7. Overlapping Generations (OLG) Model

In this section, we will study a model of pure exchange economy in which agents only live for two periods. In the first part, we will focus on the simple OLG model: the set-up, the competitive equilibrium, and identify conditions under which the First Welfare Theorem fails. When the competitive equilibrium is not Pareto optimal, we introduce a pay-as-you-go social security system to Pareto improve the competitive equilibrium. In the second part, we consider a variant of the simple OLG model in which there is population growth and productivity (endowment) growth, and do analyses similar to those in the first part to understand the differences brought about by growth.

In fact, there are multiple variants of the simple OLG model, such as the Perpetual Youth Model (continuous time), OLG with agents living for multiple periods, OLG with heterogeneity in agents, etc. This part of the note is not aimed at including all possible variants, but the discussion on the basic OLG model and on one of its variants (growing economy) can help us grasp the essence of this set of models.

7.1 Simple OLG Model

7.1.1 Set-up

• Suppose the economy has only one type of consumption good but in different periods \(t=1,2,\dots\), i.e. \(\mathbf L=\mathbb R^\infty\).
• Agents in this economy only live and consume for two periods. Each generation is indexed by the time of birth, i.e. \(\mathbf I=\{0,1,2,\dots\}\).
• Normalize the mass of agents in each period to 1.
• Consumption by generation \(i\) in period \(t\) is denoted as \(x^i_t\), and \(x^i_t=0\) for \(\forall t\ne i,i+1\).
• Utility functions:
• Agents born at dates \(t\ge1\) have the utility function \(u^i(\mathbf x^i)=v^i(x^i_i,x^i_{i+1})\).
• Agents born at date 0 can only consume goods at date \(t=1\), so any increasing function can be their utility function. WLOG, assume \(u^0(x^0_1,x^0_2,\dots)=x^0_1\).
• Generation \(i\ge1\) have positive endowments while they are alive, i.e. \(\mathbf e^i=(e^i_1,\dots,e^i_i,e^i_{i+1},\dots)=(0,\dots,e^i_i,e^i_{i+1},\dots,0)\). For generation 0, \(e^0_1>0\) and \(e^0_t=0\) for all \(t=2,3,\dots\).
• Feasible allocations constraint: \(\sum_{i\in\mathbf I}\mathbf x^i=\sum_{i\in\mathbf I}\mathbf e^i\), which means $$x^{t-1}_t+x^t_t=e^{t-1}_t+e^t_t\equiv\bar e_t,\quad\text{for }\forall t=1,2,\dots$$ where \(\bar e_t\) is the aggregate endowment of period \(t\).

7.1.2 The maximization problem of each generation

Suppose that \(\mathbf p\in\mathbb R^\infty\) is the price vector of the consumption good in every period. - Generation \(i\ge1\)'s utility maximization problem: $$\max_{x^i_i,x^i_{i+1}\in\mathbb R}u^i(x^i_i,x^i_{i+1})\quad\text{s.t.}\quad p_i x^i_i+p_{i+1}x^i_{i+1}\le p_i e^i_i+p_{i+1}e^i_{i+1}$$ - Generation 0's utility maximization problem: $$\max_{x^0_1\in\mathbb R}x^0_1\quad\text{s.t.}\quad p_1 x^0_1\le p_1 e^0_1$$

7.1.3 The competitive equilibrium

Interest rate represented by price ratios.

We can normalize the price of period-1 consumption good to be one, i.e. \(p_1=1\). The f.o.c. w.r.t. \(x^i_i\) and \(x^i_{i+1}\) gives us that $$\frac{u^i_2(x^i_i,x^i_{i+1})}{u^i_1(x^i_i,x^i_{i+1})}=\frac{p_{i+1}}{p_i}$$ where \(u^i_1\) is the partial derivative w.r.t. the first argument and \(u^i_2\) is the partial derivative w.r.t. the second argument. Since we already proved that the equilibrium is autarky, i.e. \(x^t_t=e^t_t\) and \(x^{t-1}_t=e^{t-1}_t\), we can rewrite the f.o.c. as $$\frac{p_{t+1}}{p_t}=\frac{u^t_2(e^t_t,e^t_{t+1})}{u^t_1(e^t_t,e^t_{t+1})}\tag{7.2}$$ for \(\forall t\ge1\).

