6. Optimal Growth and General Equilibrium¶
6.1. Digging deeper¶
Previously we talked about the saving mechanism, all else equal, as a key determinant of long run steady state level of wealth (capital) of an economy. In the Solow-Swan model, this mechanism is there by assumption. Now we turn attention to a discrete-time version of an optimal growth problem in which the saving rate is an equilibrium object. Why deal with this complication? There are two good reasons:
- One would like to ask: What explains saving behavior? Conceptually we would like map out the relation between people’s preferences over resource outcomes and their consumption opportunties, and, the overall economy’s savings or consumption behavior.
- This allows us to have a more disciplined structure (based on preference-theoretic welfare economics) when dealing with normative issues later in policy-relevant versions of this model.
Note
The latter point is related to an issue in econometric and policy modelling credited to Nobel laureate Robert E. Lucas, Jr as the “Lucas Critique” [Lu1976]. One is concerned with the questions: Would observed aggregate behavior change if policies or other common factors change, and if so, how does one evaluate policy experiments?
Before Lucas came along, econometricians in the twentieth century (e.g., in the Yale Cowles Commission approach) were concerned with estimating policy-relevant macroeconometric models. Most of these models —beginning from Tinbergen [Ti1939] until the 1970s—were taken to be “structural” in the sense that the structure was assumed invariant to policy and other events. Following Lucas’ critique, the basic concern for deeper structural macroeconometric modelling is the following: Postulated reduced-form econometric relationships should not be taken at face value (i.e., as “structural”) just because it holds in historical data. It may not be the case that when policies change these (reduced-form) “behavorial systems” will remain the same. Economic actors may react. For example, their best replies to policy changes and their (equilibrium) interactions may actually destabilize the measured relationship in the data.
6.2. Ramsey-Cass-Koopman’s Economy¶
We will begin our journey into the Ramsey-Cass-Koopmans (RCK) model economy. This model provides a foundation for much of modern macroeconomic theory. See [Ra1928], [Ca1965] and [Ko1965].
Let us repeat some of the similar setup from Solow-Swan. Later, we will deal with what is different. We will attack the new addition in two broad steps: First, we will ensure that we understand the mechanics of a finite-horizon optimal planning problem. Second, we go into a version of the Ramsey-Cass-Koopmans economy proper.
The following is the model’s setup in common with our Solow-Swan environment (without population or labor-augmenting technical progress):
Production function \(F: \mathbb{R}_{+}^{2} \rightarrow \mathbb{R}_{+}\) for final good (\(Y\)) using capital (\(K\)) and labor (\(L\)) such that:
(1)¶\[Y_{t} = F \left( K_{t}, L_{t} \right).\]
We also assume that there is a function \(f\) such that \(f(k) = F(k,1)\), where \(k:=K/L\). As in Solow-Swan, assume that \(f'(k)>0\) and \(f''(k)<0\).
We assume available capital at the start of date \(t \in \mathbb{N}\) is used for producing output \(Y_{t}\) and in that process a fraction \(\delta\) of capital disappears or wears out. There is a technology that converts the remaining proportion \((1-\delta) \in (0,1)\) of date-\(t\) initial capital stock (\(K_{t}\)) and investment flow \(I_{t}\) into next period productive capital, \(K_{t+1}\):
(2)¶\[K_{t+1} = (1 - \delta) K_{t} + I_{t}.\]Resource constraint for a closed economy, where total saving flows (\(S_{t}\)) must equal investment flows:
(3)¶\[Y_{t} = C_{t} + I_{t}.\]
Notice that these equations can be distilled into
Since we are focusing on a model with identical agents, let’s look at the average or per capita variables. Divide the last equation (4) by \(L_{t}\), and denoting the per-capita level of a variable \(X_{t}\) as \(x_{t} := X_{t}/L_{t}\), to get:
6.2.1. Finite Horizon¶
Before we get onto the optimal growth model, consider a simplification with a finite-period horizon for households decision making. What is new relative to Solow-Swan is the following: The representative household has a preference ordering over consumption outcomes, \((c_{0}, c_{1}, ... c_{T})\), where \(T < +\infty\). These preferences can be represented by a utility function \(U: \mathbb{R}^{T} \rightarrow \mathbb{R}\) of the form:
The function \(u: \mathbb{R}_{+} \rightarrow \mathbb{R}\) is called a per-period payoff or utility function. (In older literature, this is referred to as the felicity function.) The parameter \(\beta \in (0,1)\) is a constant weighting function, that discounts future per-period payoffs into units of the utility payoff at date-\(0\). This is conventionally called the subjective discount factor.
