April 26, 2013, Christopher D. Carroll DecentralizingRCK

Decentralizing the Ramsey/Cass-Koopmans Model

This handout shows that under certain very special conditions the behavior of an economy composed of distinct individual households will replicate the social planner’s solution to the Ramsey/Cass-Koopmans model.

1 The Consumer’s Problem

Consider first the problem of an individual infinitely lived consumer indexed by i  who has some predetermined set of expectations for how the aggregate net interest rate r
 t  and wage rate W
   t  will evolve.

At date t  household i  owns some capital kt,i  , and can in principle also borrow; designate the net debt of household i  in period t  as d
 t,i  ; since we will be examining a perfect foresight solution with perfect capital markets, the interest rate on debt must match the rate of return on assets. This means that all that really matters is the household’s total net asset position,

xt,i  =   kt,i - dt,i.                             (1)

Each household is endowed with one unit of labor, which it supplies exogenously, earning a wage rate W
  t,i  . Each household solves:

      ∫
         ∞
max        u (ct,i)e- ϑtdt
       0
(2)

subject to the budget constraint

˙xt,i =  Wt  + rtxt,i - ct,i.
(3)

Integrating the household’s dynamic budget constraint and assuming a no-Ponzi-game transversality condition yields the intertemporal budget constraint, which says that the present discounted value of consumption must match the PDV of labor income plus the current stock of net wealth:

ℙ0,i(c)  =   ℙ0,i(W  ) + x0,i.                        (4)

The formulas for these PDV’s are a bit awkward because they must take account of the fact that interest rates are varying over time. To make the formulas a bit simpler, define the compound interest factor

                  ∫  t
R - 1  =   exp (-     rτd τ),                         (5)
  t                 0
which is simply the compound interest term needed to convert a value at date t  to its PDV as of time 0.

With this definition in hand we can write the IBC as

∫ ∞
     c  R - 1dt  =   h   +  x                          (6)
 0    t,i  t           0,i     0,i
where h0,i  is human wealth,
         ∫  ∞
                    - 1
h0,i =        WtR   t  dt.                          (7)
           0

Each household solves the standard optimization problem taking the future paths of wages and interest rates as given. Thus the Hamiltonian1 is

H  (c  ,x   ,λ  ) =  u(c   ) + (r x   +  W  -  c  )λ
     t,i   t,i  t,i        t,i      t t,i      t    t,i  t,i
(8)

which implies that the first optimality condition is the usual  ′
u (ct,i) =  λt,i  . The second optimality condition is

     ˙λt,i =   ϑ λt,i - (∂H  ∕∂x )                       (9)
     ˙
     λt,i =   ϑ λt,i - λt,irt                           (10)
˙λ   ∕λ    =   (ϑ  - r )                                (11)
  t,i  t,i            t
leading eventually to the usual first order condition for consumption:
               - 1
c˙t,i∕ct,i  =   ρ   (rt - ϑ).                         (12)

Note (for future use) that the RHS of this equation does not contain any components that are idiosyncratic: The consumption growth rate will be identical for every household. The same is true of the expression for human wealth, equation (7).

2 The Firm’s Problem

Now we assume that there are many perfectly competitive small firms indexed by j  in this economy, each of which has a production function identical to the aggregate Cobb-Douglas production function. Perfect competition implies that individual firms take the interest rate ˆr
 t  and wage rate ˆW
  t  to be exogenous. Hence firms solve

  max    F (Kt,j ,Lt,j) - WˆtLt,j  -  ˆrtKt,j
{Kt,j,Lt,j}
(13)

where ˆr
 t  and Wˆ
   t  are the rental rates for a unit of capital and a unit of labor for one period. Note that, dividing by Lt,j  , this is equivalent to

       α
max   kt,j - ˆWt  - ˆrtkt,j.
{kt,j}◟ ◝◜◞
     f(kt,j)
(14)

The first order condition for this problem implies that

f′(kt,j)  =   ˆrt.                              (15)

Under perfect competition firms must make zero profits in equilibrium, which means, by fact [EulersTheorem], that:

f(kt,j)  =   ˆWt +  ˆrtkt,j.                          (16)

3 Equilibrium At a Point in Time

Thus far, we have solved the consumer’s and the firm’s problems from the standpoint of atomistic individuals. It is now time to consider the behavior of an aggregate economy composed of consumers and firms like these.

