004: Macroeconomic Theoryecondse.org/wp-content/uploads/2019/01/004-Micro...Das (Lecture Notes, DSE) Macro Jan 22-25; 2019 12 / 41 Classical/Keynesian Production Side Story (Contd.)

004: Macroeconomic TheoryMicro Foundations of Macroeconomic Systems

Mausumi Das

Lecture Notes, DSE

Jan 22-25; 2019

Das (Lecture Notes, DSE) Macro Jan 22-25; 2019 1 / 41

Micro-foundations of various Macroeconomic Systems:

So far we have discussed various macroeconomic systems (Classical,Keynesian and their different extensions).

Recall that we have presented each system as a bunch of ‘adhoc’equations that are supposed to define the macroeconomy as a whole.

We made no attempt to derive the underlying micro-behaviour ofoptimizing agents that would generate these aggregative equationsfor the macro-economy.

Question is: Can these aggregative behavioural equations bereconciled with some kind of optimal behaviour of firms/householdsat the micro level?

Put differently, can we provide some micro-foundation to theseaggregative equations?


Micro Foundations:

We now attempt to develop the micro-foundations of some of theseequations. In the process we also demonstrate some of thesemicro-foundations would work only in a dynamic set up; hence thereis a need to move from the static to a dynamic framework.Micro-foundations are thought to be necessary for two reasons:

There are many implicit assumptions that are made in formulating theaggregative relationships. A precise discussion of the micro-foundationsallows us to highlight these assumptions and also check for theirvalidity;As Robert Lucas pointed out, many of the constants in the aggregativesystems are not parameters in the true sense of the term; they captureequilibrium behaviour under certain conditions (e.g. ceratinexpectations about the policies, environment etc). As those conditionschange, people optimally change their equilibrium behaviour; hencethese ‘reduced form’terms also change their values. Sopredictions/forecasts about the economy based on these constantscould go wrong unless one actually derives these optimal values fromthe underlying micro-foundations.


Micro Foundations: Classical/Keynesian Production SideStory

We start with the micro-foundations of the production relations.Recall that the production side story for Classical and Keynesiansystems are identical: both assume that firms operate in a perfectlycompetetive market structure with a production technology whichexhibits the following properties:

Yi = F (Ni ,Ki )

Constant returns to scale (CRS)/Homegenous of degree one:F (λNi ,λKi ) = λF (Ni ,Ki )Positive but diminishing returns in each factor: FNi ,FKi > 0;FNiNi ,FKiKi < 0Both inputs are essential: F (0,Ki ) = F (Ni , 0) = 0Inada conditions:

limNi→0

FN (Ni ,Ki ) = ∞; limNi→∞

FN (Ni ,Ki ) = 0;

limKi→0

FK (Ni ,Ki ) = ∞; limKi→∞

FK (Ni ,Ki ) = 0Das (Lecture Notes, DSE) Macro Jan 22-25; 2019 4 / 41

Classical/Keynesian Production Side Story (Contd.)

Any production function that satisfies all the above properties iscalled a ‘Neoclassical’production function.

Example: Cobb Douglas Technology: Yi = Nαi K1−αi ; 0 < α < 1.

Sometimes the production function is also written as

Yi = AF (Ni ,Ki )

where A is an index of technology that captures the economy-wideproductivity level or total factor productivity (TFP). Under theneoclassical technology, the TFP term is assumed to be exogenouslydetermined.

In a static framework, A is typically assumed to be a constant.(Noticethat A is not firm-specific; it relates to the entire economy).



Suppose there are S firms in the economy, all having access to anidentical ‘Neoclassical’technology given by:

Yi = AF (Ki ,Ni ).

Recall that the firms take decisions about:

How much final output to produce;How much labour to employ.

The firms also decide how much capital to employ. However in thisstatic framework, we shall assume here that this decision is trivial:they employ the total capital stock available in the economy (K̄ )equally such that

K̄i =K̄S



Indeed if the firms are perfectly competitive (which means they areprice-takers), then a firm’s choice is rather straight-forward.

Given W and P, the optimization problem of the i-th firm is definedas:

Max .P{Ni }

AF (K̄i ,Ni )−WNi − RK̄i .