Suppose there is a benevolent planner that solves the following allocation problem:
subject to the constraint (5) holding at each date \(t \leq T\), the boundary value \(k_{0}\in (0,+\infty)\) (given) and \(k_{t+1}\geq 0\).
Let’s place more structure on the function \(u\) to facilitate more insight later:
- \(u \in \mathbf{C}^{2}(\mathbb{R}_{+})\) — i.e., it is a twice-continuously differentiable real-valued function on the domain \(\mathbb{R}_{+}\); and
- \(u'(c)>0\) and \(u''(c)<0\) for all \(c\in\mathbb{R}_{+}\), which implies that \(u\) is a strictly increasing and a strictly concave preference representation.
- \(\lim_{c \nearrow \infty} u'(c) = 0\) and \(\lim_{c \searrow 0} u'(c) = \infty\).
The first assumption allows us to work with first-order conditions. The second (strict concavity) ensures a solution to be unique. The final assumption rules out corner solutions.
Exercise Finite Horizon (1)
Show that there exists a natural upper bound on \(k\). Denote this as \(\bar{k}\). Hint: argue using (5) and the fact that the production function \(f\) is continuous, increasing and strictly concave.
From the last part, we can re-write the planning problem as
(8)¶\[\max_{(c_{0}, k_{1}, c_{1}, k_{2}, ... c_{T}, k_{T+1})} U( c_{0}, ..., c_{T} )\]subject to the constraint (5) holding at each date \(t \leq T\), and the boundary value \(k_{0}\) given and \(k_{t+1} \in [0, \bar{k}]\).
Consider \(T = 1\) (i.e., a two-period economy). Show and argue that the necessary and sufficient condition,
(9)¶\[u'(c_{0}) = \beta u'(c_{1}) \left[ 1 - \delta + f'(k_{1} ) \right],\]
along with (5) holding at each date \(t =0, 1\), and the boundary value \(k_{0}\in [0,\bar{k}]\) (given), and \(k_{t+1} \in [0, \bar{k}]\), characterize the optimal planning problem.
- Interpret in words what these conditions say.
- Can you illustrate the optimal allocation by depicting the graphs of the boundaries of indifference and budget sets in \((c_{0}, c_{1})\)-space?
- Suppose we instantiate \(u(c) = \ln(c)\) and \(f(k)=k^{\alpha}\).
- From the first-order characterization of the optimal solution, i.e., the optimal plan, argue that the optimal plan is a sequence of consumption functions \(\{ c_{t} = g_{t}(k_{t}) : t = 0, 1, ..., T\}\), where \(\{g_{t}\}_{t=0}^{T}\) are unknown functions to be determined.
- Can you solve for the optimal saving plan explicitly? What can you say about the marginal propensity to save (or consume) in this example? Explain your observation(s).
Now, let’s look at the case of \(1 < T < +\infty\). Under the same assumptions on preferences and technology, the optimal allocation described above can generalized by the following first-order conditions, for all dates \(0 \leq t \leq T-1\):
along with (5) holding at each date \(t \leq T\), \(k_{t+1} \in [0, \bar{k}]\). and given \(k_{0}\). This is called an Euler equation.
Exercise Finite Horizon (2)
Can you set up the generic \(T\)-period planner’s problem as a Lagrangean?
Show how you can obtain the first-order optimality conditions stated above. Hint: Show that at the terminal date \(T > 0\), the first-order conditions imply that
(11)¶\[\beta^{T} u'(c_{T}) k_{T+1} = 0.\](Can you interpret what the condition in the hint says?)
Together with (5) holding at each date \(t \leq T\), and the boundary value \(k_{0}\) (given), and, \(k_{t+1} \in [0, \bar{k}]\), (10) form a system of simultaneous equations. From the first-order characterization of the optimal solution, i.e., the optimal plan, argue that the optimal plan is a sequence of consumption functions \(\{ c_{t} = g_{t}(k_{t}) : t = 0, 1, ..., T\}\), where \(\{g_{t}\}_{t=0}^{\infty}\) are unknown functions to be determined.
Using the example with \(u(c) = \ln(c)\) and \(f(k)=k^{\alpha}\), can you write down a pseudocode for solving this system of equations?
Implement this using appropriate function definitions and scripting in Python. Pick \(\delta = 0.10\) and \(\alpha = 1/3\), solve and plot the optimal trajectory for consumption, capital and income (in per person units).
Note
The class of problems above are finite-dimensional convex optimization problems, which you would have encountered in static consumer theory in undergraduate economics. Mathematically, these examples are no different! In fact, each date-contingent consumption can be re-labelled as another good. In other words, we have a \(T\)-good utility maximization problem.
