Two Period Models
Econ602. Spring 2005. Lutz Hendricks
The main points of this section are:
•
Tools: setting up and solving a general equilibrium model; Kuhn-Tucker conditions; solving
multiperiod problems
•
Economic insights: Ricardian equivalence. Equivalence of competitive equilibrium and planning
problem
Math background: Lagrangean and Kuhn-Tucker (see lecture notes on mathematical methods).
The strategy of this section is to start with exceedingly simple models. We will then add more and
more complications as we go along. This will make the early models look somewhat silly, but it allows
us to build up the complexity step by step instead of plunging right into a full-blown model.
1. Static Models
This section begins our analysis of general equilibrium models. To keep the environment as simple as
possible, we start with an economy that lasts for only one period. Such models are obviously not very
useful, but we will see that the methods used to characterize their equilibria carry over to much more
complicated environments.
We begin by describing the model elements:
•
Agents: There are N identical households. For now, there are no other agents (firms, government,
…).
•
Preferences: Households value consumption of two goods according to a utility function u(c1, c2).
Of course, u is strictly increasing in both arguments and (to be able to use standard optimization
techniques) strictly quasi-concave (words in this font are defined in the math notes). Marginal
utility goes to infinity as consumption goes to zero: lim ci →0 ui (c1 , c2 ) = ∞ . The subscripts on
functions (u1, u2) denote partial derivatives. This ensures that the household consumes positive
amounts of both goods.
•
Technology: The technology is trivial: each agent receives endowments of the two goods (e1, e2).
There is no production. Endowments cannot be stored.
•
Markets: Agents trade goods in a market, where everyone behaves as a price-taker. There are no
financial assets. The prices of the two goods are p1 and p2.
That’s it: nothing else is needed to describe the economy. In a sense, this is already too much. Purists
would not want to prescribe that agents behave as price-takers. Instead, they would derive it from the
fact that N is large.
Recall the steps we have to go through in order to characterize equilibrium:
1
1. Solve the problem for each agent, taking prices as given. Find the decision rules. Here the only
agents are households.
2. Impose market clearing to determine equilibrium prices and quantities.
1.1 Household problem
Consider the problem facing one of the N households. It takes as given market prices for the two
goods, p1 and p2, and the endowments it receives, e1 and e2. The choice variables are c1 and c2. The
only constraint is a budget constraint p1 e1 + p2 e2 = p1 c1 + p2 c2 . We can normalize the price of one
good to one (numeraire), so we set p1 = 1 and simply call the relative price of good 2 p = p2 / p1. The
household then solves
max u (c1 , c2 ) s.t. e1 + p e2 = c1 + p c2
What exactly is a solution to the household problem? One way of stating a solution is as a vector of
quantities (c1, c2) for given prices and endowments. An alternative that is often more useful is to derive
optimal choices as decision rules. The household’s decision rules will be of the form “choice variables
= f(state variables).” The state variables are all relevant variables the household takes as given: p, e1,
e2.1 The choice variables in this case are c1 and c2, so we need to find consumption functions. To
derive the decision rules set up a Lagrangean:
Γ = u (c1 , c2 ) + λ (e1 + p e2 − c1 − p c2 )
For this particular problem it would actually be easier to substitute the constraint into the objective
function and solve the unconstrained problem max u (e1 + p e2 − p c2 , c2 ) , but the Lagrangean is
instructive. The first order conditions are
(1)
∂ Γ / ∂ c1 = u1 (c1 , c2 ) − λ = 0
(2)
∂ Γ / ∂ c2 = u2 (c1 , c2 ) − λ p = 0
The multiplier λ has a useful interpretation. It is the marginal utility of c1, but more importantly it is
the marginal utility of relaxing the constraint a bit, i.e. the marginal utility of wealth. The solution to
the household problem is then a vector (c1, c2, λ) that solves the FOCs together with the budget
constraint. This was, of course, a bit loose: the solution is really a triple of functions. In particular, we
can write the decision rules as ci ( p, e1 , e2 ) .
