5 General Equilibrium Examples

5.1 Introduction

This section works through two extended examples of general equilibrium. The first solves for equilibrium factor prices in a production economy with two competitive industries, showing how output prices and factor supplies together determine the wage and the rental rate of capital. The second studies a pure-exchange economy under uncertainty and uses it to derive the fundamental equation of asset pricing: every asset’s price is the expectation of its payoff weighted by a stochastic discount factor equal to agents’ marginal rate of substitution across states.

The two examples play different roles. The first is deliberately partial in one respect: output prices are exogenous and consumers are absent, so that the interaction between zero profits and factor-market clearing can be studied on its own. The second is a full general-equilibrium analysis of the exchange side of an economy: the state prices that end up pricing every asset are determined entirely within the model by preferences, probabilities, and endowments.

Reading material: MWG Ch. 15

5.2 Equilibrium and factor prices

Consider two types of firms, one produces good 1 and the other produces good 2. Each firm uses capital and labor to produce their good. The production function for firm \(i\) is \(f_{i}(k,l) = A_{i}k^{\alpha_{i}}l^{1-\alpha_{i}}\). The rental rate of capital is \(r\) and the wage rate is \(w\). The market price of good \(i\) is \(p_{i}\) and is assumed to be exogenous. Notice that in order to simplify the problem we have abstracted away the utility maximization part of the general equilibrium example.

Firm \(i\)’s profit is \(\pi_{i} = p_{i}A_{i}k^{\alpha_{i}}l^{1-\alpha_{i}} - wl_{i} - rk_{i}\).

Before doing any algebra, it is worth being explicit about what an equilibrium of this economy is.

Definition 5.1 (Equilibrium with exogenous output prices) Given output prices \((p_{1},p_{2})\) and aggregate factor supplies \((K,L)\), an equilibrium of this economy is a pair of factor prices \((w,r)\) and output levels \((q_{1},q_{2})\) such that

Optimization. Each firm’s factor demands \((l_{i},k_{i})\) minimize the cost of producing its output \(q_{i}\) at the factor prices \((w,r)\).
Zero profits. Free entry under constant returns to scale drives profits to zero: \(p_{i}q_{i} = wl_{i} + rk_{i}\) for each sector that produces.
Market clearing. Factor demands sum to factor supplies: \(l_{1}+l_{2} = L\) and \(k_{1}+k_{2} = K\).

Conditions 2 and 3 supply four equations in the four unknowns \((w,r,q_{1},q_{2})\); the factor demands that appear in them are pinned down by condition 1. The next subsections work through these pieces in order.

5.2.1 Calculate the relative market shares

Firm \(i\)’s cost-minimization problem is to choose \((l_{i},k_{i})\) to minimize expenditure \(wl_{i} + rk_{i}\) subject to producing its output: \(A_{i}k_{i}^{\alpha_{i}}l_{i}^{1-\alpha_{i}} = q_{i}\). The Lagrangian is

\[\mathcal{L} = wl_{i} + rk_{i} + \lambda\left(q_{i} - A_{i}k_{i}^{\alpha_{i}}l_{i}^{1-\alpha_{i}}\right),\]

where the multiplier \(\lambda\) on the output constraint is the marginal cost of output. Because the technology is Cobb–Douglas, marginal products take the convenient form

\[\frac{\partial f_{i}}{\partial l} = (1-\alpha_{i})\,\frac{f_{i}(k,l)}{l}, \qquad \frac{\partial f_{i}}{\partial k} = \alpha_{i}\,\frac{f_{i}(k,l)}{k},\]

so, evaluating \(f_{i} = q_{i}\) at the constraint, the first order conditions are

\[\begin{aligned} w - \lambda(1-\alpha_{i})\frac{q_{i}}{l_{i}} & = 0 \\ r - \lambda \alpha_{i} \frac{q_{i}}{k_{i}} & = 0 \\ A_{i}k_{i}^{\alpha_{i}}l_{i}^{1-\alpha_{i}} & = q_{i} \end{aligned}\]

