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Rules Rather than Discretion: The Inconsistency of Optimal Plans

Finn E. Kydland; Edward C. Prescott

The Journal of Political Economy, Vol. 85, No. 3. (Jun., 1977), pp. 473-492.

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Finn E. KydlandNorwegian School of Economics and Business Administration

Edward C. Prescot tCarnegie-Mellon University

Even if there is a n agreed-upon, fixed social objective function andpolicymakers know the timing and magnitude of the effects of theiractions, discretionary policy, namely, the selection of that decision whichis best, given the current situation and a correct evaluation of the end-of-period position, does not result in the social objective function beingmaximized. The reason for this apparent paradox is that economicplanning is not a game against nature but, rather, a game againstrational economic agents. We conclude that there is no way controltheory can be m ade applicable to economic planning when expectationsare rational.

I . IntroductionOptimal control theory is a powerful and useful technique for analyzingdynamic systems. At each point in time, the decision selected is best,given the current situation and given that decisions will be similarlyselected in the future. Many have proposed its application to dynamiceconomic planning. The thesis of this essay is that it is not the appro-priate tool for economic planning even when there is a well-defined andagreed-upon, fixed social objective function.

We find that a discretionary policy for which policymakers select the


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RULES RATHER THAN DISCRETION 475flood-control problems for which consistent policy procedures are notseriously considered. Then, for the aggregate demand managementproblem, it is shown that the application of optimal control theory isequally absurd, at least if expectations are rational. Doing what is best,given the current situation, results in an excessive level of inflation, butunemployment is no lower than it would be if inflation (possibly deflationor price stability) were at the socially optimal rate. Consistency forinfinite-period recursive economic structures is then considered. In equilib-rium, optimizing agents follow rules which specify current decisionsas a function of the current state.' Methods are developed for computingthese equilibrium decision rules for certain specialized structures. Themethods are used to evaluate alternative investment-tax-credit policiesdesigned both to stabilize and to yield optimal taxation. Among thepolicies evaluated is the suboptimal consistent policy. It'ithin the classof feedback policy rules, we found that the opt imal one depended upon theinitial conditions. Thus it was not optimal to continue with the initialpolicy in subsequent periods; that is, the optimal policy was inconsistent.

11. Consistent PolicyLet n = (n ,, n,, . . . , nT) be a sequence of policies for periods 1 to T[which may be infinite) and x = (xl, x2, . . . , xTj be the correspondingsequence for economic agents' decisions. An agreed-upon social objectivefunction

S(x1, . . . ,XT, n1, . . . , TT) (1)is assumed to exist.' Further, agents' decisions in period t depend uponall policy decisions and their past decisions as follotvs:

In such a framework an optimal policy, if it exists, is that feasible nwhich maximizes (1) subject to constraints ( 2 ) . Th e concept of consistencyis less obvious and is defined as follows:

Dejnitiorz: A policy n is consistent if, for each time period t , n,maximizes ( 1 ) , taking as given previous decisions, x, , . . . , x,-,,and that future policy decisions (n, for s > t ) are similarlyselected.

Th e original objective of this research was to demonstrate the applicability of optimal


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476 JOURNAL OF POLITICAL ECONOMYThe inconsistency of the optimal plan is easily demonstrated by a

two-period example. For T = 2, n, is selected so as to maximize

subject to

and

For a plan to be consistent, n2 must maximize (3j, given the past decisionsn,, x,, and constraint ( 4 ) . Assuming differentiability and an interiorsolution, then necessarily

The consistent policy ignores the effects of n, upon x,. For the optimaldecision rule, the first-order condition is

Only if either the effect of n, upon x, is zero (i .e ., c'Xl/?n2 = 0 ) or theeffect of changes in x, upon S both directly and indirectly through x,is zero (i.e., [?Sli;xl + ?S,dx, 2X2,/2xl] = 0 ) would the consistent policybe optimal.

Pollak (1968) resolved a planning inconsistency which arose becausedifferent generations had different preference orderings by assuming ateach stage that the policy selected was best (relative to that generation'spreferences), given the policies which will be followed in the future. Forthe T-period problem, the nT is determined which, conditional uponprevious decisions n t and xt, is best:

Once the functional relationship II, is known, the determination of thebest policy rule nT-, = II,-,(n,, . . . , n,-,, x,, . . . , x,-,) can bedetermined, and in general the consistent policy


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RULES RATHER THAK DISCRETIOX 477for future policy election.^ But as the simple example illustrated, thisprocedure is suboptimal.

