Agent model which is determined only by life-time per-adult equivalent income. To undo this, κ2 has to be decreased below φ2 in order to increase life-time per-adult equivalent income and thus life-time per-adult equivalent consumption in the Single Agent model. While in quantitative work life-time (per-adult equivalent) consumption is not a statistic of 1 primary interest, it is relevant for welfare comparisons between different economic environments. If e.g. a linear income tax rate is raised to finance wasteful government consumption, households in the economy with the lower, pre-reform life-time per-adult equivalent consumption suffer a larger welfare loss with concave utility. Moreover, this statistic highlights a key difference between the Single Agent and Demographics model. In the latter the household has total household income at its disposal to allocate between the two periods which then determines life-time per-adult equivalent consumption. In the Single Agent model, the income process is exogenously adjusted thus directly restricting life-time per-adult equivalent income (total resources available). This difference becomes particularly obvious when one would consider two types of households (A,B) experiencing differences in the timing of household income, i.e. yA 1 yA 2 6= y B 1 yB 2 . Even for the same life-time household income yA 1 +y A 2 = y B 1 +y B 2 , life-time per-adult equivalent incomes differ in the Single Agent but not in the Demographics model. This generates inequality in life-time per-adult equivalent consumption in the Single Agent model but not in the Demographics model. The derivation and implications of Proposition 1 only depend on the choice of δ2 in so far that it determines C1 and thus, for a given Y1 and Y2, the relationship between the two per-adult equivalent consumption levels. 2 Insurance Coefficients In the following section, we consult an alternative statistic closely related to the variance of log per-adult equivalent consumption. We follow Blundell, Pistaferri, and Preston (2008) and Kaplan and Violante (2010) and compute insurance coefficients for each model, a measure of how much consumption comoves with income shocks. For each model we calculate the contemporaneous correlation of changes in consumption and permanent (P ) and transitory (Tr) shocks to labor income: ψ t = 1− cov(∆ log(ct/φt), x t ) var( t ) where ct/φt is per-adult equivalent consumption at age t and x = {P, Tr}. The higher the comovement between consumption and unexpected changes in income (represented by the shocks x), the lower is the implied value of the insurance coefficient.