1
This is actually a bit more complicated than it sounds. For example, why not add another “irrelevant” variable
to the state vector, such as the position of Jupiter relative to Neptune? If this sounds like a silly idea, take a look
at the literature on sunspots. The economics literature, of course, not the astronomy literature.
2
Tip: Always explicitly state what variables constitute a solution and which equations do they have to
satisfy. You should have a FOC for each choice variable and all the constraints. Make sure you have
the same number of variables and equations. Later on, this will make it easier to assemble the
equations needed for the competitive equilibrium.
At this point, it is typically useful to substitute out the Lagrange multiplier. Take the ratio of (1) and
(2) to obtain
u2 / u1 = p .
(3)
This is the algebraic expression of the familiar tangency condition: marginal rate of substitution equals
relative price. You have seen the graph with indifference curves tangent to budget constraints many
times before. Now the solution is a pair (c1, c2) that satisfies (3) and the budget constraint.
If we assume log utility, u = log(c1 ) + β log(c2 ) , this can be solved in closed form: 1 / c1 = λ ,
β / c2 = p λ . Therefore, (3) becomes β c1 = p c2 . Substitute both into the budget constraint to solve for
λ:
e1 + p e2 = 1 / λ + β / λ ⇒ λ = (1 + β) / W ,
where W = e1 + p e2 is total wealth. Therefore,
c1 ( p, e1, e2 ) = W /(1 + β) and p c2 ( p, e1, e2 ) = W β /(1 + β) .
Tip: This is a peculiar (and often very useful) feature of log utility: the expenditure shares are
independent of p. The reason is exactly the same as that of constant expenditure shares resulting from a
Cobb-Douglas production function: unit elasticity of substitution.
Tip: Recall that taking a monotone transformation of u doesn’t change the optimal policy functions. In
β
particular, we can replace u by eu = e ln(c1 )+ln(c2 ) = c1 c2β . Convince yourself that this yields exactly the
same consumption functions.
1.2 Market Clearing
There are two markets (for goods 1 and 2). Each agent supplies the endowments ei and demands
consumption ci in those markets. Why isn’t there just one market where agents exchange good 1 for
good 2? It is better to think in terms of 2 markets in which goods are traded for units of account. I
don’t like to use the word money here because there is no such thing in this economy.
The market clearing condition is “aggregate supply = aggregate demand.” Aggregate supply is simply
the sum of individual supplies:
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Si = ∑ h=1 ei = N ei
N
where the second equality follows from the fact that all agents are identical. Similarly, aggregate
demand is found by summing consumption demands over households. To be pedantic, and inconsistent
with what we did above, let’s write consumption of household h as cih . Then
Di = ∑ h =1 cih = N ci
N
Market clearing therefore requires N ei = N ci or ei = ci. This is not surprising: all agents are identical
and therefore do not trade. More interesting is to find the market clearing price. The key is that each
agent could trade any quantity at that price, but chooses not to. The market clearing price satisfies
ci ( p, e1 , e2 ) = ei .
1.3 Definition of Equilibrium
A competitive equilibrium is an allocation (cih ; h = 1, K, N ; i = 1,2) and a price p that satisfy:
1. The cih satisfy the household optimality conditions (FOC and budget constraint).
2. The two goods markets clear (ei = ci).
Now we count equations and variables. We have 2 N consumption levels and one price. These satisfy 2
N household optimality conditions and 2 market clearing conditions. However, Walras’ law tell us that
one market clearing condition is redundant.
This was more pedantic that we would usually want to be. Given that all households face identical
problems, we would usually impose from the outset that cih = ci for all h.
Note that we could add the household’s Lagrange multiplier to the list of variables. Then we would
also have to add another equation. We would do so by defining the household optimality conditions in
“1.” as 2 FOCs plus one budget constraint. This makes no difference. We can do whatever is more
convenient.
In the log example, the price is determined by p e2 = W β /(1 + β) .
1.4 Insights
The method used to solve this model carries over to more complicated ones.
1. First, derive conditions that characterize the solution to each agent’s problem, taking prices as
given. This typically involves a number of FOCs and constraints.
