004: Macroeconomic Theory

004: Macroeconomic Theory

Lecture 11

Mausumi Das

Lecture Notes, DSE

October 14, 2014

Micro Foundations of Aggregate Macro Equations

In our analysis so far we have taken the aggregate macroeconomic equations depicting the demand and supply sides of various markets (e.g., the labour market, goods market and money market) as given.

Of course some of these equations di¤er - depending on whether one is talking about the Classical System, Keynesian System or the Neo-Keynesian System.

However since all these demand and supply functions are outcomes of actions taken by individual agents (namely, …rms and households), ideally we should be able to derive each of these from the ‘optimal decisions’undertaken by …rms and households at the micro level.

Without such explicitmicro-foundations, the behavioural macroeconomic equations described by any system would remain arbitrary and ad hoc.

We now discuss the speci…c micro-foundations of these equations.

Micro Foundations: The Labour Market

Recall that

The Labour Market under the Classical System:

Demand Equation:

W =PF_N(N,K¯) Supply Equation:

W =Pg(N);g⁰(N)>0 While

The Labour Market under the Keynesian System:

Demand Equation:

W =PF_N(N,K¯) Supply Equation:

W =W^¯

Micro Foundations: The Labour Market (Contd.)

The demand for labour comes from the …rms, who are also the producers of the …nal commodity (i.e., the suppliers in the goods market).

The supply of labour on the other hand comes from the households, who are also the consumers of the …nal commodity (and thus constitute a part of the demand side in the goods market; the other two parts being the …rms themselves (investment demand) and the government (government consumption/investment demand).

Thus the demand side in the labour market and the supply side in the goods market are determined through the same optimization exercise by the …rms.

Likewise, the supply side in the labour market and the consumption demand in the goods market are determined through the same optimization exercise by the households.

Demand for Labour & Supply of Goods (Firm-side story):

Suppose the goods market is charaterised byS identical …rms indexed by i.

Let the market structure be perfectly competitive(i.e, …rms are price takers).

Technology available to any …rmi:

Yi =F(Ni,K¯i);F_N,F_K >0;F_NN,F_KK <0.

We also assume that the …rm-speci…c production function satis…es the following two additional properties:

(i) CRS : F(λN_i,λK¯_i) =λF(N_i,K¯_i);

(ii) Inada Conditions : 8>

><

>>

:

NLimi!⁰F_Ni = Lim

Ki!⁰F_Ki = ∞;

NLim_i!∞F_Ni = Lim

K_i!∞F_Ki =0.

Firm Side Story (Contd.):

The …rms take the capital stock as given in the short term.

Thus the most important production decision entails how much labour to employ, which in turn determines the quantity of …nal goods supplied by the …rm.

In a perfectly competitive set up, the optimization problem of the i-th

…rm:

Max.Πi f^Nig

=PY_i WN_i =PF(N_i,K¯_i) WN_i. First order condition of the pro…t maximization exercise yields:

∂Πi

∂Ni

=0)^PFNi(Ni,K¯i) =W.

This looks similar to our familiar labour demand equation, except that it is the labour demand for thei-th …rm, while the macro labour demand equation relates to the aggregate demand for labour in the economy.

Firm Side Story (Contd.):

.

Note however that with identical technology, identical capital stock and facing the same real wage rate, all …rms would employ exactly the same amount of labour:

Ni : ∂Yi

∂N_i F_Ni(Ni,K¯i) = ^W

P for all i. (1) (The Inada Conditions ensure that a unique solution toNi exits for any given ^W_P ).

Thus aggregating over all …rms, we get the aggregate demand for labour as:

N =

∑

S i=1

Ni =SNi.

Notice that we have not been able to get rid of the …rm-speci…c subscripti yet. We cannotsimply sum equation (1) over all …rms to arrive at the aggregate labour demand. (Why not?) We shall come back to this point in a moment.

Firm Side Story (Contd.):

Let us now look at the corresponding supply of goods.

Each …rm supplies

Y_i =F(N_i,K¯_i).

Hence summing over all …rms, we get the aggregate supply function as :

Y =

∑

S i=1

Y_i =SY_i =SF(N_i,K¯_i) =F(SN_i,SK¯_i)(using CRS) ) ^Y =F(N,K¯).

This immediately gives us the aggregate supply of the …nal good in the economy as a function of total labour employed (N) and aggregate capital stock in the economy (K¯) - exactly as has been speci…ed in the macro equations.

Firm Side Story (Contd.):

Now let us go back to the aggregate labour demand equation.

Recall that we have notbeen able to generate the aggregate labour demand function as a relationship between the aggregate employment (N) and the real wage rate (W/P) by simply aggregating over similar equations for all …rms.

Having the aggregate production function as the sum of all individual production function now allows us to make some progress in that direction.

Notice that from the aggregate supply equation, we can derive:

∂Y

∂N_i =F_N(N,K¯)^∂N

∂N_i =F_N(N,K¯)^∂(SNi)

∂N_i =SF_N(N,K¯). Also sinceY =SY_i, we can write the same derivative as:

∂Y

∂N_i = ^∂(SY_i)

∂N_i =S∂Y_i

∂N_i =SF_Ni(N_i,K¯_i).

Firm Side Story (Contd.):

From the RHS of the above two derivations, it follows that:

F_N(N,K¯) =F_Ni(N_i,K¯_i)for all i.