6.2.2. Infinite Horizon¶
So at this point, we might be tempted to take things easy going forward. Beware: In the following, when we take \(T\) to the limit of infinity, solving for the optimal plan is not so straightforward. (For most of us, this is the part where our Road Map says “Here be Monsters”; but we will tame what may be perceived as an impossible beast into an obedient and approachable puppy soon!)
Consider \(T \rightarrow +\infty\). Every possible infinite sequence of consumption outcomes \(\{ c_{t} \}_{t \in \mathbb{N} } :=: (c_{0}, c_{1}, ... ) \in \mathbb{R}_{+} \times \mathbb{R}_{+} \times \cdots\) induces an infinite sequence of payoff flows: \(\{ u(c_{t}) \}_{t \in \mathbb{N} }\).
The planner’s problem now is given by:
subject to \(k_{0}\) given, the constraint (5) and \(k_{t+1} \in [0, \bar{k}]\) for each date \(t \in \mathbb{N}\).
Note
- This is an infinite-dimensional optimization problem.
- In the finite-horizon setting, along with the Euler equations, choosing the terminal condition on the state \(k_{T+1}=0\) are both necessary and sufficient for attaining an optimum. What about in the case of the infinite horizon economy? The problem here is that we have no terminus to speak of! We’ll see that there is one such “similar” condition for the infinite-horizon setting. It turns out that this so-called transversality condition is actually part of a set of necessary (see Kamihigashi [Ka2000]) and sufficient conditions for optimum in the infinite-horizon setting:
| [Ka2000] | Kamihigashi, T., “A Simple Proof of Ekeland and Scheinkman’s Result on the Necessity of a Transversality Condition,” Economic Theory, 15(2), 463-468, 2000. |
At this point you might consider doing the following: Set up a Lagrangean function and maximize it. The mechanical calculus user would then proceed to try solving for the optimal trajectory, just like in the previous exercise. What is the difficulty with this solution strategy? To be sure, let’s do the following problems.
Exercise Infinite Horizon (1)
- Write down the Lagrangean of the problem in (12), such that given the initial value \(k_{0}\), (5) holds and \(k_{t+1} \in [0, \bar{k}]\), at each date \(t \in \mathbb{N}\).
- Derive the necessary (and sufficient) first order conditions for an optimal allocation.
- Verify that the transversality condition with the other first order conditions, are sufficient to guarantee an optimal allocation.
- Can you solve for an optimal allocation in a manner similar to what you did for the finite-horizon setting? Why not?
6.3. Solving The RCK Economy¶
We have established that solving the RCK model with an infinite decision horizon is not possible, at least not at first glance! Fortunately, under some regularity conditions, we can find a way to solve the model. We’ll first motivate this by solving the following special-cases mechanically. Pay attention to what you see along the way while attempting these exercises!
Exercise Solving The RCK Economy (1)
Consider the infinite-horizon RCK economy, instantiated by \(u(c) = \ln(c)\) and \(f(k):= k^{\alpha}\), and by setting \(\delta = 1\).
Policy Function (Euler Operator) Approach: Suppose we are told that the optimal savings policy has the following time-invariant functional form:
\[k_{t+1} = G \times k_{t}^{\alpha}.\]
- Evaluate the Euler (functional) operator using this guess where \(G\) is an undetermined coefficient. Verify that \(G = \alpha \beta\).
- Show that your solution for the policy function, indexed by \(G\), also satisfies the transversality condition, and therefore the (Pareto) optimal infinite-sequence allocation is sufficiently described by recursive solution above.
Value Function (Bellman Operator) Approach: The policy-function guess in the last problem is actually motivated by this other approach. Now consider the value—or what economists call the indirect utility—of the planner’s problem beginning at state \(k_{0}\):
\[\begin{split}V(k_{0}) & \equiv \max_{c_{t}, k_{t+1} } \sum_{t=0}^{+\infty} \beta^{t} u(c_{t}) \\ \text{s.t. for all } & t \in \mathbb{N}: \\ & k_{t+1} = k_{t}^{\alpha} - c_{t}, \\ & k_{t} \in [ 0, \bar{k}], \\ & k_{0} \text{ given}.\end{split}\]
Show (by induction) that this problem can be written recursively in terms of the value of the indirect utility function \(V: [0,\bar{k}] \rightarrow \mathbb{R}\):
\[V(k_{t}) = \max_{k_{t+1} \in [0, \bar{k}]} \bigg{\{} \ln \left(k_{t}^{\alpha} - k_{t+1} \right) + \beta V (k_{t+1}): k_{t} \text{ given} \bigg{\}}\]
This equation which you just derived:
is called a Bellman functional equation/operator or Bellman equation in casual terms. The entire expression on the RHS is an operator or “machine”, which we called \(V \mapsto T(V)\). In words, \(T\) maps the function \(V\) back into itself on the LHS—i.e., the function \(V\) is a fixed point of this operator and it is determined as part of the solution to the planning problem! However, unlike the notion of a fixed point on the real line or in Euclidean space, this is a functional fixed point—i.e., the fixed point \(V\) is a function.