Dividing the first condition by the second eliminates \(\lambda\) and gives the cost-minimizing factor mix,

\[\frac{w}{r} = \frac{1-\alpha_{i}}{\alpha_{i}}\cdot\frac{k_{i}}{l_{i}}, \qquad\text{equivalently}\qquad \frac{k_{i}}{l_{i}} = B_{i}\,\frac{w}{r}, \qquad\text{where } B_{i} = \frac{\alpha_{i}}{1-\alpha_{i}}.\]

Note that the factor mix depends only on relative factor prices, not on the scale of production. Substituting \(k_{i} = B_{i}(w/r)\,l_{i}\) into the production constraint and solving for \(l_{i}\), then recovering \(k_{i}\), yields the conditional factor demands

\[\begin{aligned} l_{i}(w,r,q_{i}) & = \frac{q_{i}}{A_{i}}\left(\frac{r}{w}B_{i}^{-1}\right)^{\alpha_{i}} \\ k_{i}(w,r,q_{i}) & = \frac{q_{i}}{A_{i}}\left(\frac{w}{r}B_{i} \right)^{1-\alpha_{i}} \end{aligned}\]

Substituting these demands into \(wl_{i} + rk_{i}\) and simplifying gives the cost function

\[c_{i}(w,r,q_{i}) = \frac{q_{i}}{A_{i}}\left(\frac{r}{\alpha_{i}}\right)^{\alpha_{i}}\left(\frac{w}{1-\alpha_{i}}\right)^{1-\alpha_{i}}.\]

Now impose the zero-profit condition — and note why it must hold. Under constant returns, cost is linear in output, so firm \(i\)’s profit is \(\pi_{i} = \left(p_{i} - c_{i}(w,r,1)\right)q_{i}\), where \(c_{i}(w,r,1)\) is unit cost. If price exceeded unit cost, profit could be made arbitrarily large by scaling production up, which is incompatible with equilibrium; if price fell short of unit cost, any positive output would generate losses and the firm would produce nothing. Hence in any equilibrium in which good \(i\) is produced, price equals unit cost:

\[p_{i} = \frac{c_{i}(w,r,q_{i})}{q_{i}} = \frac{1}{A_{i}}\left(\frac{r}{\alpha_{i}}\right)^{\alpha_{i}}\left(\frac{w}{1-\alpha_{i}}\right)^{1-\alpha_{i}}.\]

Notice that because the technology has constant returns, unit cost does not depend on \(q_{i}\); the zero-profit condition therefore restricts the factor prices \((w,r)\) but leaves each firm’s scale undetermined at the firm level.

Because the technology is Cobb–Douglas, the cost-minimizing factor payments are constant shares of total cost: capital receives the share \(\alpha_{i}\) and labor the share \(1-\alpha_{i}\), so \(r k_{i} = \alpha_{i}\, c_{i}(w,r,q_{i})\) and \(w l_{i} = (1-\alpha_{i})\, c_{i}(w,r,q_{i})\). Using zero profit to replace cost by revenue, \(c_{i} = p_{i}q_{i}\), the equilibrium factor demands are

\[\begin{aligned} l_{i} &= \frac{(1-\alpha_{i})\, p_{i} q_{i}}{w}, \\ k_{i} &= \frac{\alpha_{i}\, p_{i} q_{i}}{r}. \end{aligned}\]

Writing \(Y_{i} = p_{i} q_{i}\) for the value of sector \(i\)’s output, factor-market clearing \(l_{1} + l_{2} = L\) and \(k_{1} + k_{2} = K\) gives the equilibrium factor prices

\[\begin{aligned} w & = \frac{(1-\alpha_{1})Y_{1} + (1-\alpha_{2})Y_{2}}{L}, \\ r & = \frac{\alpha_{1}Y_{1} + \alpha_{2}Y_{2}}{K}. \end{aligned}\]

Each factor’s price is the cost-share-weighted total value of the output it helps produce, divided by its aggregate supply. The output values \(Y_{i} = p_{i} q_{i}\) are pinned down by closing the model: the two zero-profit conditions are two equations in \((w,r)\), which together with the two market-clearing conditions determine the four unknowns \((w, r, q_{1}, q_{2})\).

