34. Bubbles Pareto Improve Decentralized Competitive Equilibrium

In this section, we will be looking at three problems. The first one is a model which has a similar structure as in overlapping generations model, but it is static instead of dynamic, so there is no time periods in it. We use that model to reach some conclusions on Pareto efficiency of decentralized equilibrium, and the algebra is the same for the next two problems. The second one is the overlapping generation model, in which we show that in some cases the competitive equilibrium is Pareto inefficient. The last one is introducing fiat money (or other intrinsically worthless asset) to the model. And the competitive equilibrium is Pareto improved when the fiat money's price is strictly positive, i.e. when there are price bubbles. So, the takeaway is that, quite counter-intuitively, price bubbles could improve the efficiency of an competitive equilibrium in some cases.

34.1 Static Model

34.1.1 Set-up

This is a static endowment economy, so there is only one period, and there is no production. There are infinitely many individuals, indexed by \(i\in\mathbb{Z}\) (integers, could also be negative), and each has measure 1. There are infinitely many types of goods, indexed by \(j\in\mathbb{Z}\). Each individual \(i\) has endowment of each good \(j\), which is denoted by \(e_{i,j}\): \(e_{i,i}=e_1>0\), \(e_{i,i+1}=e_2>0\), \(e_{i,j}=0\) for \(j\ne i\) and \(j\ne i+1\).

Preferences of individual \(i\) are represented by \(\ln c_{i,i}+\ln c_{i,i+1}\) (additively separable log function with discounting factor equal to one; in fact, we only need the concavity of the utility function, but for simplicity we assume this). Let \(q_i^{i+1}\) denote the relative price of good \(i+1\) in terms of good \(i\), i.e. the amount of good \(i\) needed to purchase one unit of good \(i+1\). The budget constraint is \(c_{i,i}+q_i^{i+1}c_{i,i+1}=e_1+q_i^{i+1}e_2\). The aggregate endowment of good \(i\) is \(e_1+e_2\), which is the same across goods. A competitive equilibrium is a set of prices \(\{q_i^{i+1}\}_{i\in\mathbb{Z}}\) such that all individuals maximize their utility and all good markets clear, i.e.

$$ c_{i,i}+c_{i-1,i}=e_1+e_2 \tag{34.1} $$

34.1.2 Consumer's utility maximization problem

$$ \max_{c_{i,i},c_{i,i+1}}\ln c_{i,i}+\ln c_{i,i+1}\quad\text{s.t.}\ c_{i,i}+q_i^{i+1}c_{i,i+1}=e_1+q_i^{i+1}e_2 \tag{34.2} $$

The Lagrangian f.o.c. give \(\dfrac{c_{i,i}}{c_{i,i+1}}=q_i^{i+1}\) (34.3), and plug into the budget constraint to obtain

$$ c_{i,i}=\frac{e_1+q_i^{i+1}e_2}{2},\qquad c_{i,i+1}=\frac{e_1+q_i^{i+1}e_2}{2q_i^{i+1}} \tag{34.4} $$

34.1.3 Prices that support competitive equilibrium

Plug (34.4) into the goods market clearing condition (34.1) and rearrange to get

$$ q_i^{i+1}=1+\frac{e_1}{e_2}\left(1-\frac{1}{q_{i-1}^i}\right) \tag{34.5} $$

Note that \(\{q_i^{i+1}\}\) supporting a competitive equilibrium is non-negative, otherwise agents will demand infinitely many of goods with negative price and thus there is no way to have an equilibrium. (34.5) has two fixed-point solutions: Solution 1 \(q_i^{i+1}=1\) \(\forall i\); Solution 2 \(q_i^{i+1}=\dfrac{e_1}{e_2}\) \(\forall i\).

Figure 23 (Relationship Between Relative Prices, paraphrased): vertical axis \(q_i^{i+1}\), horizontal axis \(q_{i-1}^i\). The curve of (34.5) is increasing and concave in \(q_{i-1}^i\) with supremum \(1+\frac{e_1}{e_2}\), intersecting the \(45^\circ\) line at two points: \(A\) (lower fixed point \(\min\{1,\frac{e_1}{e_2}\}\)) and \(B\) (upper fixed point \(\max\{1,\frac{e_1}{e_2}\}\)). Between \(A\) and \(B\), as \(i\) increases \(q\) moves toward \(B\); if \(q_i^{i+1}\) is to the left of \(A\), as \(i\) decreases the price goes negative (no equilibrium); if to the right of \(B\), since the curve has supremum, as \(i\) decreases there is a certain \(i\) with \(q_{i-2}^{i-1}\) negative (no equilibrium).