There is one problem though—we don’t know what \(V\) is, yet! If that is the case, how do we evaluate the maximum problem defining the operator \(T\)? Let’s push this example further while running on faith. (Remember, you need to get your hands dirty here to fully get it!)
Exercise Solving The RCK Economy (2)
For now, suppose that this solution has the form \(V(k) = A + B \ln (k)\), where \(A\) and \(B\) are undetermined coefficients that you need to find. Go find them.
Suppose we are not that lucky and we don’t know the guess above. Notice that the recursive problem defines a fixed-point solution in terms of the unknown \(V\)? Consider iterating on successive approximations of \(V\), denoted as the sequence \(\{ V_{n} \}_{n \geq 0}\)—i.e., these are some other functions that are meant to be alternative guesses of the true function \(V\). Specifically, we can construct these guesses iteratively, for each \(n=0,1,...\), as follows:
\[V_{n+1} (k_{t}) = \max_{k_{t+1} \in [0, \bar{k}]} \bigg{\{} \ln \left(k_{t}^{\alpha} - k_{t+1} \right) + \beta V_{n} (k_{t+1}): k_{t} \text{ given} \bigg{\}}\]Note that the index \(n\) has nothing to do with natural time \(t\)! Each \(n\) is an index to an approximant of \(V\).
Start with a guess like \(V_{0}(k_{t+1}) = 0, \forall k_{t+1}\). Work out what is \(V_1\). Do this for \(n \leq 3\). Can you derive the functional forms for the functions \((V_{1}, V_{2})\)? At each \(n\)-th approximation, you should have:
\[V_{n} (k) = A_{n} + B_{n} \ln (k).\]Along the way you would have also solved for a sequence of supporting policy functions. These should look like:
\[k_{t+1} = G_{n} k_{t}^{\alpha}.\]Show that as \(n\rightarrow \infty\), \((V_{n}, G_{n}) \rightarrow (V, G)\)—i.e., these approximating value and saving-policy functions converge to our lucky guess-and-verify solutions earlier!
Let’s get back to some economics. Take the Solow-Swan model earlier, but with \(\alpha = 1/3\), \(\delta = 1\), and \(n=g=0\), so that we can compare it with the RCK special case that we just solved for.
- Notice some similarity and some difference between the two? Comment on the Golden-rule solution in Solow-Swan previously and contrast that with our benevolent planner’s Pareto-optimal allocation above. In particular compare and discuss their respective steady-state growth paths.
- What does that imply for the economies’ rate of convergence to their respective steady-state points?
We will recall these exercises later in the chapter Dynamic Programming. For now, it pays to have worked on them at least once.
6.4. Competitive Equilibrium¶
In a manner similar to Solow-Swan we can re-write this model in terms of a decentralized economy with households, firms and markets (capital, labor and final good). The solution concept here is that of competitive or general equilibrium. In words, a competitive equilibrium in this economy is one where the consumer and firm optimize, and the resulting supply and demand in the labor, capital and final goods markets clear at every date.
Observe that in this neoclassical economy, preference sets and production sets are strictly convex. Moreover, they are represented, respectively, by continuous utility and production functions, respectively defined over compact feasible choice and production sets. In other words, there will exist a unique competitive equilibrium allocation that is also Pareto efficient (i.e., equivalent to the benevolent planner’s solution). That is, the Fundamental Welfare Theorems of general equilibrium will be at work here.
We will present two alternatives to describing competitive equilibrium here: a date-zero trading structure (i.e. Arrow-Debreu Competitive Equilibrium or ADE) and a sequential-markets trading structure (i.e. Sequential Competitive Equilibrium or SCE). In the latter setup, we will consider two alternative ownership or financial structures and label these SCE-I and SCE-II accordingly. We also contrast the competitive equilibrium allocation with that of a social planner, which is what we have previously studied. (This will be done as an exercise below.)
The main message is as follows:
- The date-zero or its alternative sequential markets trading assumption are equivalent in terms of equilibrium allocation of resources.
- The decentralization and ownership structure in these alternative economies are inconsequential. They yield the same characterization of competitive equilibrium allocation of resources.
- All these notions of competitive equilibrium yield identical allocations that are Pareto efficient.
This implies that we can solve for competitive equilibrium in such models by tackling the hypothetical social planner’s problem directly. Then, we can back out the implied competitive equilibrium relative prices as shadow prices in the planner’s economy.
