202: Dynamic Macroeconomic Theory

202: Dynamic Macroeconomic Theory

History versus Expectations:

Murphy-Shleifer-Vishny

Mausumi Das

Lecture Notes, DSE

Oct 16,Oct 30; 2015

History vis-a-vis Expectations in the Process of Development:

The growth/development literature in general attributes observed di¤erences in the growth trajectories of di¤erent countries to various historical factors.

History shows up either in terms of an initial condition (e.g., initial capotal stock, initial distribution of wealth) or in terms of an institutional set up that the eceonomy has inherited (e.g., nature of the …nancial or political institutions).

There is however a third strand of the litearture which focuses on the role of expectations in determining the development trajectory of an economy.

It has been observed that countries with similar history sometimes follow divergent growth paths. This latter strand of the literature attempts to explain these cases.

Role of Expectations in the Process of Development:

The basic theoretical underpinning of this literature is the

coordination failure models which result in multiple (Nash) equilibria.

The basic idea here is that even with a given history, an economy may face multiple possible growth trajectories due to existence of strategic complementarity among the choice variable of agents.

Among the many possible equilibrium trajectories, which one will be chosen depends crucially on agents expectations - what one believes about other’s choice of action.

Even with favouarble historical conditions, an economy mail fail to take o¤ simply because agents in this economy failed to coordinate on their actions.

Such cases highlight a special role of the government in terms of coordinating the actions of various agents to generate an outcome that is pareto e¢ cient.

Role of Expectations in the Process of Development:

(Contd.)

We shall discuss two papers that look at the process of development as a coordination game:

Murphy, Shleifer, Vishny: Industrialization and the Big Push (Journal of Political Economy, 1989)

Krugman: History Versus Expectations (Quaterly Journal of Economics, 1991)

The Murphy-Shleifer-Vishny paper explores a static model of coordination failure.

The Krugman paper starts with a static model and then extends it to a dynamic framework which allows us to precisely charaterise the role of history vis-a-vis expectations.

We shall strat with the Murphy-Shleifer-Vishny model.

Role of Expectations in the Process of Industrilization:

Murphy-Shleifer-Vishny

The basic idea is as follows:

Suppose the economy has access to two types of technologies: an IRS modern technology and a CRS cottage technology.

The …rst technology is more productive - but it also entails a …xed cost. Thus a …rm will adopt this technology only if there is su¢ cient demand.

The level of aggregate demand on the other hand depends on the actions of other …rms:

if all …rms adopt the modern technology together then that generates income which in turn creates demand for each of the products to make the adoption of the modern technolgy viable;

if no other …rms adopt the modern technology, then there is no incentive for a single …rm to adopt the technology alone - since the resulting demand will not be su¢ cient to cover the …xed cost.

Murphy-Shleifer-Vishny: (Contd)

Existence of such demand complementarities across …rms thus generates possibilities of mutiple equilibria and the consequent role of self-ful…lling (rational) expectations:

If each …rm expects that others will adopt the modern technology, then it adopts the modern technology too - hence the economy embarks on the path of industriazation

If each …rm expects that others will not adopt the modern technology, then it does not adopt the modern technology either - hence the economy remains stuck wit home production.

However, Murphy-Shleifer-Vishny shows that presence of such strategic complementarities (in this case working through demand) is necessary but not su¢ cient to generate multiple equilibria.

In fact they construct two similar models - one of which exhibits multiple, the other does not.

Murphy-Shleifer-Vishny: Model I

Consider a static (one period) closed economy.

The economy consists of a continuum of population of measure 1, represented by the unit interval [0,1].

All agents/households in this economy are identical; so we can talk in terms of a representative agent. (Note that since total population has a measure of unity, average and aggregate values in this economy would be identical).

There exists a variety of …nal goods, represented by the continuum [0,1],such that each variety is represented by an indexq 2 [0,1]. The representative agent’s preference over all these varieties of …nal goods is de…ned by a Dixit-Stiglitz ‘Love for Variety’utility function:

U =

Z1

0

logx_qdq

Murphy-Shleifer-Vishny: Model I (Contd.)

Notice that this ‘Love for Variety’utility function has similal features as the ‘Love for Varity’production funvtion that we have seen bafore.

In particular, for any variety q, ∂U

∂x_q = ¹

x_q >0; ∂²U

∂x_q² = ¹

(xq)² <0.

Morever, as x_q !^0, ∂x^∂Uq !^∞.

These two features will ensure that as long as the varieties are associated with …nite prices, the agent will consume all the varieties.

