Macroeconomics 2

• Krueger Macroeconomics Theory

General Principles for Specifying a Model

1. Households: preferences and endowments of commodities
2. Firms: production technology
3. Government: policy instruments (taxes, money supply etc.)

Simple Endowment Economy

There are 2 individuals that live forever in this pure exchange economy. There are no firms, and the government is absent as well. All information is public, i.e. all agents know everything. At period 0; before endowments are received and consumption takes place, the two agents meet at a central market place and trade all commodities, i.e. trade consumption for all future dates. In all future periods the agents in the economy just carry out the trades they agreed upon in period 0. (Arrow-Debreu Market)

Let $p_t$ denote the price, in period 0, of one unit of consumption to be delivered in period t, in terms of an abstract unit of account. We will see later that prices are only determined up to a constant, so we can always normalize the price of one commodity to 1 and make it the numeraire. The two agents are price taker.

$$u\left(c^{i}\right)=\sum_{t=0}^{\infty} \beta^{t} \ln \left(c_{t}^{i}\right)$$

Agents have deterministic endowment streams

$$e^{i}=\left\{e_{t}^{i}\right\}_{t=0}^{\infty}$$

$e_{t}^{1}= \begin{cases}2 & \text { if } t \text { is even } \\ 0 & \text { if } t \text { is odd }\end{cases}$ $e_{t}^{2}= \begin{cases}0 & \text { if } t \text { is even } \\ 2 & \text { if } t \text { is odd }\end{cases}$

Competitive Equilibrium

Households solve the following optimization problem,

$$\begin{aligned} &\max _{\left\{c_{t}^{i}\right\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^{t} \ln \left(c_{t}^{i}\right) \\ & s.t.\\ &\sum_{t=0}^{\infty} p_{t} c_{t}^{i} \leq \sum_{t=0}^{\infty} p_{t} e_{t}^{i} \\ & c_{t}^{i} \geq 0 \text { for all } t \end{aligned}$$

A (competitive) Arrow-Debreu equilibrium are prices $\hat{p}$ and allocations $\hat{c}$, that

1. households optimized at $\hat{p}$
2. Goods market clearing
   $$\hat{c}_{t}^{1}+\hat{c}_{t}^{2}=e_{t}^{1}+e_{t}^{2} \text { for all } t$$
   FOC-$c_t^i,c_{t+1}^i$
   $$\begin{aligned} \frac{\beta^{t}}{c_{t}^{i}} &=\lambda_{i} p_{t} \\ \frac{\beta^{t+1}}{c_{t+1}^{i}} &=\lambda_{i} p_{t+1} \end{aligned}$$
   By goods clearing and pluggin $c$ , solve for $p$.
   $$p_{t+1}\left(e_{t+1}^{1}+e_{t+1}^{2}\right)=\beta p_{t}\left(e_{t}^{1}+e_{t}^{2}\right)$$
   $$p_{t}=\beta^{t} p_{0}$$
   Then,
   $$c_{t+1}^{i}=c_{t}^{i}=c_{0}^{i}$$
   The left hand side of the budget constraint becomes
   $$\sum_{t=0}^{\infty} \hat{p}_{t} c_{t}^{i}=c_{0}^{i} \sum_{t=0}^{\infty} \beta^{t}=\frac{c_{0}^{i}}{1-\beta}$$
   The equilibrium allocation is then given by
   $$\begin{aligned} &\hat{c}_{t}^{1}=\hat{c}_{0}^{1}=(1-\beta) \frac{2}{1-\beta^{2}}=\frac{2}{1+\beta}>1 \\ &\hat{c}_{t}^{2}=\hat{c}_{0}^{2}=(1-\beta) \frac{2 \beta}{1-\beta^{2}}=\frac{2 \beta}{1+\beta}<1 \end{aligned}$$

Pareto Optimality

an allocation is Pareto efficient if it is feasible and if there is no other feasible allocation that makes no household worse off and at least one household strictly better off.

We now prove that every competitive equilibrium allocation for the economy described above is Pareto efficient. (First Welfare Theorem)

Negishi's Method

Now we describe a method to compute equilibria for economies in which the welfare theorem(s) hold. The main idea is to compute Pareto-optimal allocations by solving an appropriate social planners problem.

If the First welfare theorem holds then we know that competitive equilibrium allocations are Pareto optimal; by solving for all Pareto optimal allocations we have then solved for all potential equilib- rium allocations. Negishi's method provides an algorithm to compute all Pareto optimal allocations and to isolate those who are in fact competitive equilibrium allocations.