The two notions of competitive equilibrium—ADE or SCE—are not that practical however: They are stated in terms of conditions for an equilibrium allocation which is an infinite sequence of date-contingent goods. As we saw earlier, such infinite sequence problems are not directly soluble. In the last part, we look at a recursive version of the notion of competitive equilibrium due to Prescott and Mehra [PrMe1980]. It can be shown that this notion of Recursive Competitive Equilibrium, induces an infinite sequence as equilibrium allocation which will coincide with the allocation under ADE or SCE, and is thus also Pareto efficient. [1]
Footnotes
| [1] | At least in this class of competitive equilibria where the Fundamental Welfare Theorems hold, this will be true. In models in which there are market distortions—e.g. incomplete markets models—the sequential equilibrium may no longer coincide with the recursive equilibrium. They are regularity conditions and alternative means (e.g., defining auxiliary state variables) of recursifying the sequential equilibria. See Duffie, Geanakopolos, Mas-Colell and McLennan [DGMM1994] |
6.4.1. Model Primitives¶
Let’s work with a slightly more general setting where labor or leisure choice is made endogenous. This will be useful later when we consider more empirically relevant versions of such general equilibrium models.
The preference representation of the household over sequences of consumption and labor, \(\{c_{t}, n_{t}\}_{t \geq 0}\), is:
where \(u_{c}>0\), \(u_{cc} < 0\), \(\lim_{c \searrow 0} u(c, \cdot) = \infty\), \(u_{n} < 0\) and \(u_{nn} < 0\).
The production function is as described before in (1).
6.4.2. Two stories of competitive equilibrium¶
We will consider two presentations of competitive equilibrium:
- Arrow-Debreu’s once-and-for-all date-\(0\) market for contingent claims trading.
- Sequential competitive equilibrium with markets in every date.
Note
Looking ahead, a good understanding of the setup here will be beneficial for understanding the competitive equilibrium and asset pricing structure of an environment with heterogeneous agents (facing idiosyncratic risks) and/or with aggregate risks. We will deal with these later.
6.4.2.1. Date-\(0\) centralized markets¶
In this formulation due to Kenneth Arrow and Gerard Debreu, there is a centralized market that opens at the beginning of date \(t=0\). Here, agents contract on how much to transfer to each other in terms of date-contingent allocations of consumption, investment (or capital), and labor: \(\{c_{t}, i_{t}, n_{t} \}_{t \geq 0}\), once and for all. In this story, all contingent plans are determined at date \(t=0\), along with the sequence of market-determined relative prices. [2]
Footnotes
| [2] | Think about how stringent these contractual requirements are. Departure from such a strong contractual environment is summarized in the modern literature on dynamic contract theory, or better known as recursive contracts. In such environments, limited commitment to, and/or, private information underlying exchanges distort what would otherwise be complete markets equilibria. You will learn more of this in more advanced courses on economic dynamics. |
Implicit in the contracts, agents commit to executing date-contingent transfers as specified by the date-\(0\) agreement. Let \(q_{t}^{0}\) be the relative price of a date-\(t\) claim to (consumption or investment) goods in units of date-\(0\) final good. Denote \(w_{t}\) and \(r_{t}\), respectively, as the rental rates applicable to labor and capital to be delivered in date \(t\geq 0\). Then, in units of the initial final good, the sequence of (real) exchange rates are determined at date-\(0\) as \(\{q_{t}^{0}, q_{t}^{0}w_{t}, q_{t}^{0}r_{t}\}_{t\geq 0}\) in support of the forward contract \(\{c_{t}, i_{t}, n_{t} \}_{t \geq 0}\). These prices and contracted allocations are pinned down at date \(0\). After the date-\(0\) centralized market closes, agents execute the transfers as time unfolds—i.e., households create new capital, deliver work and capital to firms, firms use these inputs to produce and deliver a final good and profit back to households at each date.
From the point of view of the household at date the household’s budget constraint is:
From its perspective in the date-0 once-and-for-all centralized market, the household’s relevant budget constraint is its lifetime constraint:
where the last term on the RHS is lifetime profit in terms of the initial final good.
The household capital production technology constraint is
Exercise Competitive Equilibrium (1)
The household maximizes (14) subject to (16) and (17).