Also, if the same price is charged for all the varieties then the agent will spread his income equally over all the varieties and consume equal amount of each.

In general, if the agent has incomey, then the optimization problem of the representative agent is given by:

Max. Z1

0

logxqdq subject to Z1

0

pqxqdq =y.

Model I: Production Side Story

Production on any …nal commodi requires only labour - no capital (simpli…cation).

Each …nal good sector - producing a particular varietyq has access to two types of technologies:

(i) A modern, highly productive, IRS mass-production technology - which entails a …xed set up cost of F units of labour. Once this set up cost has been incurred, every additional unit of labour emplyed in this sector generates αunits of …nal output of variety q, where α>1.

(Why is the modern technology IRS?)

(ii) A traditional, less productive, CRS cottage technology - which does not ential any …xed cost. Every unit of labour employed in this sector generates 1 unit of …nal output of variety q.

Model I: Market Structure

Each of the modern technology is operated by a monopolist …rm who sets its own price (given the demand) so as maximise pro…t.

However houselds hold ownership shares of these …rms so that a part of the monopoly pro…t is distributed to the households in the form of dividends.

The cottage technologies can either be operated by competitive …rms of can be produced at home using own labour.

We shall assume the latter.

Model I: Wages & Prices

When a variety is produced at home using the cottage technology, a unit of the …nal commdity is produced by a unit of labour. Thus implicit wage rate in the cottage sector (in tems of the …nal commodity) is equal to 1.

Let us take labour as the numaraire: w =1.

Then implicit price of each variety under cottage production is also eqaul to unity: pq =1.

The monopolist …rm in each sector on the other hand sets its price by looking at the demand.

Notice that given the utlity function of the agent, the demand for any varietyq is derived from the following equation:

1 xq

= λpq for all q ) ^pqxq = ¹

λ for all q

Model I: Wages & Prices (Contd.)

Plugging this in the budget equation of the agent:

Z1

0

1

λdq =y ) ¹ λ =y

Thus the representation agent’s demand for a varietyq is given by:

p_qx_q =y

Since the total measure of households is equal to unity, the total demand for each variety is also represented by the same equation:

x_q = ^y

p_q (1)

Notice that the price elasticity of demand of each variety is unity; so a monopolist would like to charge an arbitrarily high price level (close to in…nity) (Why?)

Model I: Wages & Prices (Contd.)

However because of the presence of the cottage sector, it cannot charge anything other than a price equal to 1. (Otherwise nobody will buy from the monopolist.)

On the other hand, the monopolist being a price-taker in the factor (labour) market will pay the same cottage wage rate of w =1.

This implies that irrespective of whether a variety is produced by a monopolist using the modern technology, or under cottage

production, the corresponding wage rate and the prices would be the same, given by:

w = 1;

p_q = 1 for all q.

However, since the cottage sector is less productive, the level of income would di¤er in the two cases. The higher is the proportion of varieties that are produced by the monopolists, the higher would be the aggregate output.

Model I: Demand-Pro…t Interlinkage

But there is an additional constraint as well: given that the monopolist has to incur a …xed cost, will he always operate?

The answer is: No.

Given the …xed cost, a monopolist will operate if and only if the total revenue is enough not only to cover the variable cost, but also the

…xed cost.

To put it di¤erently, a monopolit would operate if and only if his net pro…t is non-negative.

His pro…t depends on the level of demand - which in turn depends on the level of income earned by the households.

Recall (from equation (1)) that the demand for each varietyq 2[0,1] is given by:

xq =y (sincepq =1)

Accordingly, pro…t of a monopolist operating in sectorq is given by:

πq =x_q 1 αx_q F

Model I: Demand-Pro…t Interlinkage (Contd.)

Simplifying, we get the following relationship betweeny andπq: πq = 1 1

α y F

= ay F wherea α 1

α <1. (2) A monopolist would operate if and only if

πq = ⁰ ) ^y = ^F_a^.

Thus for a monopolist to operate in any sector, the corresponding demand (y)has to be su¢ ciently high.

Note that if the monopolist in any sector abstains from production then the correspoding demand is served by the cottage sector.

Model I: Pro…t-Demand Interlinkage

Notice however that there is a reverse linkage from pro…t to demand working though income.

Since a part of the pro…t income goes back to the households in the form of dividends, the households’income consists of the total wage bill + the share of pro…t that is distributed.

Let θ be the share of pro…t that is distributed back to the households.