Two examples follow :The issues are obvious in many 11 ell-known problems of public policy.

For example. suppose the socially desirable outcome is not to have housesbuilt in a particular flood plain but, given that they are there, to takecertain costly flood-control measures. If the government's policy Merenot to build the dams and levees needed for flood protection and agentsknew this was the case, even if houses Mere built there, rational agents~vould ot li\,e in the flood plains. But the rational agent knows that, ifhe and others build houses there, the government will take the necessaryflood-control measures. Consequently. in the absence of a law prohibitingthe construction of houses in the flood plain, houses are built there. andthe army corps of engineers subsequently builds the dams and levees.

A second example is patent policy. Given that resources have beenallocated to inventive activity which resulted in a new product or process,the efficient policy is not to permit patent protection. For this example,few would seriously consider this optimal-control-theory solution as beingreasonable. Rather , the question would be posed in terms of the optimalpatent life (see. e.g., Nordhaus 1969,, which takes into consideration boththe incentive for inventive activity provided by patent protection and theloss in consumer surplus that results when someone realizes monopolyrents. I n other words, economic theory is used to predict the effects ofalternative policy rules, and one with good operating characteristics isselected.

111. The Inflation-Unemployment ExampleThe suboptimality of the consistent policy is not generally recognizedfor the aggregate demand management problem. The standard policyprescription is to select that policy which is best, given the currentsituation. This may seem reasonable, but for the structure considered:which we argue is a plausible abstraction of reality, such policy resultsin excessive rates of inflation without any reduction in unemployment.The policy of maintaining price stability is preferable.

There are some subtle game-theoretic issues which have not been addressed here.Peleg and Yaari (1973) criticized Pollak's solution because sometimes it did not exist and


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478 JOURNA L OF POLITICAL ECONOMYThe attempts of economists to rationalize the apparent trade-off

between unemployment and inflation in modern theoretical terms haveresulted in models with the following structure: unemployment (employ-ment) is a decreasing (increasing) function of the discrepancy betweenactual and expected inflation rates. This example assumes such a relation-ship and that it is linear:

u, = i,(x: - x,) + u*, ( 5 )where u, is unemployment in period t , iLpositive constant, x, the in-flation rate, x: the forecasted or expected inflation rate, and u* thenatural rate implied by these theories. As has been recently shown byPhelps and Taylor (1975), one need not rely upon imperfect informationacross firms about the "generality" of shock or imperfect foresight aboutthe persistence of shock over time to obtain a similar relationship. Theyobtained one by assuming price rigidities, namely, that prices and wagesare set prior to the realization of demand.

The crucial issue is what assumption to make concerning priceexpectations. The conventional approach is to assume tha t expectationsdepend in some mechanical ad hoc way upon past prices. If so, controltheory would be an appropriate tool to determine the optimal path ofunemployment and inflation. The policy decision in each period wouldconsider both current outcomes and a proper evaluation of the terminalprice expectations state variable. Such a treatment of expectations isdifficult to justify either on a priori or empirical grounds. A changein administration which reflects a change in the relative costs societyassigns to unemployment and inflation will have an immediate effectupon expectations-contrary to the implicit assumption of the proponentsof control theory. hloreover, private agents or their agents have as muchinformation about the economic structure as does the policymaker andsome information concerning the implicit objective function which ration-alizes policy selections. Therefore their forecasts of future policy be-havior will be related to actual policy selection. This does not imply thatpolicy is perfectly predicted, but then neither is the behavior of privateagents. Just partial predictability of policy is sufficient to invalidate t heuse of optimal control theory.

For this example, we shall assume that the expectations are rational,so that the mathematical expectation of inflation equals the expectedrate :


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RULES RATHER THAN DISCRETION

FIG.1 -Consistent and optimal equilibrium

which used direct measures of expectations and found that forecasterrors were systematically related to lagged sales, so we will do so. Re-sponses to this finding are that there may be biases in their measurementof expectations, and these biases are related to lagged sales. This isnot implausible, given the subtleness of the expectations concept and theimprecision of survey instruments. Further, even if there were a system-atic forecast error in the past, now that the Hirsch and Love11 resultsare part of agents' information sets, future forecast errors should not besubject to such biases.