Show that the household optimal plan, given centralized market prices \(\{q_{t}^{0}, q_{t}^{0}w_{t}, q_{t}^{0}r_{t}\}_{t\geq 0}\), is described by the first-order conditions:
\[\begin{split}\beta^{t} u_{c} (c_{t}, n_{t} ) &= \mu q_{t}^{0} \\ -\beta^{t} u_{n} (c_{t}, n_{t} ) &= \mu q_{t}^{0}w_{t} \\ q_{t}^{0} &= q_{t+1}^{0} (r_{t+1} + 1-\delta)\end{split}\]for every date \(t \geq 0\). (Note that \(\mu\) is the Lagrange multiplier on the lifetime budget constraint (16).)
Show that the first two conditions, together with the property of \(u\), imply a labor supply function that is increasing in the rate of return of labor services. Also, is labor supply increasing in consumption? Interpret what this means.
Interpret what the last condition says.
Let \(K_{t}\) and \(N_{t}\) denote aggregate capital and labor, respectively, demanded by the representative price-taking firm. The firm’s problem is
Exercise Competitive Equilibrium (2)
Show that the firm’s profit maxizing strategy is to hire capital and labor at each date according to the (implicit) demand functions
\[\begin{split}F_{K} (K_{t}, N_{t}) &= r_{t} \\ F_{N} (K_{t}, N_{t}) &= w_{t}\end{split}\]Argue that \(\Pi_{0} = 0\).
Definition (ADE)
An Arrow-Debreu competitive equilibrium (determined at date \(0\)) in this economy is a date-contingent allocation \(\{C_{t}, N_{t}, K_{t+1}\}_{t \geq 0}\) and prices \(\{q_{t}^{0}, q_{t}^{0}w_{t}, q_{t}^{0}r_{t}\}_{t\geq 0}\), such that:
- The plan \(\{c_{t}, n_{t}, k_{t+1}\}_{t \geq 0}\) is optimal for the household, given prices;
- the plan \(\{N_{t}, K_{t}\}_{t \geq 0}\) is optimal for the firm, given prices; and
- there is zero excess demand for all date-contingent transfers: \(c_{t} = C_{t}, k_{t} = K_{t}, i_{t} = I_{t}\) and \(n_{t} = N_{t}\).
Observe that in the Arrow-Debreu equilibrium, we have at each date,
and
6.4.2.2. Sequential competitive equilibrium I¶
Consider another story. Markets are open every period. In this case, the agents’ problems are sequential. In each date, they buy and sell consumption and investment goods, and they also lease and hire capital and labor.
The household problem is now to maximize (14) subject to (15) and (17). The firm’s problem is static:
Definition (SCE-I)
A sequential competitive equilibrium is an allocation \(\{ C_{t}, N_{t}, K_{t+1} \}_{t \geq 0}\) and prices \(\{ w_{t}, r_{t} \}_{t \geq 0}\), such that:
Given prices, the plan \(\{c_{t}, n_{t}, k_{t+1} \}_{t \geq 0}\) is optimal for the household—i.e., it satisfies
\[\begin{split}u_{c}(c_{t}, n_{t}) &= \beta u_{c} (c_{t+1}, n_{t+1}) (r_{t+1} + 1 - \delta), \\ -\frac{ u_{n}(c_{t}, n_{t}) }{ u_{c}(c_{t}, n_{t}) } &= w_{t}, \\ \lim_{t \rightarrow \infty} \beta^{t} u_{c}(c_{t}, n_{t}) k_{t+1} & = 0;\end{split}\]given prices, the firm is demanding capital and labor at profit maximizing levels:
\[\begin{split}F_{K} (K_{t}, N_{t}) &= r_{t}, \\ F_{N} (K_{t}, N_{t}) &= w_{t};\end{split}\]Markets clear at each date: \(c_{t} = C_{t}, k_{t} = K_{t}, i_{t} = I_{t}\) and \(n_{t} = N_{t}\).
Exercise Competitive Equilibrium (3)
- The statement ”... \(c_{t} = C_{t}, k_{t} = K_{t}, i_{t} = I_{t}\) and \(n_{t} = N_{t}\) ...” in both definitions of competitive equilibrium above is somewhat high-level. To solve the model, you need to work it a little bit more. Can you explain what the equalities actually imply, in terms of what happens in the respective markets in this example economy? Hint: Think in simple demand and supply language and then write out these conditions verbosely!
- Prove rigorously that the allocation implied in the Arrow-Debreu Equilibrium (ADE) is identical to that in the Sequential Competitive Equilibrium (SCE) of this model economy. Hint: Use the household’s transversality condition.
6.4.2.3. Sequential competitive equilibrium II¶
Consider an alternative ownership structure. Suppose, as in reality, that the average household does not directly own capital stock. The household owns shares \(\psi_{t} \in [0,1]\) in firms. The household problem now is to choose the sequence \(\{ c_{t}, n_{t}, \psi_{t+1} \}_{t \geq 0}\) to maximize (14) subject to it sequence of budget constraints:
for all \(t \geq 0\) given \(V_{0}\).