Exercise 5.1 Take \(A_{1} = A_{2} = 1\), \(\alpha_{1} = \tfrac{1}{3}\), \(\alpha_{2} = \tfrac{2}{3}\), output prices \(p_{1} = p_{2} = 1\), and factor supplies \(K = L = 1\). Solve the four equilibrium conditions for \((w, r, q_{1}, q_{2})\), and report the resulting factor demands \(l_{i}, k_{i}\) for each type of firm. (You should find a symmetric solution.)

5.3 Endowment economies and asset pricing

We now turn to a pure-exchange economy under uncertainty and use it to derive the central equation of asset pricing. There are two dates, \(t = 0\) (today) and \(t = 1\) (tomorrow); at date \(1\) exactly one of \(S\) states \(s \in \{1,\ldots,S\}\) occurs, with commonly-held probabilities \(\pi_{s} > 0\). There is a single consumption good at each date–state.

There are \(I\) individuals. Each is a price taker. Individual \(i\) has an initial endowment \(e^{i} = (e^{i}_{0}, e^{i}_{1}, \ldots, e^{i}_{S})\) — a quantity of the good today and one contingent on each state tomorrow — and preferences of the expected-utility form

\[U^{i}(c) = u^{i}(c_{0}) + \beta_{i}\sum_{s=1}^{S}\pi_{s}\, u^{i}(c_{s}),\]

with each \(u^{i}\) strictly increasing and strictly concave, and \(\beta_{i} \in (0,1]\) individual \(i\)’s time-discount factor. Preferences of this form are exactly those delivered by the von Neumann–Morgenstern theorem (Theorem 4.1) applied to lotteries over date–state consumption, with the state probabilities \(\pi_{s}\) — here objective and commonly held — playing the role of the lottery probabilities.

5.3.1 Contingent-claims equilibrium

Suppose individuals can trade, at date \(0\), claims to the good in each date–state. Normalize the price of date-\(0\) consumption to \(1\), and let \(q_{s}\) be the date-\(0\) price of one unit of the good delivered if and only if state \(s\) occurs. Individual \(i\) solves

\[\max_{c \ge 0}\ u^{i}(c_{0}) + \beta_{i}\sum_{s}\pi_{s}u^{i}(c_{s}) \quad \text{subject to} \quad c_{0} + \sum_{s} q_{s} c_{s} \le e^{i}_{0} + \sum_{s} q_{s} e^{i}_{s}.\]

Definition 5.2 A contingent-claims (Arrow–Debreu) equilibrium is a price vector \((q_{1},\ldots,q_{S})\) and an allocation \(\{c^{i}\}_{i=1}^{I}\) such that: (i) for each \(i\), \(c^{i}\) solves individual \(i\)’s problem at those prices; and (ii) all markets clear, \(\sum_{i} c^{i}_{0} = \sum_{i} e^{i}_{0}\) and \(\sum_{i} c^{i}_{s} = \sum_{i} e^{i}_{s}\) for every state \(s\).

Letting \(\lambda_{i}\) be the multiplier on individual \(i\)’s budget constraint, the first-order conditions are \(u^{i\prime}(c^{i}_{0}) = \lambda_{i}\) and \(\beta_{i}\pi_{s} u^{i\prime}(c^{i}_{s}) = \lambda_{i} q_{s}\). Eliminating \(\lambda_{i}\),

\[q_{s} = \beta_{i}\,\pi_{s}\,\frac{u^{i\prime}(c^{i}_{s})}{u^{i\prime}(c^{i}_{0})} \qquad \text{for every individual } i.\]