So, \(\{q_i^{i+1}\}\) supporting a competitive equilibrium must lie between \(\left[\min\{1,\frac{e_1}{e_2}\},\max\{1,\frac{e_1}{e_2}\}\right]\); for any arbitrary \(q_0^i\) in this interval, the \(\{q_i^{i+1}\}\) obtained from (34.5) supports a competitive equilibrium.

34.1.4 Competitive equilibrium could be Pareto inefficient

Two special competitive equilibria (the two fixed-point solutions):

• \(q_i^{i+1}=1\) \(\forall i\): \(c_{i,i}=c_{i,i+1}=\dfrac{e_1+e_2}{2}\) \(\forall i\) (34.6).
• \(q_i^{i+1}=\dfrac{e_1}{e_2}\) \(\forall i\) (assume \(e_1\ne e_2\) to be non-trivial): \(c_{i,i}=e_1\), \(c_{i,i+1}=e_2\) \(\forall i\) (34.7).

By the concavity of utility function, Jensen's inequality implies that (34.6) strictly dominates (34.7) \(\forall i\) (since (34.6) gives equal consumption in both goods while (34.7) gives unequal). So, the first competitive equilibrium (\(q=1\)) Pareto dominates the second (\(q=\frac{e_1}{e_2}\)), i.e. the second competitive equilibrium is Pareto inefficient. The standard First Welfare Theorem proof fails here because the aggregate endowment is infinite (we can go both directions of \(i\)).

34.2 Overlapping generations model

34.2.1 Set-up

This is an overlapping endowment economy, so there are infinitely many periods starting from period 0, and there is no production. Individuals are indexed by their birth period \(t\in\mathbb{Z}\) (could also be negative), and each cohort has measure 1. They live for 2 periods, and then die. There is only one type of good. Individual born in period \(t\ge0\) gets endowment \(e_1\) in his first (young) period and \(e_2\) in his second (old) period, and choose the consumption \(c_{1,t}\) when they are young and \(c_{2,t+1}\) when they are old. Individual who is old in period 0 (initial old) only gets endowment \(e_2\) in period 0, and he consumes \(c_{2,0}\) in period 0 and dies. (The first subscript 1/2 denotes age: 1 means young, 2 means old; the second subscript denotes period.) Preferences of individual born in period \(t\ge0\) are \(\ln c_{1,t}+\ln c_{2,t+1}\) (initial old: \(\ln c_{2,0}\)). Let \(q_t^{t+1}\) denote the relative price of good in period \(t+1\) in terms of good in period \(t\). The budget constraint is \(c_{1,t}+q_t^{t+1}c_{2,t+1}=e_1+q_t^{t+1}e_2\). The aggregate endowment of good each period is \(e_1+e_2\). A competitive equilibrium is a set of prices \(\{q_t^{t+1}\}_{t\in\mathbb{Z}_+}\) such that all individuals maximize their utility and good market each period clears, i.e. \(c_{1,t}+c_{2,t}=e_1+e_2\) (34.8).

34.2.2–34.2.3 Solving and the unique competitive equilibrium pinned down by initial old

The algebra is exactly the same as §34.1: \(\dfrac{c_{1,t}}{c_{2,t+1}}=q_t^{t+1}\) (34.10), \(c_{1,t}=\dfrac{e_1+q_t^{t+1}e_2}{2}\) etc. (34.11), the recursion \(q_t^{t+1}=1+\dfrac{e_1}{e_2}\left(1-\dfrac{1}{q_{t-1}^t}\right)\) (34.12), with the two fixed points \(1\) and \(\frac{e_1}{e_2}\). The difference is that we don't have negative \(c\)'s (\(t\ge0\)) and we cannot start with any arbitrary \(q_0^i\) — the initial old pins down the unique \(q_0^1\). Consider the trivial maximization problem of the initial old: \(\max\ln c_{2,0}\) s.t. \(c_{2,0}=e_2\) ⟹ \(c_{2,0}=e_2\), and by market clearing in period 0 \(c_{1,0}=e_1\); by (34.11) at \(t=0\): \(\frac{e_1+q_0^1 e_2}{2}=e_1\Rightarrow q_0^1=\frac{e_1}{e_2}\). By the fixed-point property of \(\frac{e_1}{e_2}\), the only competitive equilibrium price is \(q_t^{t+1}=\dfrac{e_1}{e_2}\) \(\forall t\ge0\), and the corresponding consumption is \(c_{1,t}=e_1\), \(c_{2,t+1}=e_2\) \(\forall t\ge0\), and \(c_{2,0}=e_2\).