The household takes as given:
- Prices \(w_{t}\),
- the value of the firm, \(V_{t}\), and
- the ex-dividend value of the firm \(V_{t} - D_{t}\). (This just says the value of the firm after dividend payouts.)
The transversality condition for the household problem now says:
There is a measure-one continuum of perfectly competitive firms with the same technology. The representative firm chooses a sequence of labor and investment flows \(\{ N_{t}, I_{t} \}_{t \geq 0}\) to maximize its total discounted stream of dividends:
The firm takes as given:
- Prices \(w_{t}\), and
- the market discount factor or pricing kernel \(M_{t,t+1}\). (This is a measure of intertemporal relative prices which will be determined in equilibrium!)
Definition (SCE-II)
A sequential competitive equilibrium in this economy is an allocation \(\{ C_{t}, \psi_{t+1}, N_{t}, K_{t+1} \}_{t\geq 0}\) and relative prices \(\{ V_{t}, w_{t}, M_{t, t+j} \}_{t \geq 0}\) such that:
- Given prices, households optimize;
- given prices, firms optimize; and
- markets clear: \(\psi_{t} = 1\), \(C_{t} = c_{t}\), \(N_{t} = n_{t}\).
Let’s look closer into the first condition of this version of sequential competitive equilibrium: Households optimize. The first-order condition with respect shares, \(\psi_{t+1}\) is
for all dates \(t \geq 0\).
What do these sequential first-order conditions say? The LHS gives us the household’s marginal utility valuation of the firm’s ex-dividend value. The RHS is the present discounted marginal utility valuation of the firm’s cum-dividend value. In an equilibrium, the household/equity-investor necessarily trades off its forgone consumption today for more consumption tomorrow at a marginal rate of substitution at precisely the intertemporal rate of return of the firm, measured by the rate \((V_{t}-D_{t})/V_{t+1}\). Let’s see what this optimal trade-off condition for household shares investment implies. Rearranging we get
This just says that, at an optimal investment plan for the household, the value of the firm from the household/investor’s point of view is just the dividend flow in one period that can be claimed by holding the shares today, plus the present discounted continuation value of the firm, where the discount factor is given by the household’s marginal rate of substitution of consumption across the two dates.
After iterated forward substitution of (19) and invoking the sufficient transversality condition (18) for the household, we get
In words, this says that the value of the firm (to the household) equals its total discounted flow of dividend payments. Again, note that the discounting is in terms of the household’s preference-based marginal rate of substitution of consumption across dates.
Exercise Sequential competitive equilibrium II (1)
- Show how you derived the expression (20).
We are not done yet. We need to consider the firm’s optimality condition, since this is also part of the equilibrium requirement. The firm’s first-order conditions for all \(t\geq 0\) are
Note that mathematically, \(Q_{t}\) is the Lagrange multiplier on the firm’s capital accumulation constraint. Economically, this has an interpretation of the shadow price of capital—i.e., the marginal value of the firm given an unit increase in the firm’s capital stock. This is also commonly known as Tobin’s Q.
Exercise Sequential competitive equilibrium II (2)
Can you set up the firm’s infinite sequence or Lagrangian problem and derive these first-order conditions?
[Ha1982] What if the capital accumulation constraint is instead the following:
\[K_{t+1} = (1-\delta) K_{t} + \phi (I_{t}, K_{t})\]where \(\phi_{1} < 0\), \(\phi_{11} < 0\) and \(\Phi(0, \cdot) = 0\)? Can you interpret the purpose of the function \(\phi\) in words? Given the assumptions, what does this mean? Derive the first order conditions of the firm under this alternative model.
Assume that \(\Phi(I,K) := \frac{\phi}{2} (I/K)^{2}\). What do the first-order conditions look like?
Finally we can show that the allocation in this version of the SCE is identical to that in the ADE and the previous treatment of the SCE.
Exercise Sequential competitive equilibrium II (3)
- Using all the conditions defining SCE in this economy, show that all three notions of competitive equilibrium—ADE, SCE I and SCE II—have identical characterizations of the competitive equilibrium allocation.
| [Ha1982] | Hayashi, Fumio, “Tobin’s Marginal q and Average q: A Neoclassical Interpretation,” Econometrica, 50(1), 213-224, 1982. |
6.5. Recursive Competitive Equilibrium¶
Observe that in all dynamic characterizations of competitive equilibria above—ADE, SCE-I and SCE-II—equilibrium choices made by firms and households are consistent with each other through market clearing conditions. The margins at which these allocations are determined are functions of aggregate relative prices determined simultaneously through market clearing. Moreover, these relative prices depend only on the aggregate state variable \(K_{t}\):
If so, off-equilibrium we may have individual value functions and decision functions being dependant on individual state \(k_{t}\), but because agents behave competitively (i.e. take prices as given) we would have that in equilibrium their decisions are only functions of the aggregate state.