2. State the market clearing conditions.
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3. Make sure the number of unknowns equals the number of independent equations, keeping in mind
that Walras’ law renders one market clearing condition redundant.
4. Solve. The rest is either just algebra or simply intractable.
It is typically useful to write out the definition of equilibrium fairly carefully: “a list of variables (…)
that satisfy …” Make sure the number of variables is the same as the number of equations. It is also
useful to be careful about the state variables: what are the givens that we need to know in order to
solve an agent’s problem. These typically include prices, endowments, asset holdings, etc.
Is it silly to have a model in which nobody trades because all households are identical? It depends on
the application. The main reason for studying these models is that they are tractable (all households
can be identical; we can study a representative household). Whether anything is lost by making that
assumption depends on the problem one is interested in.
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2. An Intertemporal Model
Nothing prevents us from reinterpreting the previous model as a two-period model. Assume that there
is only one physical commodity, but there are two dates (1 and 2). The utility function is the same as
before [write it as u (c1 ) + β u (c2 ) , but it is not essential that it be separable]. The good is not storable.
How then can agents trade? They obviously need to trade intertemporally. There are two possible
arrangements.
First, there may be markets at date 1 at which agents can buy and sell goods at all future dates (in this
case only at date 2, but there could of course be more dates). This is called the Arrow-Debreu setup.
In this example, it means there is a market in which I can sell goods today in order to receive units of
account, which can then be used to buy goods for delivery tomorrow. Here, the price p has the
interpretation “giving up p goods today buys one unit tomorrow.” Note that the equilibrium description
is exactly the same as in the one period model. Whether the goods refer to different physical
commodities or to the same commodity at different dates makes no difference. This result holds
generally.
It may appear that this approach is in trouble when there is uncertainty because it requires the agents to
decide how much they wish to consume at all future dates. But the approach is easily extended to cover
the case of uncertainty by defining a commodity to be indexed by date and state of the world (e.g. “an
umbrella tomorrow, if it rains”). The micro course will handle these issues in full glory.
Alternatively, there could be a sequence of markets. At each date, agents can buy and sell one period
bonds. Giving up one unit of consumption today buys a bond that promises (1+r) units of consumption
tomorrow. Note the close relationship between the Arrow-Debreu price p and the interest rate r. If we
define p = 1/(1+r) the agents’ budget constraints and the description of the equilibrium is the same in
both arrangements. This is also a general result: the two setups can used interchangeably and yield the
same allocation, if markets are complete, which essentially means that for each possible state of the
world at each date, there exists an asset that pays precisely in that state/date.2
Adopting the sequence of markets approach, we can write the household problem as
max u (c1 ) + β u (c2 ) s.t. b = e1 − c1 ; c2 = b (1 + r ) + e2
In the first period, the household “saves” e1 – c1 units of account, for which he buys b bonds, which
cost 1 unit of account a piece. In the second period, the household receives the principal and interest on
the bonds purchased and uses it together with the endowment to buy c2.
2
We will not go into the details of what complete markets mean. Suffice it to note that in the models considered
here markets are almost always complete.
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Here, I have taken the liberty of normalizing all prices to one! Why can I do that? I can normalize p1
and p2 because I can choose the units of account in both periods. In other words, the price p1 in this
economy is meaningless. It says: you need to give up p1 date 1 units of account to buy one unit of c1.
Similarly for p2. Note that I can use different units of account at different dates. This would not be the
case if there was a way to carry units of account from one period to the next (as in the case where the
unit of account is a commodity like money). In this economy bonds allow me to transfer units of
account from period to period, but the bonds have a real rate of return which is endogenous. Their
nominal return simply adjusts to get the same equilibrium real return no matter how I choose p2 or p1.
And I can set the price of a bond to 1 by choosing units for bonds. If you don’t believe any of this,
simply set up the model with prices at every date that may differ from 1. You will find that all prices
drop out and the equilibrium is the same no matter how you choose them.