We can define the net interest rate of period \(t\) as \(r_t\): $$\frac{1}{1+r_t}\equiv\frac{p_{t+1}}{p_t}\tag{7.3}$$ for \(\forall t\ge1\). Plug in the expression for \(r_t\) to rewrite (7.3): $$r_t=\frac{u^t_1(e^t_t,e^t_{t+1})}{u^t_2(e^t_t,e^t_{t+1})}-1\tag{7.4}$$ for \(\forall t\ge1\). Notice also that $$p_t=\frac{p_{t-1}}{1+r_{t-1}}=\frac{1}{1+r_{t-1}}\cdot\frac{p_{t-2}}{1+r_{t-2}}=\frac{1}{1+r_{t-1}}\cdot\frac{1}{1+r_{t-2}}\cdots\frac{p_1}{1+r_1}=\prod_{i=1}^{t-1}\frac{1}{1+r_i}\tag{7.5}$$ or equivalently \(\frac{1}{p_t}=\prod_{i=1}^{t-1}(1+r_i)\).

7.1.4 Competitive equilibrium with log, time-separable utility

The autarky solution, equilibrium interest rate and price. We want to check whether the autarky comparative equilibrium is Pareto optimal (i.e. satisfies the First Welfare Theorem). In order to have a simple and explicit expression for the related variables in equilibrium, let's consider the following log and time-separable utility function: $$u^t(c^t_y,c^t_o)\equiv(1-\beta)\ln c^t_y+\beta\ln c^t_o,\quad\text{for some }\beta\in(0,1),\ \text{for }\forall t\ge1$$ We also normalize the endowments in each period to one and make them the same across generations, i.e. \(e^t_t=1-\alpha\), \(e^t_{t+1}=\alpha\), for some \(\alpha\in(0,1)\), for \(\forall t\ge1\).

Proposition 7.1 gives us that the competitive equilibrium consumption must be autarky, i.e. $$c^{t\star}_y=1-\alpha\ \text{for }\forall t\ge1,\qquad c^{t\star}_o=\alpha\ \text{for }\forall t\ge1$$ In the competitive equilibrium, (7.2) and (7.3) gives us that $$\begin{aligned}\frac{1}{1+r_t}&=\frac{p_{t+1}}{p_t}=\frac{\beta}{1-\beta}\frac{c^{t\star}_y}{c^{t\star}_o}=\frac{\beta}{1-\beta}\frac{1-\alpha}{\alpha}\\\Rightarrow r_t&=\frac{1-\beta}{\beta}\frac{\alpha}{1-\alpha}-1=\frac{\alpha(1-\beta)-\beta(1-\alpha)}{\beta(1-\alpha)}=\frac{\alpha-\beta}{\beta(1-\alpha)}\equiv\bar r\end{aligned}\tag{7.6}$$ Note that \(\bar r\) is the interest rate whose implied prices make "never trade" optimal to every generation. Rewrite equation (7.5): for \(\forall t\ge1\), $$p_t=\prod_{i=1}^{t-1}\frac{1}{1+\bar r}=\left(\frac{1}{1+\bar r}\right)^{t-1}=\left(\frac{\beta}{1-\beta}\frac{1-\alpha}{\alpha}\right)^{t-1}$$ which pins down the equilibrium prices.