To complete the model, a theory of policy selection is needed. Hereit is assumed tha t there is some social objective function which rationalizespolicy choice :

S(x,, u,) .If the rationalization is not perfect, a random term must be introducedinto the function. The consistent policy maximizes this function subject


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480 JOURNAL OF POLITICAL ECONOMYbest, given the current situation. The indifference curves imply that thesocially preferred inflation rate is zero, which seems consistent with thepublic's preferences. W e of course recognize that inflation is a tax onreserves and currency, and a more informed public might prefer somepositive or negative inflation rate. If so, x, need only be interpretedas deviation from the optimal rate. The outcome of a consistent policyselection clearly is not optimal. If the policymakers were compelled tomaintain price stability and did not have discretionary powers, the re-sulting equilibrium would have no higher unemployment than the con-sistent policy. T he optimal equilibrium is point 0 , which lies on a higherindifference curve than the consistent-equilibrium point C.

It is perhaps worthwhile to relate our analysis to that of Taylor's(19751, in which he found that the optimal monetary policy was randomin a rational-expectations world. Similar results would hold for our proh-lem if uncertainty in the social objective function had been introduced.Both for his structure a nd for ours, the optimal policy is inconsistent, andconsequently it is not optimal for the policymaker to continue with hisoriginal policy rules.

IV . Consistent Planning for the Infinite HorizonThe method of backward induction cannot be applied to infinite-periodproblems to determine a consistent policy because, unlike the finite-period problem, there is no final period with which to begin the induc-tion. For recursive structures, however, the concept of consistency canbe defined in terms of policy rules. Suppose that the economy at timet can be described by a vector of state variables y , , a vector of policyvariables n,, a vector of decision variables xi for the economic agents,and a vector of random shocks E, which are temporally independent.The movement over time of these variables is given by the system ofequations

J t + I = F ( y t , TI,, x,, 4 .Let the feedback policy rule for future periods be

For certain structures: rational economic agents will in the future followa rule of the formx, = df(y,; nf). (7)


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RULES RATHER T H A N DISCRETION 481Again, it is important to note that expectations of future policy affectcurrent decisions. For example: the effect of an increase in the invest-ment tax credit will depend upon the expected future levels of theinvestment tax credit.

If, in addition, the social objective function is of the form

and the objective is to minimize its expected value, the optimal valuefor n, will depend upon both 3, and IIf, the policy rule which will beused in the future. I n other words, the best policy rule for the currentperiod IIC(y, s functionally related to the policy rule used in the futureI I f ( j ) , say nc = g (n* ) .X stationary policy rule II is consistent if it is a fixed point of mappingg, for then it is best to use the same policy rule as the one expected to beused in the f ~ t u r e . ~

Suppose policymakers and agents do not have a clear understandingof the dynamic structure of the economy. Over time, agents will gropefor and most likely converge to the equilibrium rules of forms (6) and( 7 ) . Policymakers taking the decision rules of agents as given, whenevaluating alternative decisions, typically would consider the trade-offof current outcomes relative to the desirability or value of the end-of-period state. Assuming that their valuation of the terminal state isapproximately correct, they will be selecting the approximately consistentpolicy, assuming also that agents have approximately rational expecta-tions. Thus it seems likely that the current practice of selecting that policywhich is best, given the current situation, is likely to converge to theconsistent but suboptimal policy.

It is hard to fault a policymaker acting consistently. The reason thatsuch policies are suboptimal is not due to myopia. The effect of thisdecision upon the entire future is taken into consideration. Rather, thesuboptimality arises because there is no mechanism to induce futurepolicymakers to take into consideration the effect of their policy, via theexpectations mechanism, upon current decisions of agents.

This is the solution concept used by Phelps and Pollak (1968) for an infinite-periodsecond-best growth problem when different generations had inconsistent preferences.