Prescott and Mehra [PrMe1980] took their cue from this insight—which also generalizes to many other models—and devised a manageable or recursive way to describe what would otherwise be the ADE or SCE. Recall that the ADE and SCE are described in terms of infinite sequences, which do not lend these characterizations to practical solution. They called their recursive formulation of competitive equilibria Recursive Competitive Equilibria (or RCE).
If we let \(X := X_{t}\) and \(X' := X_{t+1}\), then the household problem can be written down as
(21)¶\[\begin{split}V(k,K) &= \max_{c, n, k'} u(c,n) + \beta V(k', K') \\ & \text{s.t.} \\ k' & = r(K) k + w(K) n + (1-\delta)k - c \\ K' & = G(K)\end{split}\]
where the mapping \(G\) is taken as given by the household, and has the interpretation of its perceived law of motion for the (equilibrium) aggregate state, \(K\).
Since in this model, agents are homogeneous, we know in equilibrium \(k = K\). The following defines the notion of a RCE in this economy:
Definition (RCE)
A recursive competitive equilibrium in this economy is:
a value function \((k,K) \mapsto V(k,K)\),
decision functions \((k,K) \mapsto G^{x}(k,K)\), where \(x \in \{ c, n, k \}\) and:
\[\begin{split}c &= G^{c} (k,K), \\ n &= G^{n}(k,K), \\ k' &= G^{k}(k,K),\end{split}\]a perceived law of motion \(K \mapsto G(K)\) where \(K' = G(K)\), and
price functions \(r(K)\) and \(w(K)\),
such that:
given prices \(w(K)\), \(r(K)\), and the perceived law of motion \(K \mapsto G(K)\), \((k,K) \mapsto V(k,K)\) satisfies the household Bellman equation (21) and \((k,K) \mapsto G^{x}(k,K)\), where \(x \in \{ c, n, k \}\) are the maximizers in the Bellman equation problem;
the price functions satisfy the firm’s profit maximization conditions:
\[\begin{split}r(K_{t}) &= F_{K} (K_{t}, N_{t}); \\ w(K_{t}) &= F_{N} (K_{t}, N_{t});\end{split}\]Markets clear:
- \(k = K\) at all dates
- \(G^{k}(K,K) = G(K)\)
- \(G^{c}(K,K) = C\)
- \(C^{n}(K,K) = N\)
- \(K' - (1-\delta) K + C = F(K, N)\).
Observe that in a RCE, the perceived law of motion coincides with the equilibrium or actual law of motion: \(K' = G^{k}(K,K) = G(K)\). This is our first encounter with a concept that is generalized later as a rational expectations equilibrium when we consider a similar environment with individual and/or aggregate risk.
Exercise Recursive Competitive Equilibrium (1)
Follow up from the special-case RCK model where \(\delta = 1\), \(u(c,n) = \ln(c)\) and \(F(K,N) = K^{\alpha}N^{1-\alpha}\).
- Define a recursive competitive equilibrium for this special-case RCK economy.
- Suggest how you would solve for this notion of competitive equilibrium.
- Compute the competitive equilibrium allocation and prices using Python for the parametrized example earlier.
Exercise Recursive Competitive Equilibrium (2)
Show that the RCE described in this section also characterizes identical competitive equilibrium allocation as the ADE or the SCE.
Let’s return to the hypothetical benevolent social planner for this economy.
- Write down the planner’s problem.
- Show that the planner’s allocation is a competitive equilibrium allocation.
| [Ca1965] | Cass, D., “Optimum Growrh in An aggregate Model of Capital Accumulation,” Review of Economic Studies, 32(3), 233-240, 1965. |
| [Ko1965] | Koopmans, T., “On the Concept of Optimal Economic Growth,” Academiae Scientiarum Scripta Varia, 28(1), 225-287, 1965. |
| [Ti1939] | Tinbergen, J., Statistical Testing of Business Cycle Theories, Geneva: League of Nations, 1939. |
| [PrMe1980] | (1, 2) Prescott, Edward C., and Rajnish Mehra, “Recusive Competitive Equilibirum: The Case of Homogeneous Households,” Econometrica, 48(6), 1365-1379, 1980. |
| [DGMM1994] | Duffie, Darrell, John Geanakopolos, Andreu Mas-Colell and Andrew McLennan, “Stationary Markov Equilibria,” Econometrica, 62(4), 745-781, 1994. |