PDF Lecture 12 Mausumi Das - Delhi School of Economics

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004: Macroeconomic Theory

Lecture 12

Mausumi Das

Lecture Notes, DSE

October 16, 2014

Das (Lecture Notes, DSE) Macro October 16, 2014 1 / 25

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Household Side Story: Labour-Leisure Choice &

Consumption-Savings Choice

In the last class we had de…ned the utility maximization problem of householdh as:

Max.

f^C^1h^,C^1h^,ˆ^L^hg^log(C_1h) +βlog(C_2h) +v(L^ˆ_h), subject to,

C_1h+ ^P^˜ P

C_2h

(1+r˜)+ ^W

P Lˆ_h = ^W

P + ^y^1h

P + ^y^2h P(1+r˜)^, which yielded the following FONCS:

(i⁰) C_2h

βC = (1+r˜) ^P P˜ ;

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Labour Supply by the Households:

Then using FONC (i⁰) in the life-time budget constraint of agent, we got the reduced-formlife-time budget contraint in terms ofC_1h and Lˆ_h as follows:

(1+β)C_1h+ ^W

P Lˆ_h = ^W

P +yˆ_h, whereyˆ_h ^y_P^1h + ^P_P^˜ ₍₁₊^y^2h

˜

r)P^˜ is a known constant.

On the other hand, using FONC (i⁰) in the utility function fo the agent, we got the reduced-form utility function in terms ofC_1h and Lˆ_h alone as follows:

U(C_1h,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), whereK βlogh

β(1+r˜) ^P_P_˜ ⁱ is a known constant.

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Labour Supply by the Households (Contd.):

We are now going to use these two reduced-form expressions to derive the optimal labour-leisure choice of the household.

Since the functional form of v(L^ˆ_h)has not been speci…ed (except that it has been assumed to be concave), we cannot derive explicit solutions.

However, we can still analyse it diagrammatically and comment on the nature of the relationship, depending on various assumptions about the utility function.

We now plot the utility function and the budget function (for given values ofW,P,r˜ andP˜), withC_1h on the vertical axis andLˆ_h on the horizontal axis.

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Labour Supply by the Households (Contd.):

Slope of the reduced-form utility function:

dCh

dLˆ_{h U}₌U¯

= ¹

(1+β)^C^h^v

0(L^ˆh).

Slope of the reduced-form budget line for a given real wage rate

W P 0:

dC_h dLˆ_h ^W

P=(^WP)₀

= ¹

(1+β) W

P ₀. Moreover, from the budget equation,

whenLˆ_h =0,C_h=

W P +yˆ_h (1+β) ^; whenC_h=0,Lˆ_h=1+ ^y^ˆ^h

W P

.

However, sinceLˆ_h 1, the other corner of the budget line is reached atLˆ_h =1, where C_h = ^y^ˆ^h

(1+β) >0.

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Equilibrium Labour-Leisure Choice for a Given Real Wage Rate:

Let the equilibrium occur at the tangency point of the two curves (interrior optima), as denoted by E₀ in the diagram below.

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Equilibrium Labour-Leisure Choice for an Increase in Real Wage Rate:

We are interested to know how household’s optimal choice of leisure (& therefore labour supply) changes when there is a change in the real wage rate.

Suppose the real wage rate now increasesto a higher value ^W_P ₁. There are two associated changes in the budget line:

It becomes steeper;

Its intercept with the vertical axis increases too.

In other words, the real income of the household increases; at the same time the relative price of leisure increases too.

Thus there will be two opposite impacts on the leisure choice:

Income e¤ect would imply that the household would like to enjoy more leisure (assuming leisure is a ‘normal’good);

Substitution e¤ect would imply that the households would like to enjoy less leisure.

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Equilibrium Labour-Leisure Choice when Substitution E¤ect Dominates:

The optimal leisure choice depends on which e¤ect dominates.

In what follows, we shall assume that the substitution e¤ect dominates the income e¤ect (which requires additional restrictions on thev function).

Under this assumption,as the real wage rate goes up, optimal value of leisure falls, as shown in the diagram below.

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Households’Labour Supply Equation:

Thus (when all these assumptions hold) we can write the household’s optimal leisure choice as a negative function of W

P ,and corresspondingly its labour supply as a positive function of W

P : Nˆ_h 1 Lˆ_h =f W

P ; f⁰ >0.