This generates the following FONC from the i-th firm:

AFN (K̄i ,Ni ) =WP.

The above equation implicitely defines the labour demand of the i-thfirm as a function of the real wage rate (WP ), its capital share (K̄i )and the aggregate TFP Index (A):

Ni = f̂(WP, K̄i ,A

).



Aggregating over all firms, we get the corresponding aggregate outputsupplied:

Y S : Y = S .Yi = S .AF (K̄i ,Ni ) = AF (SK̄i ,SNi ) (by CRS)

As long as S .Ni = N (which we are going to prove shortly), the latterdescribes the aggregate production function for the economy.

Notice that the assumption of CRS plays a very important role ingenerating the equivalence between S .Yi and AF (K̄ ,N).

Also recall that we have assumed that total capital is divided equallyacross all firms.Question: will such equivalence still hold when capital is dividedarbitrarily across the firms? (We shall come back to this questionsoon).



We still have to prove that S .Ni = N, i.e., the labour demand that isgenerated from the aggregate production function is identical to thelabour demand function generated by summing up the labourdemands coming from all the firms.The labour demand function in the Classical (and Keynesian) systemwas earlier defined in terms of the aggregate production function:

ND : AFN (K̄ ,N) =WP

On the other hand, the labour demand function generated byaggregating all the firms’demand for labour is given by

ND = S .Ni where Ni = f̂(WP,K̄S,A)such that

Ni : AFN

(K̄S,Ni

)=WP

Question: How do we know that these two expressions areequivalent?



In proving this equivalence, we can directly apply the followingproperty of a homogenous function:“Let f (x1, x2, ....., xn) be a differentiable function of n variables that is

homogeneous of degree k. Then each of its partial derivatives∂f∂xi

(for i = 1, ..., n) is homogeneous of degree k − 1.”Here however we follow a more circuitous route.We exploit another characteristic of the production function whichfollows from its CRS property:

Consider any function: Y = AF (N,K ).

If it is CRS, thenYN= AF

(1,KN

)= Af (k), where k ≡ K

N. Thus

another way to write the production function is: Y = N.Af (k).From this latter specification,

∂Y∂N

= A[f (k) +Nf ′(k)

∂k∂N

]= A

[f (k) +Nf ′(k)

(− KN2

)]= A

[f (k)− kf ′(k)

]Das (Lecture Notes, DSE) Macro Jan 22-25; 2019 10 / 41

Micro Foundations: Production Side Story (Contd.)

In other words, for a CRS production function, the marginalproduct of labour is a function of the capital-labour ratioemployed.

(Question: How about the marginal product of capital? Can we derive

a similar relationship between∂Y∂K

and k through the function f (k)

and/or its derivatives? Try this as a homework. )



Given this property, let us now go back to the two questions askedearlier.

Let us take up the second question first:

Are ND : AFN (N, K̄ ) =WPand ND = SNi : AFN

(Ni ,

K̄S

)=WP

equivalent?

Notice that AFN (N, K̄ ) is the marginal production labour associatedwith the aggregate production function. Hence applying the aboveproperty, the ND equation from the aggregate production functionreduces to:

A[f (k)− kf ′(k)

]=WP

(i)

where the relevant k here is the aggregate capital-labour ratio:K̄N.



On the other hand, AFN

(Ni ,

K̄S

)is the marginal product of labour

relevant for the i-th firm. Hence applying the above property, thelabour demand equation for each firm i reduces to :

A[f (ki )− ki f ′(ki )

]=WP

(ii)

where the relevant ki here is the firm-specific capital-labour ratio:K̄iNi.

Since equation (i) and equation (ii) are identical equation (though thevariable is different) the solutions to these two equations must also bethe same, i.,e

ki = k = C (some constant)



In other words, even though firms are employing different levels ofcapital and labour than the aggregate economy, the capital-labourratio in all cases must be the same, denoted by some constant C .(This property is called Scale-Neutrality).Given this relationship, we can now write the labour demand coming

out of the aggregate profuction function as ND :K̄N= C ⇒ N = K̄

C.

On the other hand, labour demand coming from an individual firm

can be written as Ni :K̄iNi= C ⇒ Ni =

K̄iC.