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482 JOURNAL OF POLITICAL ECONOMYV. The Investment-Tax-Credit ExampleIn this section an equilibrium framework is developed for evaluating aclass of investment-tax-credit policies. The assumed technological struc-ture is similar to the one used by Jorgenson (1963), though increasingcosts associated with rapid adjustment in capacity are assumed. A firmuses k, units of capital and n, units of labor to produce an output Ak;njl-"I.Output price is p,, and the real wage is assumed to be a constant, say 1.Investment planned in period t and carried out that period and thenext, xi, does not become productive until period t + 2. The relationshipbetween current productive capital, planned investment, and futureproductive capital is

where 6 is the constant physical rate of depreciation. Investment costsassociated with x,, the new investment plans in period t , occur in bothperiod t and period t + 1. This reflects the fact that time is requiredto expand capacity, and investment expenditures occur over the entiretime interval. The fraction of the investment effort induced by plan xiin the current period is 4, and the fraction induced in the subsequentperiod is 1 - 4. The investment rate in period t is then

z , = dx, + (1 - 4)x t - , .Following Haavelmo (1960), Eisner and Strotz (1963), Lucas (1967),Gould (1968), an d Treadway (1969), we assume that the investmentexpenditures are an increasing convex function of the rate of capitalexpansion z,. In order to insure constant returns to scale in the longrun, the function is assumed to have slope equal to the price q of capitalgoods at z, = 6k,. Making the quadrat ic approximation, the investmentexpenditures in period t are then

i, = qz, + y(z, - 6kJ2,where y is positive. Observe that i, depends upon investment plans in boththe current and previous periods a nd that , if xi is constant over time andsufficient to maintain the capital stock, i, = qx,.

The gross cash inflow during period t is

In period t a tax rate 0 is applied to sales less labor costs and depreciation.


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RYLES RATHER THAN DISCRETION 483The view is that the adjustment cost term reflects costs internal to thefirm and therefore not eligible for the investment tax credit.

The net cash inflow in period t is (8) less (9) plus (10). The objectiveof the firm is to maximize the expected present value of this net cashinflow stream. Maximizing each period's cash flow over the period's n,,the objective function to be maximized becomes

where j. = [I , ( 1 - r,][A(l - rj]':" is output per unit of capital and jis the discount factor.The inverse aggregate demand function is assumed to be linear. Lettingcapital letters denote the aggregates of the corresponding variables forthe individual firms, the inverse demand function is of the form

where b is a positive constant, a , is a stochastic demand shift parameter.and K , is the aggregate capital stock for the firms. LVe assume that a , issubject to the first-order autoregressive process

where the E , are positive independent random variables with mean p andvariance c,.

For the economy to be in equilibrium, the expected and actual distri-bution of the random elements must be equal. Here we are assumingrat ional expectations of Muth (1961, and of Lucas and Prescott (197 1).Brock (1972) has characterized such expectations schemes as being self-fulfilling. We are implicitly assuming that the economy is Hicksian in thesense that a single consumer could have the implicit excess demand func-tion. For such economies wealth effects net out, and our equilibriumyields the same allocation as the Arrow-Debreu state preference equilib-rium upon extension of their analysis to infinite-dimensional space.

In the Appendix we develop direct methods for computing the com-petitive equilibrium, given the policy rule. Also discussed in the Appen-dix is the stability of the equilibrium.

TVe are also implicitly assuming a per unit rather than a percentage tax credi t.' If policy does not depend upon agents' decisions, the competitive equilibrium is


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484 JOURNAL OF POLITICAL ECONOMYThe policymakers are choosing values for n in each time period so as

to minimize the value of some social preference function. We use a quad-ratic approximation of a function which includes the terms likely to carryany weight. The assumed form is

where each o i , i = 1, . . . , 6, indicates the relative weight on each of thecomponents. The terms g,, g,, and g, are targets for real output of theindustry, for real investments, and for the total of the two, respectively.Reasons for including the tax credit in the loss function are that theamount paid may have to be collected elsewhere in the form of taxes andinefficiencies generally caused by such measures.