Aggregating over all households we get the aggregate labour supply function as:

N =

∑

H h=1

Nˆ_h =H.Nˆ_h.

Since Nˆ_h is an increasing function of the real wage rate for every householdh,the aggregate labour supply must be an increasing function of the real wage rate too:

N =h W

P ;h⁰ >0.

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Classical Labour Supply Equation:

Inverting the above function, we retrieve the classical labour supply equation for the aggregate economy:

W

P =h ¹(N) g(N); g⁰ >0.

In other words,

W =P.g(N); g⁰ >0.

This is precisely the equation that was speci…ed under the Classical system earlier.

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Household’s Consumption-Savings Choice:

After determining the optimal leisure choice of the household, let us now go back to its consumption-savings choice.

Note that since Lˆ_h is now optimally determined, we know precisely the total income of the household (wage as well as non-wage income):

yh = ^W

P 1 Lˆ_h +yˆh

= ¹

(1+β) 2 66 66 4

W

P 1 Lˆ_h + ^y^1h

| {z P }

¯ y1h

P

+ ^y^2h P(1+r˜)

3 77 77 5

where ^y^¯_P^1h represents the agent’s total …rst period income - labour as well as non-labour (in real terms).

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Household’s Consumption-Savings Choice (Contd.):

Then from the budget constraint, current consumption of the household:

C_1h = ¹ (1+β)

¯ y_1h

P + ¹

(1+β) y_2h P(1+r˜)

= ¹

(1+β)

¯ y1h

P +C^¯h

(Marginal propensity to consume out of current income: ₍₁₊¹

β) <1.) [Question: How about the corresponding Savings function?]

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Aggregate Consumption Function:

Since all households are identical they enjoy the same consumption level.

Thus aggregating over all households, aggregate consumption function in the economy:

C =

∑

H h=1

1 (1+β)

¯ y1h

P +C^¯h = ¹ (1+β)

∑

H h=1

¯ y1h

P +HC¯h

= ¹

(1+β)^Y +C^¯

whereY represents aggregate income.

Thus indeed we have got back the aggregate consumption function, as speci…ed in the macro systems:

C =C^¯ +cY,

wherec = ₍₁₊¹_β₎ is the MPC for the aggregate economy.

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Keynesian Labour Supply Equation: Micro Foundations

We have now retrieved the Classical labour supply equation from the corresponding micro behaviour of agents.

How about the Keynesian labour supply function?

Recall that the Keynesian labour supply is perfectly elastic (at a given nominal wage rate).

To generate that, we have to change the speci…cation of the utility function a bit.

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Keynesian Labour Supply Equation (Contd.):

Let us go back to the optimization problem of the household, as speci…ed at the very beginning:

Max.U(C_1h,C_2h,Lˆ_h) subject to

C_1h+ ^P^˜ P

C_2h

(1+r˜)+ ^W

P Lˆ_h = ^W

P + ^y^1h

P + ^y^2h P(1+r˜)^. Let us now assume the following speci…cation of the utility function:

U(C_1h,C_2h,Lˆ_h) (C_1h)^β.(C_2h)¹ ^β+v Lˆ_h , 0< β<1.

Note that there is no explicit discount factor anymore. However the powers βand (1 β) capture the weightage given to current consumption and future consumption respectively.

In particular, β> ¹₂ would imply a bias towards current consumption.

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Keynesian Labour Supply Equation (Contd.):

The new speci…cation yields the following FONCs:

(i⁰⁰) βC_2h

(1 β)C_1h = (1+r˜) ^P P˜ ; (ii⁰⁰) β(C_1h)^β ¹.(C_2h)¹ ^β

v⁰(L^ˆ_h) = ^P W.

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Keynesian Labour Supply Equation (Contd.):

As before, using FONC (i⁰⁰) in the life-time budget constraint of agent, we get thereduced-formlife-time budget contraint in terms of C_1h and Lˆ_h as follows:

1

βC_1h+ ^W

P Lˆ_h = ^W

P +yˆ_h, whereyˆ_h ^y_P^1h + ^P_P^˜ ₍₁₊^y^2h

˜

r)P˜ is a known constant.