Using the above solution, the labour demand coming out of

aggregation over all firms is: ND = SNi = SK̄iC= S

K̄SC

=K̄C.

Thus that the two labour demand functions will be exactly the same.This result has of course been derived here under the assumption that

K̄i =K̄S.

What if total capital stock is NOT divided equally?Das (Lecture Notes, DSE) Macro Jan 22-25; 2019 14 / 41


The ’Scale-Neutrality’property also gives us the clue as to whathappens when firms differ in terms of their share of capital.

To see that, let us now assume that capital is not distributed equallyacross all the firm.

For simplicity, let us assume that there are two sets of firms: one setof firms (S1 in number) is given a capital stock of K1 while the otherset (S2 in number) is given a capital stock of K2 < K1 such that

S1 + S2 = S ,

and S1K1 + S2K2 = K̄ .



As we have seen earlier firms belonging to the first group will have alabour demand equation given by:

Ni : AFN (K1,Ni ) =WP

i.e., A[f (ki )− ki f ′(ki )

]=

WP; ki ≡

K1Ni

(i)

On the other hand, firms belonging to second group will have alabour demand equation given by:

Nj : AFN (K2,Nj ) =WP

i.e., A[f (kj )− kj f ′(kj )

]=

WP; kj ≡

K2Nj

(ii)

Since equations (i) and (ii) are identical, their solution set must alsobe exactly the same. Moreover, since f (k)− kf ′(k) is a monotonicfunction of k taking values bewteen (0,∞) (prove this), we end upwith a unique solution for both, given by ki = kj = C .



Thus we can write the individual demand for labour functions comingfrom each set of firms as:

Ni =K1C;

Nj =K2C.

Aggregating over all firms, the aggregate labour demand function willnow be given by:

ND = S1N1 + S2N2

= S1K1C+ S2

K2C

=1C[S1K1 + S2K2]

=K̄C



Note that the labour demand function that we have obtained byaggregating over firms with unequal distribution of capital is exactlyidentical to the labour demand function that we had obtained earlierwhen total capital stock was divided equally across all firms.Moreover, it also coincides with the demand for labour that wasderived by using the ‘ad-hoc’aggregate production function.The upshot of this exercise is that when the production function isCRS and there is perfect competition, the size of the firm does notmatter:

Different firms may employ different levels of capital and labour, butthe capital-labour ratio for every firm is identical.As a result when we aggregate over all firms, the micro-foundedlabour demand function and the corresponding supply curve thatwe derive by aggregating optimal decisions of ‘atomistic’firmsoperating under perfect competition would indeed be identical tothose obtained from the aggregative relationship specified in theClassical/Keynesian system.


Aggregate Production Function: Neoclassical or not?

The above analysis also highlights the importance of the assumptionof a ‘Neoclassical’technology.Marvin Frankel (an American economist) pointed out (AER, 1962)that even when each ‘atomistic’micro firm faces a productionfunction which is ‘Neoclassical’in nature, there is no reasonwhy the aggregate production function would also be strictly‘Neoclassical’, especially if the TFP term is endogenouslydetermined.Consider an economy with S identical firms - each having access toan identical firm-specific technology:

Yi = ĀF (Ki ,Ni ) ≡ Ā (Ki )α (Ni )1−α ; 0 < α < 1.

Note that the firm-specific production function exhibits all theneoclassical properties. The term Ā represents the current state ofthe technology in the entire economy, which is treated as exogenousby each ‘atomistic’firm.


Production Function: Neoclassical or not? (Contd.)

Frankel then relates the Ā term to the aggregate capitallabour-ratio in the economy - due to ‘knowledge spillovers’and‘learning by doing’:

Ā = g(KN

); g ′ > 0; where K = SKi ; N = SNi .

The idea is as follows:Productivity depends on how quickly workers can adapt themselves tonew machines. This is the process of learning by doing.When the aggregate capital stock in the economy is very high inrelation to its total labour stock, everybody gets greater opportunity tofamiliarise themselves with the machines; hence overall productivity ishigher.Moreover, there is knowlege spillovers - workers can learn from oneanother (without everybody spending time to go through theinstruction manuals).Both these factors would imply that Ā would be an increasing functionof the aggregate capital-labour ratio.