The two period budget constraints can then be combined into a present value budget constraint:
e1 + e2 /(1 + r ) = c1 + c2 /(1 + r ) . The first-order conditions are
u ' (c1 ) = λ , β u ' (c2 ) = λ /(1 + r )
u ' (c1 ) = β (1 + r ) u ' (c2 )
Combining them yields
which is known as an Euler equation. It describes the intertemporal tradeoff faced by the household:
giving up one unit of consumption today costs u ' (c1 ) . Next period, the household gains (1+r) units of
consumption, but these are discounted at rate β. The Euler equation states that a small reallocation
between consumption today and tomorrow along the budget line must leave utility unchanged. That is,
contemplate giving up dc1 = ε at date 1. The utility cost is u ′(c1 ) ε . Tomorrow, this allows to consume
an additional dc2 = ε (1 + r ) leading to a utility gain of β u ′(c2 ) ε (1 + r ) . Setting both equal yields the
Euler equation. The same condition would hold with more than two periods.
Good 2
e1 (1+r)
c2
c1
e1
Good 1
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One implication of this model is the Permanent Income Hypothesis. A household’s optimal
consumption path only depends on total wealth W, not on the individual endowments separately (his
savings do!). That is, the timing of income over the life-cycle should not affect consumption in any
way. This prediction fails empirically (Carroll and Summers 1991). Another implication is Ricardian
Equivalence: any policy that only changes the timing of lump-sum tax payments over the life-cycle
(but leaves the present value unchanged) should have no effect on consumption. We will talk about
this in detail later on.
3. An Example With Trade
So far there has never been trade in equilibrium because all agents were identical. Now we give up this
assumption and assume instead that there are N agents who receive endowment e1 when young, but
nothing when old, and N agents who receive e2 when old but nothing when young. Just to be pedantic,
we will go through all the steps again. We first need to solve the problems for all agents. Now we have
two types of agents: households who receive early endowments and those who receive late
endowments.
3.1 Households
A household with early endowment solves
max u (c1I ) + β u (c2I ) s.t. e1 = c1I + p c2I .
A household with late endowments solves
max u (c1II ) + β u (c2II ) s.t. p e2 = c1II + p c2II .
We could now write out separate first-order conditions for each household type, but it is easier to write
a generic problem for household type s as
max u (c1s ) + β u (c2s ) s.t. W s = c1s + p c2s .
where wealth levels are
W I = e1 and W II = p e2 .
Assuming log utility, we know that the decision rules are
(4)
c1s = W s /(1 + β) and p c2s = W s β /(1 + β) ,
A solution to the household problem of type s is then a pair (c1s , c2s ) that satisfies (4).
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3.2 Market clearing
Aggregate demand for good i is now
Di ( p,...) = ∑ s = I ∑ h=1 cis ( p, e1s , e2s )
II
N
= N ciI ( p, e1I , e2I ) + N ciII ( p, e1II , e2II )
Similarly, aggregate supply is Si ( p,...) = N eiI + N eiII . Market clearing, in the special case considered
here, then reduces to
N ei = N ciI + N ciII .
(5)
3.3 Competitive Equilibrium
A CE is an allocation (cis ; i = 1,2; s = I , II ) and a price p that satisfy:
•
2 optimality conditions for each household type (4 equations)
•
2 market clearing conditions
We have 5 variables and 6 equations, one of which is redundant by Walras’ law.
In the log utility case:
e1 = e1 /(1 + β) + p e2 /(1 + β) = W /(1 + β)
e2 = W β /(1 + β) ⋅ (1 / p ) ,
where W = e1 + p e2 . This is not surprising: If every household spends the same fraction of its
endowment on good 1 (c1h = W h /(1 + β)) , then aggregate spending on good 1 is that same fraction of
the aggregate endowment, N W.
Taking ratios yields the market clearing price: p = β e1 / e2 . This makes sense: The price for good 2 is
higher if there is more demand for it (β ↑) or less supply of it. Equilibrium consumption levels are then
c1I = e1 /(1 + β) and c2I = e1 β /(1 + β) ⋅ (e2 / β e1 ) = e2 /(1 + β)
c1II = p e2 /(1 + β) = e1 β /(1 + β) and c2II = e2 β /(1 + β) .