On the other hand, using FONC (i⁰⁰) in the utility function fo the agent, we get thereduced-form utility function in terms ofC_1h and Lˆh alone as follows:

U(C_1h,Lˆ_h) = (1 β)

β (1+r˜) ^P P˜

1 β

C_1h+v Lˆ_h )

= K C^ˆ _1h+v(L^ˆ_h), whereKˆ ⁽¹ ^β⁾

β (1+r˜) ^P_P_˜ ¹ ^β is a known constant.

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Keynesian Labour Supply Equation (Contd.):

Using the budget constraint to eliminate C_1h,we get the following unconstrained optimization problem, de…ned in terms of Lˆ_h alone:

Max f^L^ˆ^hg^.

Kˆβ yˆ_h+ ^W

P (1 Lˆ_h) +v(L^ˆ_h). We now make some further assumptions about the v function.

Recall that earlier we had speci…ed the v function as concave:

v⁰ >0;v⁰⁰<0.

Now we change that assumption and make the v function linear in Lˆ_h :

v(L^ˆ_h) =γLˆ_h,

where γ>0 is the constant marginal utility associated with leisure.

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Keynesian Labour Supply Equation (Contd.):

Notice that the new objective function is given by:

Max f^L^ˆ^hg^.

Kˆβ yˆ_h+ ^W

P (1 Lˆ_h) +γLˆ_h.

From the speci…cation of the objective function, it is clear that the household would enjoy zero leisure and be willing to work full-time for any real wage rate ^W_P =^γ.

The corresponding labour supply curve will be perfectly elastic at γ, and will be vertical (at N_h =1) thereafter.

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Keynesian Labour Supply Equation (Contd.):

Aggregating over all households, we get an aggregate labour supply schedule which looks very similar:

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Keynesian Labour Supply Equation (Contd.):

We have almost derived the perfectly elastic Keynesian labour supply equation - except that the labour supply function derived by us is horizontal at the real wage rate γ, not a nominal wage rate.

In fact as long households care for real values of goods and services, the labour supply decisions will depend on real wage rate (not withstanding its slope) - unless we impose some additional

‘behavioural’assumptions on part of the households (e.g., money illusion; constant price expectations etc.), which are not derived from the utility maximization exercise of rational agents.

In other words, the Keynesian system - as is speci…ed in terms of macro equations - cannot be fully retrieved from the optimizing behaviour of perfectly rational agents.

This is one of the weaknesses of the Keynesian system. This is also a primary source of critisism of the Keynesian macro system, which is why increasingly macro models start from optimizing agents and then go to the aggregative macro analysis, rather than it being the other way round.

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Consumption & Savings Functions under the Keynesian Micro Speci…cation:

Once the agents make their labour-leisure choice, once again the income of the household is fully determined.

As before we can the determine the value ofC_1h from the corresponding budget equation:

1

βC_1h = ^W

P (1 Lˆ_h) +yˆ_h =y_h ) ^C1h =βy_h.

Proceeding ws before we can break up the terms within y_h to write current consumption as a function of current income and a constant

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Keynesian Consumption & Savings Functions:

Aggregating over all households, we shall again get back the aggregate consumption function, as speci…ed in the macro systems:

C =cY +C^¯;

wherec =β is the MPC for the aggregate economy.

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Two Comments & Two Questions:

The consumption and savings functions that we have derived here - which depend linearly on aggregate income - is very much an artifact of the logarithmic/Cobb-Douglas form of theu function that we have assumed here. Even a slight generalization of theu function would not generate this result.

To see that, suppose the utility function is given by:

U(C_1h,C_2h,Lˆ_h) (C_1h)¹ ^σ

1 σ +β(C_2h)¹ ^σ

1 σ +v Lˆ_h , σ6=1 ; 0< β<1.

This above utility function (called a CRRA variety of utility function) is a close cousin of the logarithmic utility function and has many similar properties (including the fact that the ratio of marginal utilities of C_1h and C_2h is a function of the ^C_C^1h

2h ratio). And yet the resulting consumption function of a household will be di¤erent from

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Two Comments & Two Questions (Contd.):

Finally, we have seen that micro foundations for the Keynesian system generated a labour supply schedule whicj is perfectly elastic at some real wage rate γ, and becomes vertical (atN =H) for any ^W_P >γ.

We can call this scenario a case of ‘real wage rigidity’rather than the Keynesian ‘nominal wage rigidity’.

Draw the corresponding AS schedule for this labour supply function. (Will it still be upward sloping?)

Comment on the e¤ectiveness of monetary and …scal policies in this case.