Without much loss of generality, let us assume:

g(KN

)= B

(KN

)β; β > 0,

Corresponding Aggregate Production Function (which can beobtained by summing over all firms):

Y = ∑Yi = S[Ā (Ki )

α (Ni )1−α]

= Ā (SKi )α (SNi )

1−α

= Ā (K )α (N)1−α .

Notice however that Ā is the total factor productivity term - which isgiven for each firm, but not so for the aggregate economy.Replacing the value of Ā in the aggregate production technology:

Y = Ā (K )α (N)1−α = B (K )α+β (N)1−α−β .



Notice that the aggregate production technology is indeed‘Neoclassical’only in the special case where α+ β < 1, but nototherwise!To carry forward the micro-founded production relations to theClassical/Keynesian aggregative production side story, we musttherefore assume that there is no such spill over/externality at theaggregate level.(Is that empirically true? We shall attempt to answer this questionlater when we discuss the empirics of output dynamics).


Micro Foundations: Neo-Keynesian Production Side Story

The production side story becomes a little more interesting for theNeo-Keynesian case.

Recall that in this case the output supply function is perfectly elasticat some price level P̄.

What kind of micro-founded firm-side story would support thisaggreagtive behaviour?

It is obvious that we now have to move away from the perfectlycompetitive set up (i.e., price-taking behaviour by firms) and allow forsome form of market imperfection (i.e., price-setting behaviour byfirms).

An obvious way to motivate this is to assume that the productionfunction exhibits IRS (increasing returns to scale) such that perfectcompetition is not sustainable in equilibrium and monopoly emergesas the natural outcome.

This is the route that we are going to follow now.


Neo-Keynesian Production Side Story: (Contd.)

The simplest production function that exhibits IRS is a linear onewith a fixed cost.Let the technology be represented by the following productionfunction:

Y ={

0α (N − F̄ )

if N 5 F̄if N > F̄

where α is the constant marginal product of labour employed in theactual production process and F̄ is the fixed cost defined in terms ofunits of labour.This production function implies that F̄ quantity of labour is requiredto set up the production unit before actual production can take place.Thereafter every additional unit of labour employed produces α unitsof output.(We are ignoring the role of capital for the time being, but capital canbe easily brought in either as a part of the fixed cost or the variablecost)Question: Why is this production function IRS?



Production is now carried out by a monopolist producer who knowsthe exact demand schedule.Given the demand function, the monopolist producer optimallychooses the price level to maximise his profit:

Max .{P}

Π = P.Y D (P)−WN.

Notice that to produce Y D amount of output, the monopolist

producer has to employY D

αunits of labour in actual production. In

addition, he has to employ F units of labour to set up the productionunit.

Thus, N =Y D

α+ F . Plugging this in the optimization problem of the

monopolist:

Max .{P}

Π = P.Y D (P)−W[Y D (P)

α+ F

].



Corresponding FONC:

dΠdP

= Y D (P) +[P − W

α

]dY D

dP= 0

⇒[P − W

α

]= −Y

D (P)dY DdP

⇒[P − Wα

]P

= −YD (P)dY DdP P

=1e

where e ≡ − dY DdPPY D is the price elasticity of demand.

Rearranging:

P =Wα

(e

e− 1

)



In other words, if the nominal wage is constant and the demandfunction exhibits constant price elasticity of demand (which is greaterthan unity), then the monopolist producer will indeed optimallycharge a constant price level P̄ irrespective of the level of demand:

P =Wα

(e

e− 1

)Notice however that the level of profit earned by the monopolist is:

Π = P.Y D (P)−W[Y D (P)

α+ F

]=

(P − W

α

)Y D (P)−WF

=

(1

e− 1

)WαY D (P)−WF

Hence the monopolist will operate iff the demand is suffi ciently high:

Y D = α (e− 1) FDas (Lecture Notes, DSE) Macro Jan 22-25; 2019 27 / 41


From now on we shall assume that demand is high enough(presumably because G is suffi ciently high); otherwise productionprocess will crash to zero.

Under that condition, the monopolist firm always earns a positiveprofit.

But this generates an additional conceptual problem, which is at oddswith the way we had specified our macro frameworks earlier.

Recall that we had assumed that the entire output produced in theeconomy eventually goes back to the households who are the ownersof all factors of production.