Optimal savings. Alternatively, we can solve this problem by solving for the optimal saving. Construct the saving of generation \(t\) in period \(t\) as \(s^t_t\equiv e^t_t-x^t_t\). Thus, $$\begin{cases}x_t=(1-\alpha)-s^t_t\\x_{t+1}=\alpha+(1+\bar r)s^t_t\end{cases}$$ Then, we can rewrite the maximization problem w.r.t. \(s^t_t\): $$\max_{s^t_t}(1-\beta)\ln\left((1-\alpha)-s^t_t\right)+\beta\ln\left[\alpha+(1+\bar r)s^t_t\right]$$ The f.o.c. w.r.t. \(s^t_t\) is $$\begin{aligned}&\beta(1+\bar r)\frac{1}{\alpha+(1+\bar r)s^{t\star}_t}=(1-\beta)\frac{1}{1-\alpha-s^{t\star}_t}\\\Rightarrow{}&\beta(1+\bar r)(1-\alpha-s^{t\star}_t)=(1-\beta)(\alpha+(1+\bar r)s^{t\star}_t)\\\Rightarrow{}&s^{t\star}_t(1+\bar r)\underbrace{[\beta+(1-\beta)]}_{=1}=\beta(1+\bar r)(1-\alpha)-\alpha(1-\beta)\\\Rightarrow{}&s^{t\star}_t=\beta(1-\alpha)-\frac{\alpha(1-\beta)}{1+\bar r}\end{aligned}\tag{7.7}$$ $$\begin{aligned}[\text{add and subtract }(1-\alpha)]\quad s^{t\star}_t&=\beta(1-\alpha)+[(1-\alpha)-(1-\alpha)]-\frac{\alpha(1-\beta)}{1+\bar r}\\&=(1-\alpha)-(1-\alpha)(1-\beta)-\frac{\alpha(1-\beta)}{1+\bar r}\\&=(1-\alpha)-(1-\beta)\left[(1-\alpha)+\frac{\alpha}{1+\bar r}\right]\end{aligned}\tag{7.8}$$

Another way to solve for the equilibrium interest rate. Since we know the competitive equilibrium is autarky, the interest rate must be such that the optimal saving is zero. Set equation (7.7) equal to zero to solve for \(\bar r\): $$0=s^{t\star}_t=(1-\alpha)\beta-\frac{\alpha(1-\beta)}{1+\bar r}\Rightarrow\frac{\alpha(1-\beta)}{1+\bar r}=(1-\alpha)\beta\Rightarrow1+\bar r=\frac{\alpha(1-\beta)}{(1-\alpha)\beta}\Rightarrow\bar r=\frac{\alpha-\beta}{(1-\alpha)\beta}$$ which is exactly the same as the equilibrium interest rate obtained when we solve for the f.o.c. w.r.t. \(x^t_t\) and \(x^t_{t+1}\).

7.1.5 Best symmetric allocation

We call it a symmetric allocation if every generation has the same endowment both when they are young and when they are old: $$\begin{cases}x^t_t=c_y&\text{for }\forall t\ge1\\x^t_{t+1}=c_o&\text{for }\forall t\ge1\end{cases}$$ Meanwhile, the feasibility constraint for any period should also be satisfied, which requires that the total endowment in any period adds up to 1, i.e. \(x^t_{t+1}+x^{t+1}_{t+1}=1\), which, together with the market clearing condition, gives us that \(c_y+c_o=1\).

Now, the utility maximization problem for generation \(t\ge1\) is homogeneous, which is $$\max_{c_y,c_o}(1-\beta)\ln c_y+\beta\ln c_o\quad\text{s.t.}\quad c_y+c_o=1$$ Set up the Lagrangian $$\mathcal L=(1-\beta)\ln c_y+\beta\ln c_o+\lambda(1-c_y-c_o)$$ The f.o.c. below is sufficient to solve the problem due to the concavity of the utility function: $$[c_y]:\ \frac{1-\beta}{c^*_y}=1,\qquad[c_o]:\ \frac{\beta}{c^*_o}=1,\qquad[\lambda]:\ c^*_y+c^*_o=1$$ which has the solution: $$\begin{cases}c^*_y=1-\beta\\c^*_o=\beta\end{cases}$$ Therefore, \(c^*_y=1-\beta\) and \(c^*_o=\beta\) is the best symmetric allocation.