This fortunately adds up to the endowments as it should. Note the extremely odd outcome: household I
receives fraction 1/(1+β) of both goods, regardless of his relative endowment (the beauty of log
utility…).
At this point it is useful to review how this analysis fits into the general setup presented earlier.
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1. The description of the economy is our starting point: 2 N households with log utility and a particular
endowment pattern.
2. We then solved the problems of all agents, which in this case means: the problems of two types of
households. Since we had already done that more generally before, we simply wrote down the policy
functions (4). Both budget constraints are redundant in this case, not because of Walras’ law, but
because they are “built into” the decision rules.
3. We next stated the market clearing conditions (5).
4. We then defined CE and characterized it.
A technical detail: We usually talk about N as the “number” of households. Strictly speaking, to make
this model work, we need infinitely many households, so that each one is small and acts as a price
taker. For practical purposes, we may simply assume a large, finite N. A common alternative is to
assume that there is a continuum of households of measure N. This is convenient because it allows us
to normalize this measure to N = 1. We then have a single representative household of each type. This
is what we will assume in the future.
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4. Adding Production
The next step is to add production. The economy again lasts for two periods and is populated by N = 1
identical households. The only financial market is the bond market with interest rate r as described
earlier. In a bit more detail, the primitives of the economy are:
Preferences:
u (c1 ) + β u (c2 )
Endowment: e1 received at date 1
Technology: Storing k at date 1 yields f(k) at date 2. It is common to impose Inada conditions on f.
This means f ' (0) = ∞ , f ' (∞) = 0 , f ' > 0, f ′′ < 0 . As we will see quite often, Inada conditions rule out
corner solutions (k = 0).
Markets: Households consume and produce (store) using technology f. In addition to the goods
markets at both dates, there is a bond market at date 1, where households can buy or sell one period
bonds with interest rate r. The interest is, of course, to be determined in equilibrium.
4.1 The household problem
The household maximizes u (c1 ) + β u (c2 ) subject to the budget constraints
e1 = c1 + k + b and c2 = b (1 + r ) + f (k ) ,
taking the endowment e1 and the interest rate r as given. The present value budget constraint is:
[c2 − f (k )] /(1 + r ) = e1 − c1 − k
Lagrangean:
FOC:
Γ = u (c1 ) + β u (c2 ) + λ (e1 − c1 − k − [c2 − f (k )] /(1 + r ))
u ' (c1 ) = λ
β u ' (c2 ) = λ /(1 + r )
f ' (k ) = 1 + r
The solution to the household problem now consists of (functions) (c1, c2, λ, k, b) that satisfy the 3
FOCs and the 2 budget constraints. The only added item is the FOC for k which simply requires that
the two assets the household has access to should have the same rate of return.
4.2 Equilibrium
A competitive equilibrium is defined as a list of variables (λ, c1, c2, b, k, r) that satisfy the household
optimality conditions and market clearing. There are 3 market clearing conditions: one for bonds and
two for goods. Let’s first turn to bond market clearing. The market clearing condition is b = 0. More
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precisely, indexing each household by h = 1, …, N the market clearing condition is ∑ b h = 0 . But we
know that all bh are the same so that N b = 0. Of course, we know that there is no trade in equilibrium
because all households are identical, but we still need the market clearing condition to figure out the
price (r).
The budget constraints then imply goods market clearing at both dates: e1 = c1 + k (endowments can be
eaten or stored) and c2 = f (k ) . Without trade, households are essentially operating under autarky,
although they all could trade, if they only wanted to. This means we have 6 independent equations (5
from the household and b = 0) that solve for the 6 unknowns. Note that Walras’ law in this case makes
both goods market clearing conditions redundant.
4.3 A log-utility example
To obtain a closed form solution, assume u (c) = ln(c) and f (k ) = k θ . A competitive equilibrium is
then a list (λ, c1, c2, k, r) which satisfies
1 / c1 = λ , β (1 + r ) / c2 = λ , θ k θ−1 = 1 + r , e1 = c1 + k , k θ = c2
Simple algebra allows to solve for k:
β (1 + r ) k −θ = 1 /(e1 − k )
⇒ β θ k θ−1 k −θ = β θ / k = 1 /(e1 − k )
⇒ e1 − k = k /(θ β)
⇒ k = e1 /[1 /(θ β) + 1] = e1 θ β /(1 + θ β)
The comparative statics results are intuitive: k rises (r falls) as households become more patient
(higher β) or have larger period 1 endowments.