Our examples assume that a passive investment-tax-credit stabiliza-tion policy had been pursued in the past and that the function describinginvestment behavior was equilibrium, given this passive policy. They alsoassume that econometricians have estimated the investment relationship8and that the policymaker uses control theory to determine which policyrule is optimal under the incorrect assumption that the equilibriuminvestment function is invariant to the policy rule used. Subsequent tothe implementation of this policy rule, the economy moves to the newequilibrium investment function. Econometricians revise their estimateof the investment function, arguing that there has been structural change,and the policymaker uses optimal control to determine a new policy rule.The change in policy induces still another change in the investmentfunction, which in turn induces a change in the policy rule, once theshift in the investment function is recognized. This iterative process, wethink, captures the essence of what is actually happening. We observethat econometricians are continually revising their estimates of the struc-ture on the basis of which new policies are devised and are continuallysurprised to find tha t the structure has changed.

Our results can be summarized as follovvs:Factor shares, the capital output ratio, tax rates, and capital consump-

tion allowances were used to deduce not unreasonable values for param-eters of technology and preferences with the exception of 4, the distributedlag coefficient of investment, and p , the autoregressive parameter of the


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RULES RATHER THAN DISCRETION 485aggregate demand relation9 Only these parameters and the parametersof the objective function were subject to variation. Space constraintspreclude more than a brief summary of the results.1

Typically the iterative process of the policy rule change inducinginvestment function change inducing policy rule change, etc., did con-verge. Given that it converges, the limiting policy rule is consistent in thesense described in Sections I1 and IV. In all cases for which it did con-verge, we searched for and found linear feedback policy rules which weresuperior to this consistent rule, typically by a substantial amount.

For one example (wl= 2, o2= 4, w3 = 1, o4 = 10, o5= 20,o, = 10, and p = 0.6) , the application of optimal control initially im-proved the performance of the economy relative to the assumed objectivefunction. For the first two iterations, the economy was subject to lessfluctuation and fluctuated about a preferred point. After the third iter-ation, however, performance deteriorated, and the consistent policy towhich the process converged was decidedly inferior to the passive policy forwhich the investment tax credit was not varied. The difference in per-formance corresponded roughly to the variables being 10 percent onaverage away from their targets.

For another example (w,= 1, o2= 2, o3= 1, o, = 10, o, = 3,0, = 20, and p = -0 .6), the iterative process did not converge.Changes in the policy rule induced ever larger changes in the investmentfunction. The variables fluctuated about their targeted values but fluc-tuated with increased amplitude with each iteration. This is a verydisturbing result, for it indicates that current practice, if continued, couldconceivably result in even greater fluctuations than are now beingexperienced.

There are two lessons we learned from the examples. First, the use ofoptimal control theory is hazardous and could very well increase economicfluctuations or even make a stable economy unstable. Second, even whenit does work reasonably well, it can be improved upon by following someother simple feedback rule.

This is not a n argument that economic fluctuations are either desirableor unavoidable. That our economy has experienced periods of reasonablestability is evidence that much of the fluctuation is avoidable. Rather,it is a plea for the use of economic theory to evaluate correctly the per-formance of a policy rule before it is implemented. We emphasize that


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486 JOURNAL OF POLITICAL ECONOMYoptimal control theory can not be made applicable to economic planningby taking into consideration the way changes in the policy rule change thebehavioral equation of the model when expectations are rational.

VI. DiscussionThe analysis has implications in other situations as well. Kydland (1975a)has explored the implications for a dynamic oligopoly problem with adominant firm. Like the policymaker, the dominant firm takes into con-sideration the reaction of the other agents in selecting its decision. Pre-cisely the same paradox arises.

The analysis also has implications for constitutional law. A majoritygroup, say, the workers, who control the policy might rationally chooseto have a constitution which limits their power, say, to expropriate thewealth of the capitalist class. Those with lower discount rates will savemore if they know their wealth will not be expropriated in the future,thereby increasing the marginal product and therefore wage and loweringthe rental price of capital, at least for most reasonable technologicalstructures.

Still another area is the current energy situation. We suspect thatrational agents are not making investments in new sources of oil in theanticipation that price controls will be instituted in the future. Cur-rently there are those who propose to tax away "excessive" profits ofthe oil companies with the correct argument that this will not affectpast decisions. But rational agents anticipate that such expropriationsmay be made in the future, and this expectation affects their currentinvestment decisions, thereby reducing future supplies.