7.1.6 Relationship between the autarky competitive equilibrium allocation and the best symmetric allocation

The unique competitive equilibrium (denoted as \(c^{t\star}_y\) and \(c^{t+1\star}_o\)), which is autarky: $$\begin{cases}c^{t\star}_y=1-\alpha&\text{for }\forall t\ge1\\c^{t+1\star}_o=\alpha&\text{for }\forall t\ge1\end{cases}$$ And the best symmetric allocation (denoted as \(\hat c^{t\star}_y\) and \(\hat c^{t\star}_o\)): $$\begin{cases}\hat c^{t\star}_y=1-\beta&\text{for }\forall t\ge1\\\hat c^{t\star}_o=\beta&\text{for }\forall t\ge1\end{cases}$$ We can observe that the competitive equilibrium allocation is indeed a feasible symmetric allocation. So by definition of "best", the best symmetric allocation is strictly preferred by agents \(t\ge1\) unless \(\alpha=\beta\) (in which case generations \(i\ge1\) are indifferent between the two allocations).

To decide whether the best symmetric allocation Pareto dominates the competitive equilibrium, we only need to compare the utility of the initial old for the two allocations: - If \(\beta>\alpha\) (\(\bar r<0\)): \(c^{1\star}_o=\alpha<\beta=\hat c^{1\star}_o\). So the initial old strictly prefers the best symmetric allocation, which means that all agents are strictly better off in the best symmetric allocation. Or equivalently, the best symmetric allocation Pareto dominates the competitive equilibrium allocation. Most importantly, the First Welfare Theorem fails here. - If \(\beta=\alpha\) (\(\bar r=0\)): \(c^{1\star}_o=\alpha=\beta=\hat c^{1\star}_o\). The two allocations are exactly the same. - If \(\beta<\alpha\) (\(\bar r>0\)): \(c^{1\star}_o=\alpha>\beta=\hat c^{1\star}_o\). The initial old strictly prefers the competitive equilibrium allocation to the best symmetric allocation. So neither one Pareto dominates the other. The First Welfare Theorem holds.

7.1.7 Pay-as-you-go social security system

In the previous part, we discussed that in the case \(\beta>\alpha\), the competitive equilibrium allocation is not Pareto optimal. Now let's consider the way to Pareto improve.

Consider the pay-as-you-go social security system in which we tax the young \(\tau\) in each period and pay all the collected money to the old in that period. With the invention of such a policy, every agent's post-tax endowments becomes $$\begin{cases}e^t_t=(1-\alpha)-\tau&\text{for }\forall t\ge1\\e^t_{t+1}=\alpha+\tau&\text{for }\forall t\ge0\end{cases}$$ Following the same argument, we know that the post-tax endowments are the new autarky equilibrium allocation. And by definition, this post-tax endowments are symmetric allocation. Since we know the best symmetric allocation is \(\hat c^{t\star}_y=1-\beta\) and \(\hat c^{t\star}_o=\beta\), by setting \(\tau=\beta-\alpha\), we are moving every generation \(t\ge1\) to the best symmetric allocation, which is a Pareto improvement to all of them. So if \(\beta>\alpha\), then this social security system is also a Pareto improvement to the initial old. Thus, the pay-as-you-go social security system, if set up correctly, will Pareto improve the allocation efficiency in the case of \(\beta>\alpha\), which fails the First Welfare Theorem that the competitive equilibrium cannot be Pareto improved.

The First Welfare Theorem fails when \(\beta>\alpha\) because in the proof of the First Welfare Theorem, we need the finite total expenditure, i.e. $$\mathbf p\cdot\sum_{i\in\mathbf I}\mathbf x^i<\infty\tag{7.9}$$ so that we can reach a contradiction for the competitive equilibrium not to be Pareto optimal. But this finite constraint fails when \(\beta>\alpha\) because we have infinitely many generations and a negative equilibrium interest rate cannot discount all future endowments to a finite number, which fails the condition (7.9). But when \(\beta<\alpha\), the condition (7.9) holds, thus the First Welfare Theorem holds.

On the top of the previous problem, we now allow the population size and productivity (endowment amount) to gradually grow.