We assume that the population of households and firms is distributed along the unit interval and the population masses sum to one, as per Aggregation. Thus, aggregate assets at time t  can be defined as the sum of the assets of all the individuals in the economy at time t  ,

         ∫
            1
Xt   =       xt,idi                             (17)
           0
while per capita assets are aggregate assets divided by aggregate population,
xt   =   Xt ∕1.                               (18)

Similarly, normalizing the population of firms to one yields

kt  =   Kt  ∕1.                               (19)

Up to this point, we have allowed for the possibility that different households might have different amounts of net worth. We now impose the assumption that every household is identical to every other household. This assumption rules out the presence of any debt in equilibrium (if all households are identical, they cannot all be in debt - who would they owe the money to?). Indeed, in this case, the aggregate capital stock per capita will equal the aggregate level of net worth, kt =  xt  .2

Thus, households’ expectations about Wt  and rt  determine their saving decisions, which in turn determine the aggregate path of kt  .

There is one important subtlety here, however. In writing the consumer’s budget constraint, we designated rt  as the net amount of income that would be generated by owning one more unit of net worth (e.g. capital). But if we have depreciation of the capital stock, the net return to capital will be equal to the marginal product minus depreciation. The discussion of the firm’s optimization problem did not consider depreciation because the firms do not own any capital; instead, they make a payment ˆrt  to the households for the privilege of using the households’ capital. Thus the net increment to a household’s wealth if the household holds one more unit of capital will be

rt  =   ˆrt - δ.

There is no depreciation of labor, so the labor market equilibrium will be

Wt   =   ˆWt

     =   f(kt) - ˆrtkt.

4 The Perfect Foresight Equilibrium

We assume that every household knows the aggregate production function, and understands the behavior of all the other households and firms in the economy. Understanding all of this, suppose that households have some set of beliefs about the future path of the aggregate capital stock per capita {k } ∞
   t t=0   . This belief about kt  will imply beliefs about wages and interest rates as well {W   ,r }∞
    t  t t=0   .

The final assumption is that the equilibrium that comes about in this economy is the “perfect foresight equilibrium.” That is, consumers have the sets of beliefs such that, if they have those beliefs and act upon them, the actual outcome turns out to match the beliefs.

Note now that using the fact that xt =  kt  in the perfect foresight equilibrium we can rewrite the household’s budget constraint as

˙kt  =   rtkt + Wt  -  ct                              (20)

    =   (ˆrt - δ)kt +  Wt  -  ct.                      (21)

Reproducing from (12),

˙ct∕ct  =   ρ - 1(rt - ϑ )                            (22)
             - 1
       =   ρ   (ˆrt - δ -  ϑ).                        (23)

Now compare these to the equations derived for the social planner’s problem (with population growth and productivity growth zero) in a previous handout:

k˙t  =   f(kt) -  δkt -  ct                            (24)

    =   ˆrtkt +  Wt -  δkt -  ct                       (25)
    =   (ˆr  - δ )k  + W   -  c                        (26)
          t       t      t    t
and
˙c ∕c   =   ρ- 1(f′(k ) - δ -  ϑ ).                     (27)
 t  t               t

Since the equilibrium value of ˆ     ′
rt = f (kt)  , (27) = (23). And (21) is identical to (26). Thus, aggregate behavior of this economy is identical to the behavior of the social planner’s economy!

This is a very convenient result, because it means that if we are careful about the exact assumptions we make we can often solve a social planner’s problem and then assume that the solution also represents the results that would obtain in a decentralized economy. The social planner’s solution and the decentralized solution are the same because they are maximizing the same utility function with respect to the same factor prices (rt  and Wt  ).

When will the decentralized solution not match the social planner’s solution? One important case is when there are externalities in the behavior of individual households; another possible case is where there is idiosyncratic risk but no aggregate risk; basically, whenever the household’s budget constraint or utility function differs in the right ways from the aggregate budget constraint or the social planner’s preferences, there can be a divergence between the two solutions.

References

   AIYAGARI, S. RAO (1994): “Uninsured Idiosyncratic Risk and Aggregate Saving,” Quarterly Journal of Economics, 109, 659–684.

   CARROLL, CHRISTOPHER D. (1992): “The Buffer-Stock Theory of Saving: Some Macroeconomic Evidence,” Brookings Papers on Economic Activity, 1992(2), 61–156, http://econ.jhu.edu/people/ccarroll/BufferStockBPEA.pdf.

   __________  (2000): “Requiem for the Representative Consumer? Aggregate Implications of Microeconomic Consumption Behavior,” American Economic Review, Papers and Proceedings, 90(2), 110–115, http://econ.jhu.edu/people/ccarroll/RequiemFull.pdf.

   KRUSELL, PER, AND ANTHONY A. SMITH (1998): “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy, 106(5), 867–896.