That had completed the circular flow of income for the economy,which allowed us to write the consumption demand (coming from thehouseholds) as a function of the total income (Y ).



It also fitted well with a perfectly competetive market structureswhere the firms in equilibrium were earning zero profits. After payingboth the factors their respective returns, there was nothing left withthe firms.This also explains why the firms had to borrow when the wanted toinvest and hence the borrowing cost (r) was an importantdeterminant of the investment demand.Now, with a monopolist producer which is earning positive profit, theneat chain of reasoning is broken:

Even if the households collective own the firm (as shareholders), whatproportion of that profit is given back to households as dividend andhow much is retained)?If the firm can retain at least part its profit, then in order to invest whymust it borrow from the market?How can consumption be a function of aggregate output (Y ) if a partof the output (retained profit) does no go back to the households?How can investment be a function of the borrowing cost (r) alone?


Micro-Founadtions: Top Down vis-a-vis Bottom UpApproach

It is not easy to answer these questions without simultaneously askingdeeper questions (and making further assumptions about) the basic(ad-hoc) macro structure that we had assumed earlier.And if we try to do that, these micro-foundations begin to look just as‘ad-hoc’as the ‘ad hoc’behavioural equations that we started with.Instead of trying to provide micro-foundations from top down, aninternally consistent and more logical approch would be to build thesemodels from the bottom (at the micro-level) leading all the to the top(at the macro level).And that is exactly what the more recent (DGE/DSGE) macromodels do: they build the entire macro framework from first principles- specifying the market structure and optimization problem of thefirms/households to simultneously arrive at their (and therefore theeconomy’s) supply/demand, employment/leisure,consumption/savings decisions.


Micro-foundations: An Empirical Justification

Dynamic General Equilibrium (DGE) models are also immune to theLucas’Critique. In fact they originated as a response to Lucas’Critique.As we have already discussed, traditionally, macroeconomic analysiswas based on some aggregative behavioural relationship (e.g.,Keyensian Savings Function - which postulates a relationship betweenaggregate income and aggregate savings; Labour Demand curvewhich postulated a negative relationship between employment andreal wages.).Often one would construct detailed behavioural equations for themacroeconomy and would try to estimate the parameters of theseequations using time series data to come up with estimated values ofthe coeffi cients..To be sure some of these equations would be dynamic in nature,entailing some ‘assumed’beliefs about futute. But optimization overtime was not considered to be important or even relevant.


Lucas Critique: Optimization Comes to the Fore

Lucas (1976) argued that aggregative macro models which areestimated to predict outcomes of economic policy changes are uselesssimply because the estimated parameters themselves depend on theexisting policies. As the policy changes these coeffi cients themsleveswould change, thereby generating wrong predictions!

His solution was to build macroecnomic models with clear andspecific microeconomic foundations - models that are explicitly basedon households’optimization exercises.


Lucas Critique: Optimization Comes to the Fore (Contd.)

Such models will be based on true parameters - primitives like tastes,technology etc - which are independent of the government policies.

Moreover such models would explicitly take into account agents’expectations about government policies and how those expecationsmay change as policies change..

Predictions based on such microfounded models would be moreaccurate than the aggregative models which club all these trueparameters as well as other policy-dependent parameters together.


Micro Foundations: Household Side Story - TheConsumption Function

We now turn to the last example of providing ad-hoc microfoundations: the micro-foundations of the consumption function inthe Classical/Keynesian system.In the process I shall also illustarte how Lucas’critique works.Recall that we had specified the aggregate consumption function as:C (Y ); 0 < C ′(Y ) < 1. The marginal propensity to consume out ofincome (C ′(Y )) less than unity implies that households don’tconsume their entire income; they save a part.Accordingly, the counterpart of the Keynesian consuption function isthe Keynesian savings function: S(Y ) ≡ Y − C (Y ); 0 < S ′(Y ) < 1.Let us consider a linear from of the Keynesian savings function:

St = α1 + α2Yt + et

An aggregative macro model would take the above behaviouralrelationship as given and would estimate the coeffi cients α1 and α2from data.


Micro-foundation of Keynesian Savings Function:

Let us now try to retrieve this postulated aggregative relationshipfrom households’optimization exercise.