34.2.4 Competitive equilibrium could be Pareto inefficient (three cases)

• Case 1 (\(e_1
• Case 2 (\(e_1=e_2\)): the aggregate endowment is infinite, and the standard proof for First Welfare Theorem does not apply; however, we can still show that in this case the unique competitive equilibrium is Pareto efficient by the following argument: for agent born in any period, his utility is strictly increasing in the total endowment he received in two periods (by local non-satiation condition); for each agent, the allocation between two periods are already optimally determined by setting \(c_{1,t}=c_{2,t+1}=\frac{e_1+e_2}{2}\), so increasing the utility of any agent can only be realized by giving him more goods in total; given the prices \(q_t^{t+1}=\frac{e_1}{e_2}=1\), goods are equally one-to-one transferred (in principle, but in reality there is no trade) among periods, so increasing the utility of any agent by giving him more goods in total must also decrease the total goods of someone else in the economy and thus decrease that guy's utility, which is not a Pareto improvement. So Pareto efficient.
• Case 3 (\(e_1>e_2\)): the competitive equilibrium is Pareto inefficient. Consider the allocation \(c_{1,t}=c_{2,t+1}=\dfrac{e_1+e_2}{2}\) \(\forall t\ge0\) (34.13), which Pareto dominates the unique competitive equilibrium allocation \(c_{1,t}=e_1,c_{2,t+1}=e_2\). By Jensen's inequality and concavity of utility function, any agent born in date \(t\ge0\) is strictly better off, and we also check for the initial old to find that his consumption in the unique competitive equilibrium is \(e_2\) whereas in the new allocation (34.13) is \(\frac{e_1+e_2}{2}>e_2\) (since \(e_1>e_2\)), so the initial old is also strictly better off.

34.3 Introduce an asset with price bubbles

In the overlapping generations model (\(e_1>e_2\)), the unique competitive equilibrium allocation \(c_{1,t}=e_1,c_{2,t+1}=e_2\) is Pareto improved by \(c_{1,t}=c_{2,t+1}=\frac{e_1+e_2}{2}\), which is impossible to emerge because it is not an competitive equilibrium. But, in this subsection, we can show that with the introduction of fiat money (or other intrinsically worthless assets), the system can reach the Pareto efficient competitive equilibrium where \(c_{1,t}=c_{2,t+1}=\frac{e_1+e_2}{2}\).

We continue to solve the problem in the overlapping generations set-up, but let's introduce the intrinsically worthless asset: suppose there is an asset in the economy that has zero dividends stream, and we call it fiat money; denote the price of fiat money in period \(t\) by \(p_t\ge0\); suppose the fiat money has measure 1, and is endowed only to the initial old, who uses it and passes it down to the next generation; denote the holding of fiat money of agent \(t\) born in period \(t\) by \(a_t\). Then, rewrite the utility maximization problem for agents born in period \(t\ge0\): \(\max\ln c_{1,t}+\ln c_{2,t+1}\) s.t.

$$ c_{1,t}+p_t a_t=e_1\ \text{(34.14)},\qquad c_{2,t+1}=e_2+p_{t+1}a_t\ \text{(34.15)} $$

One obvious solution is that \(p_t=0\) \(\forall t\) (non-monetary equilibrium), then the solution is exactly the unique competitive equilibrium of the overlapping generations model. But we are interested in some monetary equilibrium where \(p_t\) is strictly positive, in which case we have price bubbles.

Focus on the case \(e_1>e_2\). Consider a competitive equilibrium with initial fiat money price (bubble) \(p_0=\dfrac{e_1-e_2}{2}\): plug into (34.20) to get \(\frac{p_0}{p_1}=1\Rightarrow p_1=p_0\), then iteratively by (34.19), \(p_t=\dfrac{e_1-e_2}{2}\) \(\forall t\ge0\). The corresponding consumption is \(c_{1,t}=c_{2,t+1}=\dfrac{e_1+e_2}{2}\) (which is Pareto efficient by the argument in case 2). So, the introduction of fiat money and the existence of its price bubbles improve the competitive equilibrium if the asset has a correct initial positive bubble price \(p_0=\frac{e_1-e_2}{2}\).