Denote \(N_t\) as the number of young at time \(t\), and thus \(N_{t+1}=(1+n)N_t\) for \(\forall t\ge1\) where \(n\) is the constant growth rate of the population. Normalize \(N_0=1\). Denote \(g\) as the constant growth rate of "productivity" (i.e. endowments), i.e. $$\begin{cases}e^{t+1}_{t+1}=(1+g)e^t_t\\e^{t+1}_{t+2}=(1+g)e^t_{t+1}\end{cases}$$ where we normalize \(e^0_1=\alpha\) and \(e^1_1=(1+g)(1-\alpha)\). So for \(\forall t\ge1\), the aggregate endowment constraint is $$\begin{aligned}N_{t-1}x^{t-1}_t+N_t x^t_t&=N_{t-1}e^{t-1}_t+N_t e^t_t=N_{t-1}(1+g)^{t-1}\alpha+N_t(1+g)^t(1-\alpha)\\\Rightarrow x^{t-1}_t+(1+n)x^t_t&=(1+g)^{t-1}\alpha+(1+n)(1+g)^t(1-\alpha)=(1+g)^{t-1}[\alpha+(1+n)(1+g)(1-\alpha)]\end{aligned}$$

7.2.1 Competitive equilibrium and equilibrium interest rate

Following exactly the same argument as in Proposition 7.1, the competitive equilibrium is autarky, i.e. $$\begin{cases}x^{t\star}_t=(1+g)^t(1-\alpha)&\text{for }\forall t\ge1\\x^{t\star}_{t+1}=(1+g)^t\alpha&\text{for }\forall t\ge1\end{cases}$$ We can calculate the equilibrium interest rate by the f.o.c. w.r.t. \(x^t_t\) and \(x^t_{t+1}\) for generation \(t\ge1\): $$\frac{1}{1+r_t}=\frac{p_{t+1}}{p_t}=\frac{\beta}{1-\beta}\frac{x^{t\star}_t}{x^{t\star}_{t+1}}=\frac{\beta}{1-\beta}\frac{(1+g)^t(1-\alpha)}{(1+g)^t\alpha}$$ which is exactly the same as the one without growth if we cancel out the \((1+g)^t\) on the RHS. So the same \(\bar r\) satisfies this equation, i.e. $$\bar r=\frac{\alpha-\beta}{\beta(1-\alpha)}$$

7.2.2 Optimal saving and equilibrium interest rate

As before, denote the consumption of agent \(t\) in the two periods with saving as $$\begin{cases}x^t_t=(1+g)^t(1-\alpha)-s^t_t\\x^t_{t+1}=(1+\bar r)s^t_t+(1+g)^t\alpha\end{cases}$$ We can rewrite the agent's utility maximization problem w.r.t. \(s^t_t\): $$\max_{s^t_t}(1-\beta)\ln\left((1+g)^t(1-\alpha)-s^t_t\right)+\beta\ln\left[(1+\bar r)s^t_t+(1+g)^t\alpha\right]$$ The f.o.c. w.r.t. \(s^t_t\) gives us $$s^{t\star}_t=(1+g)^t\left(\beta(1-\alpha)-\frac{\alpha(1-\beta)}{1+\bar r}\right)=(1+g)^t\left[(1-\alpha)-(1-\beta)\left((1-\alpha)+\frac{\alpha}{1+\bar r}\right)\right]\tag{7.10}$$ Again, calculate the equilibrium \(\bar r\) from the optimal saving being zero: \(0=s^{t\star}_t\). Since it is just \((1+g)^t\) times the f.o.c. without growth, the same \(\bar r\) should solve the problem, i.e. $$\bar r=\frac{\alpha-\beta}{\beta(1-\alpha)}\tag{7.11}$$