In doing so notice that

Since we are modeling households’savings behaviour and typicallysavings are done for the purpose of future consumption, we must definethe optimization problem over consumptions on at least two dates -current and future.Since future consumption would also depend on future income andfuture prices which are currently unknown, expectations immediatelyenter into the picture.


Micro-foundation of Keynesian Savings Function: (Contd.)

Assume that the economy consists of a finite number (H) ofhouseholds. Each household is indexed by h ∈ [1, 2, ...,H ].Let us define a 2-period utility maximization problem of the h-thhousehold as:

Max .{cht ,cht+1}

log(cht ) + β log(cht+1)

subject to,

(i) cht + sht =

I htPt;

(ii) cht+1 = (1+ ret+1)s

ht +

(I ht+1

)ePet+1

.

where I ht and(I ht+1

)eare the current and expected future income of

the household in nominal terms.



From (i) and (ii) we can eliminate sht to derive the life-time budgetconstraint of the household as:

cht +cht+1

(1+ r et+1)=I htPt+

(I ht+1

)ePet+1(1+ r

et+1)

From the FONCs:cht+1βcht

= (1+ r et+1).

Solving we get:

cht =1

(1+ β)

[I htPt+

(I ht+1

)ePet+1(1+ r

et+1)

]



Thus

sht =β

(1+ β)I htPt− 1(1+ β)

[ (I ht+1

)ePet+1(1+ r

et+1)

]Aggregating over all households:

St ≡H

∑h=1

sht =β

(1+ β)

H∑h=1

I ht

Pt− 1(1+ β)

H∑h=1

(I ht+1

)e(1+ r et+1)

Noting that

H∑h=1

I ht

Pt≡ Yt (the aggregate current output in the

economy) and

H∑h=1(I ht+1)

e

P et+1≡ Y et+1, one can write the above

micro-founded aggregate savings function as:

St =β

(1+ β)Yt −

1(1+ β)

[Y et+1

(1+ r et+1)

]Das (Lecture Notes, DSE) Macro Jan 22-25; 2019 38 / 41


Note that the above micro-founded savings function would indeedresemble the postulated Keynesian savings behaviour providedhouseholds’expecations about future variables ( as captured byY et+1

(1+r et+1)) remain constant over time.

Indeed an aggregative model would equate β(1+β) to α2 and

− 1(1+β)

[Y et+1

(1+r et+1)

]to α1.

While the coeffi cient α2 is based on true parameters (primitives) andwould therefore be unaffected by policy changes, coeffi cient α1 is not.

Any policy that changes the household’s expectations about its futureincome or future rate of interest rate would affect α1.

Thus predicting outcomes of such a policy change based on theestimated values of the aggregative equations (derived from datawhen the policy was not yet announced) would be wrong.


Ad Hoc Micro Foundations: A Critique

In order to justify the aggregative macro systems (of either type) wehave now constructed a variety of different micro-foundations, each ofwhich would justify only a few equations of the aggregative systemunder special assumptions; they do not simultaneously consider allequations of the system.

This approach to provide micro-foundations seems just as ad hoc asthe aggregative macro systems themselves!!

More importantly, each micro-foundation is based on certain set ofassumptions and there is no obvious reason why all these differentassumptions specified for different types of agents (households, firms)will be internally consistent with one another!

As we had noted earlier, the micro-foundations for a neo-Keynesian ASequation (based on a single monopolist producer) may clash with themicro-foundations of an aggregate consumption function or investmentfunction!!


Ad Hoc Micro Foundations: A Critique (Contd.)

A logical apparoch is to specify an internally consistent and unifiedgeneral equilibrium set up in which individual decisions of all agentsare based on their respective optimization exercises and theseindividual decisions are then coordinated through the markets tocharacterize the macroeconomy.

That is exactly the approach that modern dynamic generalequilibrium (DGE) macroeconomic theory follows, which we take upas our next topic.


Micro Foundations of Macroeconomic Equations:

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004: Macroeconomic Theoryecondse.org/wp-content/uploads/2019/01/004-Micro...Das (Lecture Notes, DSE) Macro Jan 22-25; 2019 12 / 41 Classical/Keynesian Production Side Story (Contd.)