VII. Summary and ConclusionsWe have argued that control theory is not the appropriate tool fordynamic economic planning. It is not the appropriate tool because cur-rent decisions of economic agents depend upon expected future policy,and these expectations are not invariant to the plans selected. We haveshown that, if in each period the policy decision selected is the one whichmaximizes the sum of the value of current outcomes and the discountedvaluation of the end-of-period state, the policy selected will be consistentbut not optimal. This point is demonstrated for an investment-tax-credit


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RULES RATHER THAN DISCRETION 487The structures considered are far from a tested theory of economic

fluctuations, something which is needed before policy evaluation isundertaken. The implication of this analysis is that, until we have sucha theory, active stabilization may very well be dangerous and it is bestthat it not be attempted. Reliance on policies such as a constant growthin the money supply and constant tax rates constitute a safer course ofaction.

When we do have the prerequisite understanding of the business cycle,the implication of our analysis is that policymakers should follow rulesrather than have discretion. The reason that they should not have dis-cretion is not that they are stupid or evil but, rather, that discretionimplies selecting the decision which is best, given the current situation.Such behavior either results in consistent but suboptimal planning or ineconomic instability.

If we are not to attempt to select policy optimally, how should it beselected? Our answer is, as Lucas (1976) proposed, that economic theorybe used to evaluate alternative policy rules and tha t one with good opera-ting characteristics be selected. In a democratic society, it is probablypreferable that selected rules be simple and easily understood, so it isobvious when a policymaker deviates from the policy. There could beinstitutional arrangements which make it a difficult and time-consumingprocess to change the policy rules in all but emergency situations. Onepossible institutional arrangement is for Congress to legislate monetaryand fiscal policy rules and these rules to become effective only after a2-year delay. This would make discretionary policy all but impossible.

AppendixLety be the state variables and x the decision variables for the firm. There is alinear relationship between the next period's state variables, y,+ ,, and the currentxi and y, :

Yt+l = f ( x t , ~ , ) . ( A l lThe movement of economy-wide state variables Y and aggregate (or per firm)decision variables X are described by the same linear function:

We also include a vector of autonomous shocks, W, which are subject to a first-order autoregressive process,


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488 JO UR NAL OF POLITICAL ECONOMYThe firm's objective is to maximize

where n, is a vector of policy variables assumed to be given by a sequence oflinear policy rules,

n, = n t ( Y r ,WE), t = 0, . . . , T, (A41which in equilibrium are correctly anticipated by the firm.

The cash-flow function R is quadrat ic. T he decisions x, ar e selected sequentiallyconditional on y,, X,, Y,, and W,. Let v,(y,, Y,, W,) be the value of the firm attime t. The v , functions satisfy the recursive relationship

subject to constraints (A1)-(A4)and one additional constraint. T o explain thislast constraint, note that, since x, is chosen so as to maximize the valuation attime t , if v, is quadratic and the right-hand side of (A5) concave in x,, the x,which maximizes the right-hand side of (A5) ,taking as given X,, Y,, W,, and themotion of the economy-wide state variables, will be linearly related toy,, X,, Y,,W,, and n, :

.yt = dt(y ,, Xt, Yr, wrrC:, (A6)In order for the economy to be in equilibrium, we have to impose the constraintthat, when firms behave according to (A6) ,the aggregate or per firm X, is indeedX,. Therefore1'

Xt = dt(Yt,X,,Y,, W,, n,),which can be rewritten as

x, = D,(Y,, W,, n,) . (A71As the constraints are all linear and the right-hand side of (A5)quadratic, thefunction o, is quadratic, given that v,+ is quadratic. The function u T + is the nullfunction and therefore trivially quadratic, so by induction all the v, are quadratic.The equilibrium per firm decision function for each time period t is given by (A7).