34.4 Brief discussion on other possible settings

We continue to use the notation in overlapping generations model, and focus on the case of \(e_1>e_2\) such that it is possible to have positive price for intrinsically worthless asset.

• 34.4.1 Introduce an additional asset with dividends: in addition to the intrinsically worthless asset (holding \(m_t\), price \(p_{m,t}\)), introduce a valuable asset that yields a constant dividend \(d>0\) in every period (holding \(a_t\), price \(p_{a,t}\)), both endowed to the initial old generation with measure 1. Market clearing: good \(c_{1,t}+c_{2,t}=e_1+e_2+d\), valuable asset \(a_t=1\), worthless asset \(m_t=1\). The only possible equilibrium is the non-monetary (no bubble) equilibrium \(p_{m,t}=0\) \(\forall t\): valuable asset market clearing \(a_t=1\) requires affordability \(p_{a,t}\le e_1\); constant positive dividends mean the only way to have its price finite is to impose strictly positive (net) real interest rate \(r_t>0\) \(\forall t\); then the discounted value of total endowment is finite ⟹ standard First Welfare Theorem argument applies ⟹ competitive equilibrium must be Pareto efficient ⟹ price of intrinsically worthless asset is zero in all periods. The non-monetary equilibrium f.o.c. give \(p_{a,t}=\dfrac{e_1}{2+\frac{e_2}{p_{a,t+1}+d}}\) (34.22), which is increasing in the sum of the asset's value in the next period \((p_{a,t+1}+d)\).
• 34.4.2 Growing population with constant per capita endowment: we want to have an economy where both the valuable asset and the intrinsically worthless asset have positive price. We break the finite endowment condition necessary for First Welfare Theorem proof, and simultaneously have affordable prices for the valuable asset. Construct: population of each generation growing at rate \(1+n\) for \(n>0\); people only live two periods, but the asset (e.g. tree) is alive forever; initial old generation has unit mass (measure 1 population) endowed with unit asset whose dividends are constant \(d\) each period; each new generation \(t\) is born with \(n(1+n)^{t-1}\) units of extra asset, and the previous generation sells all of their assets to the new generation so that every generation have the same asset per capita. In this case, we could have the real interest rate \(r_t>0\) so the asset price is finite and affordable (required by the asset market clearing for competitive equilibrium), but we could still have infinite aggregate endowment when \(n\ge r_t\) \(\forall t\), so the proof of First Welfare Theorem breaks down and the competitive equilibrium is Pareto inefficient, which means the price of intrinsically worthless asset could be strictly positive to Pareto improve the competitive equilibrium, i.e. \(p_{m,t}>0\) and \(p_{a,t}>0\) could coexist.
• 34.4.3 Two types of infinitely living agents: two kinds of endowment: high \(e_1\) and low \(e_2\) (\(e_1>e_2\)); Type A receives \(e_1\) in odd periods and \(e_2\) in even periods, Type B receives \(e_2\) in odd periods and \(e_1\) in even periods; asset is intrinsically worthless, agent \(i\)'s holding of asset in period \(t\) is \(a_{i,t}\), the asset is traded one-on-one with consumption goods so the relative price in period \(t+1\) is \(q_t^{t+1}\); two types have the same preferences \(\max\sum\beta^t\ln c_{i,t}\) s.t. \(c_{i,t}=e_{i,t}+a_{i,t}-q_t^{t+1}a_{i,t+1}\), \(a_{i,t}\ge0\), \(\beta e_1>e_2\). Two kinds of equilibria: autarky (the restriction \(a_{i,t}\ge0\) prevents agents from borrowing, can be regarded as incomplete market; the relative price \(q_t^{t+1}\) is not low enough to make the low endowment agent want to buy asset; the constraint \(a_{i,t}\ge0\) makes the low endowment agent not want to trade \(a_{i,t}=0\), and the other type also won't trade, so autarky \(a_{A,t}=a_{B,t}=0\) \(\forall t\)); the second equilibrium Pareto improves the autarky (the relative price of the asset is such that both types want to trade each period; by revealed preference argument, it Pareto improves the autarky equilibrium).