7.2.3 Best symmetric allocation

Since each generation has different endowments, the definition of the best symmetric allocation has to change to $$\begin{cases}c^t_y=\hat c_y(1+g)^t\\c^t_o=\hat c_o(1+g)^{t-1}\end{cases}$$ where \(\hat c_y+\hat c_o=1\). Then, the feasibility constraint for the symmetric allocation is $$\begin{aligned}N_t\hat c_y(1+g)^t+N_{t-1}\hat c_o(1+g)^{t-1}&=N_t(1-\alpha)(1+g)^t+N_{t-1}\alpha(1+g)^{t-1}\\\Rightarrow\frac{N_t}{N_{t-1}}\hat c_y(1+g)^t+\hat c_o(1+g)^{t-1}&=\frac{N_t}{N_{t-1}}(1-\alpha)(1+g)^t+\alpha(1+g)^{t-1}\\\Rightarrow(1+n)(1+g)\hat c_y+\hat c_o&=(1+n)(1+g)(1-\alpha)+\alpha\end{aligned}\tag{7.12}$$ The best symmetric allocation in this set-up can be obtained from solving the following problem for agent \(t\ge1\): $$\max_{\hat c_y,\hat c_o}(1-\beta)\ln\left[\hat c_y(1+g)^t\right]+\beta\ln\left[\hat c_o(1+g)^t\right]\quad\text{s.t. (7.12)}$$ The Lagrangian is $$\mathcal L=(1-\beta)\ln\left[\hat c_y(1+g)^t\right]+\beta\ln\left[\hat c_o(1+g)^t\right]+\lambda\left[(1+n)(1+g)(1-\alpha)+\alpha-((1+n)(1+g)\hat c_y+\hat c_o)\right]$$ The f.o.c. is $$[\hat c_y]:\ \frac{1-\beta}{\hat c_y}=\lambda(1+n)(1+g),\qquad[\hat c_o]:\ \frac{\beta}{\hat c_o}=\lambda$$ Replace the \(\lambda\) in \([\hat c_y]\) with \(\frac{\beta}{\hat c_o}\): $$\frac{1-\beta}{\hat c_y}=\frac{\beta}{\hat c_o}(1+n)(1+g)\Rightarrow\hat c_o=\frac{\beta}{1-\beta}(1+n)(1+g)\hat c_y$$ Plug this relationship into the feasibility constraint (7.12): $$\begin{aligned}(1+n)(1+g)\hat c_y+\frac{\beta}{1-\beta}(1+n)(1+g)\hat c_y&=(1+n)(1+g)(1-\alpha)+\alpha\\\Rightarrow\frac{1}{1-\beta}(1+n)(1+g)\hat c_y&=(1+n)(1+g)(1-\alpha)+\alpha\\\Rightarrow\hat c^*_y&=(1-\beta)\frac{(1+n)(1+g)(1-\alpha)+\alpha}{(1+n)(1+g)}\\\Rightarrow\hat c^*_o&=\beta[(1+n)(1+g)(1-\alpha)+\alpha]\end{aligned}$$