4.4 Firms as separate agents
So far the household has been a consumer-producer. More commonly, it is assumed that households
consume and supply factors of production (capital and labor) to firms. In many models both
approaches lead to identical outcomes.
Adding firms in the previous model involves the following modifications. Assume there is a
continuum of firms of measure one, which is modeled as a single representative firm. The firm rents
capital (kF) from the single representative household at rental price q to maximize current period
profits:
π( k F ) = f ( k F ) − q k F .
The FOC is f ′(k F ) = q . Note the trick: if the firm does not own any assets, its problem is static.
Convince yourself that Inada conditions on f ensure an interior solution. A solution to the firm’s
problem is a pair (π, kF) that satisfies the firm’s FOC and the definition of profits.
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We also need to specify what fraction of capital depreciates in production. In general, renting k to the
firm results in output of f(k) + (1−δ) k.
There is a unit measure of households. Each owns the same fraction κ of the representative firm and is
not allowed to trade this ownership (alternatively one could introduce a stock market in which shares
of the firm are traded). Therefore, each household receives the same fraction κ of aggregate profits,
which it takes as given. In particular, the household’s k does not affect the amount of profits it receives.
How much capital to rent to firms and how many shares to own are two separate decisions. The
household budget constraints are therefore
c1 = e1 − k and k (1 + r ) + κ π = c2 ,
which implies a present value budget constraint of e1 − c1 = (c2 − κ π) /(1 + r ) . Note that household
problem is isomorphic to one with bonds and second period endowment κ π. Since all households are
identical and the mass of firms equals that of households, κ = 1. The FOCs are unchanged. A solution
to the household problem is a vector (c1, c2, k) that satisfies 2 FOCs and one budget constraint.
Exercise: Modify the model to add an equity market in which households can buy and sell shares of
the firm at date 1. Derive an equation that characterizes the value of the firm.
A competitive equilibrium is a list of prices (r, q) and quantities (c1, c2, k, kF, π) such that each
agent’s FOCs and constraints are satisfied and markets clear. Which markets? There is a rental market
for capital which clears if kF = k. Goods market clearing at date 1 requires c1 = e1 − k . In period 2, the
firm supplies f (k F ) and households demand c2. In addition, households eat the left-over capital.
Therefore: f (k ) + (1 − δ) k = c2 .
We therefore have 7 variables that satisfy the following conditions
•
the firm’s FOC and the definition of π (2 equations)
•
the household optimality conditions (3 equations)
•
3 market clearing conditions
How many of these equations are independent? Clearly period 1 goods market clearing is the same as
the period 1 budget constraint. Imposing capital market clearing on the period 2 budget constraint
yields period 2 goods market clearing. So both goods market clearing conditions are redundant.
One equation is still missing, which is essentially an accounting identity that relates the rental price of
capital paid by the firm (q) to the rental rate received by the households (r). The household receives
(1−δ) k in left-over capital and rental payments of q k. Therefore (1 + r ) k = (1 − δ) k + q k or r = q − δ.
Common special cases are no depreciation (δ = 0) or full depreciation (δ = 1). Sometimes depreciation
is “wrapped into” the production function. I.e., the production function becomes
g (k ) = f (k ) + (1 − δ) k , which changes the definition of q: q = g ′(k ) = f ′(k ) + 1 − δ . Otherwise this is
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equivalent to having no depreciation, except that g does not obey Inada conditions. For our purposes,
the default assumption is the first one: investing k yields (1+r) k = (1+q−δ) k income next period.
Note: Not all models assume that consumption goods can be converted one-for-one into capital, so that
the relative price of capital (the purchase price, not q) does not equal one. We will look at such models
much later (see the section on multi-sector models).
5. Reading
CM ch. 1
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