If the social objective function is quadratic of the form

the determination of the consistent policy is straightforward. Let

given that policy is selected consistently and that the economy is competitive.Thu s the function u, gives the total expected value of the social objective function


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489CLES RATHER THAN DISCRETIONsubject to (421, (X3) . and (h7).f u , + ~is quadra tic , a quadrat ic function is beingminimized subject to linear constraints. Therefore u, must be quadratic if u,,,is quadratic. As u r+ , = 0 and is thus trivially quadratic, all the u , are quadraticby backward induction. and the consistent policy is a linear function of Y and W:

n, = n t ( Y t , I.1;).I t is perhaps worth~vhile o make the connection between the structure just

analyzed and the one described in Section V. Th e state vector is

and the linear equations governing its movement over time are

Th e equations governing the economy-wide variables are the same. Furthermore,i tr ,+, r a,,, = pa, + E , =RT1', + qt

Th e revenue function R for the firm is

which has the assumed form, given that

and

Finally, the social objective function S given byu1(j.Kr - g11 + w2(Z, - g2) + w3( i lK ,+ Z,- g,)

+ ~ 4 ( P f- pi-112 f wgnfZt + ~ 6 7 1 :also has the quadratic form, given the assumed definitions of the variables,

Computations for the Injnite-Period ProblemEquilibrium decision rules for agents were determined as the limit of first-perioddecision rules as the life of the economy went to infinity. There is an interestingan d as yet unsolved problem as to the uniqueness of the equilibrium.12 For theseexamples the equilibrium associated with a stationary policy rule did appear tobe unique, for when we used the method of successive approximation in the valuespace (i.e., the a function in [A5]) the value function an d therefore decision rulesconverged to the same limit for a number of initial approximations. For someunreasonable policy rules and finite T, there were no competitive equilibria.

Consistent solutions were computed in two different ways. T he first determinedthe first-period consistent policy for T-period problems and the limit determined


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You have printed the following article:


Finn E. Kydland; Edward C. Prescott

The Journal of Political Economy, Vol. 85, No. 3. (Jun., 1977), pp. 473-492.

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[Footnotes]

3 On the Existence of a Consistent Course of Action when Tastes are Changing

Bezalel Peleg; Menahem E. Yaari

The Review of Economic Studies, Vol. 40, No. 3. (Jul., 1973), pp. 391-401.

Stable URL:http://links.jstor.org/sici?sici=0034-6527%28197307%2940%3A3%3C391%3AOTEOAC%3E2.0.CO%3B2-V

4On Second-Best National Saving and Game-Equilibrium Growth

E. S. Phelps; R. A. Pollak

The Review of Economic Studies, Vol. 35, No. 2. (Apr., 1968), pp. 185-199.

Stable URL:

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7 Investment Under Uncertainty

Robert E. Lucas, Jr; Edward C. Prescott

Econometrica, Vol. 39, No. 5. (Sep., 1971), pp. 659-681.

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12Contraction Mappings in the Theory Underlying Dynamic Programming

Eric V. Denardo

SIAM Review, Vol. 9, No. 2. (Apr., 1967), pp. 165-177.

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Eric V. Denardo

SIAM Review, Vol. 9, No. 2. (Apr., 1967), pp. 165-177.

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Adjustment Costs in the Theory of Investment of the Firm

J. P. Gould

The Review of Economic Studies, Vol. 35, No. 1. (Jan., 1968), pp. 47-55.

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Investment Under Uncertainty

Robert E. Lucas, Jr; Edward C. Prescott

Econometrica, Vol. 39, No. 5. (Sep., 1971), pp. 659-681.Stable URL:

http://links.jstor.org/sici?sici=0012-9682%28197109%2939%3A5%3C659%3AIUU%3E2.0.CO%3B2-0

Rational Expectations and the Theory of Price Movements

John F. Muth

Econometrica, Vol. 29, No. 3. (Jul., 1961), pp. 315-335.

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On the Existence of a Consistent Course of Action when Tastes are Changing

Bezalel Peleg; Menahem E. Yaari

The Review of Economic Studies, Vol. 40, No. 3. (Jul., 1973), pp. 391-401.

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On Second-Best National Saving and Game-Equilibrium Growth

E. S. Phelps; R. A. Pollak


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Consistent Planning

R. A. PollakThe Review of Economic Studies, Vol. 35, No. 2. (Apr., 1968), pp. 201-208.

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Rational Expectations, the Real Rate of Interest, and the Natural Rate of Unemployment

Thomas J. Sargent; David Fand; Stephen Goldfeld

Brookings Papers on Economic Activity, Vol. 1973, No. 2. (1973), pp. 429-480.

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On Rational Entrepreneurial Behaviour and the Demand for Investment

A. B. Treadway


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