The state price \(q_{s}\) equals each individual’s (probability-weighted) marginal rate of substitution between consumption in state \(s\) tomorrow and consumption today. In particular, in equilibrium every individual has the same MRS in every state — and this is precisely the first-order condition for Pareto optimality, which we can verify directly. Fix positive welfare weights \(\mu_{1},\ldots,\mu_{I}\) and consider the planner’s problem

\[\max_{\{c^{i} \ge 0\}}\ \sum_{i}\mu_{i}\left[u^{i}(c^{i}_{0}) + \beta_{i}\sum_{s}\pi_{s}u^{i}(c^{i}_{s})\right] \quad\text{subject to}\quad \sum_{i}c^{i}_{0} \le \sum_{i}e^{i}_{0}, \quad \sum_{i}c^{i}_{s} \le \sum_{i}e^{i}_{s} \ \text{ for all } s.\]

Any solution to this problem is Pareto optimal: an allocation that made someone better off and no one worse off would raise the planner’s objective. Letting \(\eta_{0}\) and \(\eta_{s}\) denote the multipliers on the resource constraints, the first-order conditions are \(\mu_{i}\,u^{i\prime}(c^{i}_{0}) = \eta_{0}\) and \(\mu_{i}\,\beta_{i}\,\pi_{s}\,u^{i\prime}(c^{i}_{s}) = \eta_{s}\). Eliminating the welfare weight \(\mu_{i}\),

\[\beta_{i}\,\pi_{s}\,\frac{u^{i\prime}(c^{i}_{s})}{u^{i\prime}(c^{i}_{0})} = \frac{\eta_{s}}{\eta_{0}} \qquad \text{for every individual } i.\]

At a Pareto optimum, all individuals share a common MRS in every state, with the multiplier ratio \(\eta_{s}/\eta_{0}\) playing exactly the role that the state price \(q_{s}\) plays in equilibrium; since each \(u^{i}\) is strictly concave, these first-order conditions are also sufficient. The equilibrium allocation therefore satisfies the conditions for Pareto optimality — a first glimpse of the welfare theorems, which we take up in general form in the next chapter.

5.3.2 The fundamental asset-pricing equation

Now introduce assets. Asset \(j\) is a claim to a state-contingent payoff \(d_{j} = (d_{j1},\ldots,d_{jS})\) tomorrow, with price \(P_{j}\) today. An arbitrage is a trading strategy that costs nothing today (or less than nothing), requires no net payment in any state tomorrow, and yields a strictly positive amount either today or in some state. Equilibrium prices must admit no arbitrage: any individual with strictly increasing utility would demand such a trade without bound, and markets could not clear. The payoff of asset \(j\) can be replicated by buying \(d_{js}\) units of the state-\(s\) contingent claim for each \(s\), at a total cost of \(\sum_{s} q_{s} d_{js}\). If \(P_{j}\) exceeded this, selling the asset and buying the replicating portfolio would pocket the difference today while covering every future obligation exactly; if \(P_{j}\) fell short, the reverse trade would do the same. Ruling out both, the asset’s price must equal the value of its payoff,

\[P_{j} = \sum_{s=1}^{S} q_{s}\, d_{js}.\]

Define the stochastic discount factor (or pricing kernel) \(m_{s} = q_{s}/\pi_{s}\) — the state price per unit of probability. Then

\[\boxed{\,P_{j} = \sum_{s}\pi_{s}\, m_{s}\, d_{js} = \mathbb{E}[m\, d_{j}].\,}\]

From the first-order conditions, \(m_{s} = \beta_{i}\,u^{i\prime}(c^{i}_{s})/u^{i\prime}(c^{i}_{0})\): the discount factor is any individual’s discounted marginal-utility ratio. This one equation prices every asset.

Two special cases make it concrete. A risk-free asset pays \(1\) in every state, so its price is \(P_{f} = \sum_{s}\pi_{s} m_{s} = \mathbb{E}[m]\) and the gross risk-free return is \(R_{f} = 1/\mathbb{E}[m]\). For a risky asset with gross return \(R_{j} = d_{j}/P_{j}\), the pricing equation reads \(1 = \mathbb{E}[m R_{j}]\); combining this with \(R_{f} = 1/\mathbb{E}[m]\) and \(\mathbb{E}[m R_{j}] = \mathbb{E}[m]\,\mathbb{E}[R_{j}] + \operatorname{Cov}(m, R_{j})\) gives

\[\mathbb{E}[R_{j}] - R_{f} = -R_{f}\,\operatorname{Cov}(m, R_{j}).\]

An asset commands a positive risk premium precisely when its return covaries negatively with the discount factor — that is, when it pays off poorly in the “expensive” states, those with high marginal utility (typically, low aggregate consumption). This is the consumption-based account of why risky assets earn more than the risk-free rate.