Consider the following social planner problem, for a Pareto weight $\alpha\in[0,1]$,

$$\begin{aligned} & \max _{\left\{\left(c_{t}^{1}, c_{t}^{2}\right)\right\}_{t=0}^{\infty}} \alpha u\left(c^{1}\right)+(1-\alpha) u\left(c^{2}\right) \\ & = \max _{\left\{\left(c_{t}^{1}, c_{t}^{2}\right)\right\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^{t}\left[\alpha \ln \left(c_{t}^{1}\right)+(1-\alpha) \ln \left(c_{t}^{2}\right)\right] \\ & \text { s.t. } \\ &c_{t}^{i} \geq 0 \text { for all } i, \text { all } t \\ & c_{t}^{1}+c_{t}^{2}=e_{t}^{1}+e_{t}^{2} \equiv 2 \text { for all } t \end{aligned}$$

The social planner maximizes the weighted sum of utilities of the two agents, subject to the allocation being feasible. The weight $\alpha$ indicates how important agent 1's utility is to the planner, relative to agent 2's utility.

FOC with multiplier $\mu_t/2$

$$\begin{aligned} \frac{\alpha \beta^{t}}{c_{t}^{1}} &=\frac{\mu_{t}}{2} \\ \frac{(1-\alpha) \beta^{t}}{c_{t}^{2}} &=\frac{\mu_{t}}{2} \end{aligned}$$

With constraints gives,

$$\begin{aligned} c_{t}^{2} &=2(1-\alpha)=c_{t}^{2}(\alpha) \\ c_{t}^{1} &=2 \alpha=c_{t}^{1}(\alpha) \end{aligned}$$

the social planner divides the total resources in every period according to the Pareto weights.

The Lagrange multipliers are given by

$$\mu_{t}=\frac{2 \alpha \beta^{t}}{c_{t}^{1}}=\beta^{t}$$

if we wouldnít have done the initial division by 2 we would have to carry the 1/2 around from now on, ther esults below wouldn't change at all,though.

Hence for this economy the set of Pareto efficient allocations is given by

$$P O=\left\{\left\{\left(c_{t}^{1}, c_{t}^{2}\right)\right\}_{t=0}^{\infty}: c_{t}^{1}=2 \alpha \text { and } c_{t}^{2}=2(1-\alpha) \text { for some } \alpha \in[0,1]\right\}$$

$$\text { By picking } \lambda_{1}=\frac{1}{2 \alpha} \text { and } \mu_{t}=p_{t} \text { the FOC in competitative equilibrium are identical. }$$

1. Solve the social planners problem for Pareto efficient allocations indexed by the Pareto weights $(\alpha,1-\alpha)$
2. Compute transfers, indexed by $\alpha$, necessary to make the efficient allocation affordable. As prices use Lagrange multipliers on the resource constraints in the planners' problem.
3. Find the Pareto weight(s) $\hat{\alpha}$ that makes the transfer functions 0.
4. The Pareto efficient allocations corresponding to $\hat{\alpha}$ are equilibrium allocations; the supporting equilibrium prices are (multiples of) the Lagrange multipliers from the planning problem.

Sequential Markets Equilibrium

we show that the same allocations as in an Arrow-Debreu (AD) equilibrium would arise if we let agents trade consumption and one-period bonds in each period. We will call a market structure in which markets for consumption and assets open in each period Sequential Markets and the corresponding equilibrium Sequential Markets (SM) equilibrium.

A one period bond is a promise (contract) to pay 1 unit of the consumption good in period t + 1 in exchange for $\frac{1}{1+r_{t+1}}$ units of the consumption good in period t.

$q_{t} \equiv \frac{1}{1+r_{t+1}}$ the relative price of one unit of the consumption good in period t + 1 in terms of the period t consumption good.

Let $a_{t+1}^i$ denote the amount of such bonds purchased by agent i in period t and carried over to period t + 1:

Definition 7 A Sequential Markets equilibrium is allocations $\left\{\left(\hat{c}_{t}^{i}, \hat{a}_{t+1}^{i}\right)_{i=1,2}\right\}_{t=0}^{\infty}$, interest rates $\left\{\hat{r}_{t+1}\right\}_{t=0}^{\infty}$ such that

1. For $i=1,2$, given interest rates $\left\{\hat{r}_{t+1}\right\}_{t=0}^{\infty}\left\{\hat{c}_{t}^{i}, \hat{a}_{t+1}^{i}\right\}_{t=0}^{\infty}$ solves
   $$\begin{aligned} & \max _{\left\{c_{t}^{i}, a_{t+1}^{i}\right\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^{t} \ln \left(c_{t}^{i}\right) \\ & \text { s.t. } \\ & c_{t}^{i}+\frac{a_{t+1}^{i}}{\left(1+\hat{r}_{t+1}\right)} \leq e_{t}^{i}+a_{t}^{i} \\ & c_{t}^{i} \geq 0 \text { for all } t \\ & a_{t+1}^{i} \geq-\bar{A}^{i} \quad \text{ guarantee existence of equilibrium. } \end{aligned}$$
2. For all $t \geq 0$
   $$\begin{aligned} &\sum_{i=1}^{2} \hat{c}_{t}^{i} =\sum_{i=1}^{2} e_{t}^{i} \\ &\sum_{i=1}^{2} \hat{a}_{t+1}^{i}=0 \end{aligned}$$