7.2.4 Relationship between the autarky competitive equilibrium allocation and the best symmetric allocation

The unique competitive equilibrium (autarky): $$\begin{cases}\hat c^{t\star}_y=(1+g)^t(1-\alpha)&\text{for }\forall t\ge1\\\hat c^{t+1\star}_o=(1+g)^t\alpha&\text{for }\forall t\ge1\end{cases}$$ And the best symmetric allocation: $$\begin{cases}\hat c^{t\star}_y=(1+g)^t(1-\beta)\dfrac{(1+n)(1+g)(1-\alpha)+\alpha}{(1+n)(1+g)}&\text{for }\forall t\ge1\\\hat c^{t+1\star}_o=(1+g)^t\beta[(1+n)(1+g)(1-\alpha)+\alpha]&\text{for }\forall t\ge1\end{cases}$$ The competitive equilibrium allocation is indeed a feasible symmetric allocation. By definition of "best", the best symmetric allocation is strictly preferred by agents \(t\ge1\) unless \(\alpha=\beta\). To decide whether the best symmetric allocation Pareto dominates the competitive equilibrium, compare the utility of the initial old: $$\begin{aligned}\hat c^{1\star}_o>c^{1\star}_o&\Leftrightarrow(1+g)^0\beta[(1+n)(1+g)(1-\alpha)+\alpha]>(1+g)^0\alpha\\&\Leftrightarrow\beta[(1+n)(1+g)(1-\alpha)+\alpha]>\alpha\\&\Leftrightarrow(1+n)(1+g)(1-\alpha)+\alpha>\frac{\alpha}{\beta}\\&\Leftrightarrow(1+n)(1+g)(1-\alpha)>\frac{\alpha(1-\beta)}{\beta}\\&\Leftrightarrow(1+n)(1+g)>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\\&\Leftrightarrow(1+n)(1+g)>1+\bar r\end{aligned}$$ - If \((1+n)(1+g)>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\) (i.e. \((1+n)(1+g)>1+\bar r\)): \(c^{1\star}_o<\hat c^{1\star}_o\). The initial old strictly prefers the best symmetric allocation, so all agents are strictly better off. The best symmetric allocation Pareto dominates the competitive equilibrium allocation. The First Welfare Theorem fails. - If \((1+n)(1+g)=\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\) (i.e. \(=1+\bar r\)): \(c^{1\star}_o=\hat c^{1\star}_o\). The two allocations are exactly the same. - If \((1+n)(1+g)<\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\) (i.e. \(<1+\bar r\)): \(c^{1\star}_o>\hat c^{1\star}_o\). The initial old strictly prefers the competitive equilibrium allocation. So neither one Pareto dominates the other. The First Welfare Theorem holds.

7.2.5 Social security

In the previous part, we discussed that in the case \((1+n)(1+g)>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\), the competitive equilibrium allocation is not Pareto optimal. Now let's consider the way to Pareto improve it.

Consider the pay-as-you-go social security system in which we tax the young in each period and pay all the collected money to the old in that period. With this policy, the post-tax endowments becomes $$\begin{cases}e^t_t=(1+g)^t(1-\alpha)-(1+g)^t\hat\tau&\text{for }\forall t\ge1\\e^t_{t+1}=(1+g)^t\alpha+(1+g)^t\hat\tau&\text{for }\forall t\ge0\end{cases}$$ Following the same argument, we know that the post-tax endowments are the new autarky equilibrium allocation, which is a symmetric allocation. Since we know the best symmetric allocation, by setting $$\begin{aligned}(1+g)^t\hat\tau&=(1+g)^t(1-\alpha)-(1+g)^t(1-\beta)\frac{(1+n)(1+g)(1-\alpha)+\alpha}{(1+n)(1+g)}\\\Rightarrow\hat\tau&=(1-\alpha)-(1-\beta)\frac{(1+n)(1+g)(1-\alpha)+\alpha}{(1+n)(1+g)}\\&=(1-\alpha)-(1-\beta)(1-\alpha)-(1-\beta)\frac{\alpha}{(1+n)(1+g)}\\&=\beta(1-\alpha)-(1-\beta)\frac{\alpha}{(1+n)(1+g)}\end{aligned}$$ we are moving every generation \(t\ge1\) to the best symmetric allocation, which is a Pareto improvement to all of them. So if \((1+n)(1+g)>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\), then the social security system is also a Pareto improvement to the initial old. If set up correctly, it will Pareto improve the allocation efficiency in the case \((1+n)(1+g)>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\), which fails the First Welfare Theorem.

The First Welfare Theorem fails when \((1+n)(1+g)>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\) because in the proof we need the finite total expenditure, i.e. $$\mathbf p\cdot\sum_{i\in\mathbf I}\mathbf x^i<\infty\tag{7.13}$$ But this finite constraint fails when \((1+n)(1+g)>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\) because we have infinitely many generations whose endowments grow at a constant rate of \((1+n)(1+g)\), so the equilibrium gross interest rate (i.e. \(1+\bar r\)) lower than \((1+n)(1+g)\) cannot discount all future endowments to a finite number, which fails the condition (7.13). But when \((1+n)(1+g)<\frac{\alpha(1-\beta)}{\beta(1-\alpha)}\), the condition (7.13) holds, thus the First Welfare Theorem holds.