But by equation (1), we already know that each of this …rm-speci…c derivativesF_Ni(N_i,K¯_i) are equal to ^W_P.

This immediately tells us that a similar realtionship must hold for F_N(N,K¯)as well. That is,

F_N(N,K¯) = ^W P

) ^PFN(N,K¯) =W.

Thus we have now been able to genearte the aggregate demand for labour - exactly as has been speci…ed in the macro systems.

(Question: Can we not obtain this relationship directly by di¤erenting the aggregate production function and equating the corresponding ‘social’

marginal product of labour to the real wage rate? Why not?)

Supply of Labour & Consumption Demand for Goods (Household Side Story):

Suppose there are H identical households indexed by h - each constisting of a single member.

Households derive utility from consumption as well as from leisure (L).ˆ Let us assume that household members live exactly for two periods:

so they care about their current period consumption (C₁) and their future consumption (C₂).

The second period consumption has been introduced to account for savings. (Otherwise why would households save?)

Utility function of any householdh:

U(C_1h,C_2h,Lˆ_h) u(C_1h) +βu(C_2h) +v(L^ˆ_h); u⁰,v⁰ >0; u⁰⁰,v⁰⁰ <0.

The constant β2(0,1)is the discount factor representating households’bias towards present consumption vis-a-vis future consumption (positive rate of time preference).

Household Side Story (Contd.):

A member of any householdh works only in the …rst period of his life.

He has a …xed endowment of total time in the current period, normalized to unity.

Out of this time endowment, he optimally decides how much to spend on leisure (Lˆ_h) and how much to work (1 Lˆ_h).

Working generates a current wage income at the market wage rate W (per unit of time): W 1 Lˆ_h .

In addition, the household may also have some other sources of …rst period income (e.g, pro…ts, rents, interests etc.), which we club under the generic term y_1h and treat as exogenous.

Household Side Story (Contd.):

The household spends its total …rst period earnings on current consumption (C_1h) and savings (S_1h), which generates the current period budget constarint of the household as follows:

PC_1h+S_1h =W 1 Lˆ_h +y_1h.

Since in the second period the agent does not work, he has no wage income in the second period. His second period consumption

therefore can come only from the interest earnings on his …rst period savings and other sources of second period income (e.g, pro…ts, rents etc.), which we again club under the generic termy_2h and treat it as exogenous.

This generates the second period budget constraint of the household as follows:

PC˜ _2h = (1+r˜)S_1h+y_2h.

(P˜ andr˜ are the next period’s (expected) price level and interest rate respectively -treated as exogenous in the current period.)

Household Side Story (Contd.):

Note that we can eliminateS_1h from the two budget equations to derive the life-time budget constraint of the agent as

PC_1h+ ^{PC˜ 2h}

(1+r˜)+WLˆ_h = W +y_1h+ ^y2h (1+r˜) i.e.,C_1h+ ^P˜

P

C_2h

(1+r˜)+ ^W

P Lˆ_h = ^W

P + ^y1h P + ^y2h

(1+r˜)P (Interpretation?)

The agent maximizes his utility, given by:

U(C_1h,C_2h,Lˆ_h) u(C_1h) +βu(C_2h) +v(L^ˆ_h), subject to the the above life-time budget constraint.

Labour Supply by the Households:

The FONCs:

(i)u⁰(C_1h) = λP; (ii)βu⁰(C_2h) = ^λP˜

(1+r˜)^; (iii) v⁰(L^ˆh) = λW Let us assume that u(C) =logC.

Then the FONCs would reduce to:

(i⁰) C_2h

C_1h = β(1+r˜) ^P P˜ ; (ii⁰) 1

C_1h.v⁰(L^ˆ_h) = ^P W.

Labour Supply by the Households (Contd.):

Using FONC (i⁰) in the life-time budget constraint of agent, we get:

(1+β)C_1h+ ^W

P Lˆ_h = ^W

P + ^y1h

P + ^P˜ P

y2h

(1+r˜)P^˜ i.e., (1+β)C_1h+ ^W

P Lˆ_h = ^W

P +yˆ_h (say) (2)

whereyˆ_h ^y_P^1h + ^P_P^˜ ₍₁₊^y2h

˜

r)P^˜ represents the entire present discounted stream of non-labour income of the agent in real terms.

The above equation can be interpreted as a reduced-form life time budget contraint, which de…nes the resource constraint of the household in terms ofC_1h andLˆ_h alone (when C_2h has already been optimally determined.)

Labour Supply by the Households (Contd.):

Likewise, Using FONC (i⁰) in the utility function fo the agent, we get the reduced-form utility function in terms of C_1h and Lˆ_h alone

(assuming C_2h has already been optimally determined) as follows:

U(C_1h,C_2h,Lˆ_h) = logC_1h+βlog β(1+r˜) ^P

P˜ C_1h +v(L^ˆ_h)

= K+ (1+β)logC_1h+v(L^ˆ_h), (3) whereK βlogh

β(1+r˜) ^P_P˜ ⁱ is a known constant.

We can now use the reduced-form utility function (3) and the

reduced-form budget equation (2) to analyse the household’s opmital choice between current consumption and leisure.

In particular, we are interested to know how the household’s optimal labour supply (1 Lˆ_h) responds to changes in the current real wage (^W_P).

We shall take up this issue in the next class.