Example 5.1 (A two-state economy) Let \(S = 2\) with \(\pi_{1} = \pi_{2} = \tfrac{1}{2}\), and consider a representative individual — equivalently, \(I\) identical individuals each owning the share \(1/I\) of the aggregate endowment, who then have no reason to trade with one another — with \(u(c) = \ln c\) and \(\beta = 0.96\). The endowment is \(e_{0} = 1\) today; tomorrow it is \(e_{1} = 0.8\) in state \(1\) (a recession) and \(e_{2} = 1.2\) in state \(2\) (a boom). In equilibrium consumption equals the endowment, so the state prices are

\[q_{s} = \beta\,\pi_{s}\,\frac{u^{\prime}(e_{s})}{u^{\prime}(e_{0})} = \beta\,\pi_{s}\,\frac{e_{0}}{e_{s}}, \qquad\text{so}\qquad q_{1} = \frac{0.48}{0.8} = 0.6, \qquad q_{2} = \frac{0.48}{1.2} = 0.4,\]

and the stochastic discount factor is \(m_{1} = q_{1}/\pi_{1} = 1.2\) and \(m_{2} = q_{2}/\pi_{2} = 0.8\): consumption claims are expensive in the recession state, where marginal utility is high, and cheap in the boom state.

The risk-free asset costs \(P_{f} = \mathbb{E}[m] = q_{1} + q_{2} = 1\), so \(R_{f} = 1\). Next price “equity,” the claim to the aggregate endowment \(d = (0.8,\, 1.2)\): its price is \(P = q_{1}(0.8) + q_{2}(1.2) = 0.96\), its state-contingent returns are \(R_{1} = 0.8/0.96 \approx 0.833\) and \(R_{2} = 1.2/0.96 = 1.25\), so \(\mathbb{E}[R] \approx 1.042\) and the risk premium is \(\mathbb{E}[R] - R_{f} \approx 4.2\%\). The covariance formula confirms this: \(\operatorname{Cov}(m,R) = \mathbb{E}[mR] - \mathbb{E}[m]\,\mathbb{E}[R] = 1 - 1.042 \approx -0.042\), so \(-R_{f}\operatorname{Cov}(m,R) \approx 4.2\%\). Equity pays off poorly exactly when consumption is scarce, so it must offer a premium over the risk-free rate to be willingly held.

Exercise 5.2 In the economy of Example 5.1, replace log utility with the CRRA utility \(u(c) = c^{1-\gamma}/(1-\gamma)\) with \(\gamma = 2\), keeping \(\beta = 0.96\) and the same endowments and probabilities.

Compute the state prices, the stochastic discount factor, and the gross risk-free return.
Compute the price, expected return, and risk premium of the claim to the aggregate endowment. Compare with the log-utility case (which is \(\gamma = 1\)): does higher risk aversion raise or lower the premium? Why?
Price the counter-cyclical asset with payoff \(d = (1.2,\, 0.8)\) and explain the sign of its risk premium.

5.3.3 Complete markets

If the traded assets span \(\mathbb{R}^{S}\) — equivalently, if the \(S\) Arrow securities (asset \(s\) paying one unit in state \(s\) and zero otherwise) can be replicated — then the asset-market (Radner) equilibrium implements the contingent-claims equilibrium: the state prices \(q_{s}\) are uniquely determined and, by the argument above, every individual’s MRS is equalized, so the equilibrium allocation is Pareto optimal. When markets are incomplete, individuals’ marginal rates of substitution need not coincide, the discount factor is no longer unique, and the allocation is generally not Pareto optimal.