Then aggregate household income (which is also the income of the representative household) is the given by:

y =W +θΠ

This circular linkage from Income ! ^Demand ! ^Pro…t !^Income creates the potential of multiple equilibria here.

Model I: Pro…t-Demand Interlinkage (Contd.)

Let each agent be endowed withL¯ units of labour, which he supplies inelastically to the market.

Recall that each unit of labour earns a wage rate ofw =1, irrespcetive of whether it is employed in cottage production of modern production.

Then

W =L.^¯

Also letn proportion of the sectors be operated by the respective monopolist producers. (The exact value of n will evetually be determined endogeneously within the model).

Then

Π=nπ_q

Putting these together, we get another relationship between y and πq:

y =L^¯ +θnπ_q (3)

Model I: Pro…t as a function of no. of modern sectors in operation

From (2) and (3), we can write aggregate income as a function ofn (the proportion of sectors which aremordernised):

y = L^¯ +θn(ay F) ) ^y(n) = ^{L¯ θnF}

1 θna (A)

Corresponding pro…t of each of the monopolist in operation:

πq = ay(n) F

) ^πq(n) =aL¯ θnF

1 θna F

) ^πq(n) = ^{aL¯ F}

1 θna (B)

Model I: Pro…t as a function of no. of modern sectors in operation (Contd.)

Equation (B) above shows that the pro…t of each potential

monopolist in operation depends on the proportion of modern …rms which are under operation.

Notice however that the denominator of the RHS is always positive (sinceθ andaare all positive fractions).

Thus the nature of the relationship depends crucially on the numerator.

In particular:

1 when aL¯ F >0, dπq(n) dn >0;

2 when aL¯ F <0, dπq(n) dn <0.

Thus there is a externality from one modern …rm to another: as the proportion of modernised …rms goes up, the pro…t of each modernised

…rm is a¤ected.

Model I: No. of Modern Sectors in Operation in Equilibrium

How many sectors will be modernised in equilibrium?

Recall that a monopolist will operate as long as he earns a non-negative pro…t.

Now there are two cases here:

In Case 1, aL¯ F >0.

We already know that in this case, dπq(n) dn >0.

Moreover,πq(0) =aL¯ F >0 andπq(1) = ^{aL¯ F} 1 θa >0.

Thus a monopolist will always be willing operate, quite independent of what valuentakes.

Therefore in equilibrium,n =1.

Model I: No. of Modern Sectors in Operation in Equilibrium (Contd.)

In Case 2, aL¯ F <0.

We already know that in this case, dπq(n) dn <0.

Moreover,πq(0) =aL¯ F <0 andπq(1) = ^{aL¯ F} 1 θa <0.

Thus a monopolist will stay away from operating, quite independent of what valuentakes.

Therefore in equilibrium,n =0.

Note that multiple equilibria would have been realized in Case 1 if we had πq(0) =aL¯ F <0 while πq(1) = ^{aL¯ F}

1 θa >0.But obviously that cannot happen here.

Also note that multiple equilibria can neven happenin Case 2, even when πq(0) =aL¯ F >0 and πq(1) = ^{aL¯ F}

1 θa <0.(Why?)

Model I: Limitations

Thus we see even when there are pro…t interlikages bewteen …rms working through demand externalities, that may not be su¢ cient to generate multiple equilibria.

Depending on the parametric conditions, we get two di¤erent equilibrium values of n. But the equilibrium value is unique.

Each monpolist …rm will decide either to operate or not operate - quite independent of what others are doing.

Thus there is no role of expecations here. Neither is there any scope for coordination-driven multiple equilibria.

This is because here the demand externality works only through the pro…t channel: if pro…t is positive to begin with, that creates more demand and therefore even higher pro…t - eventually leading to n =1. On the other hand, if pro…t is negative to begin with, that creates less demand and therefore even lesser pro…t - eventually leading ton =0.

Model I: Limitations (Contd.)

This tells us that for the multiple equilibria story to operate we need an additional channel of demand externality that works independent of the pro…t channel.

Murphy-Shleifer-Vishny builds a second model, where this additional channel is provided by a demand externality that works through the wages.

For this purpose the …rst model is modi…ed to incorporate a wage premium for the modern sector workers.

We now discuss the details of the second model.

Model II: Introducing Factory Wage Premium

Let us assume that working in factories under modern production has certain disutilities associated with it (due to displacement cost, impersonal non-family environment etc.).

The utility function of the representative agent is now given as follow:

U = 8>

>>

<

>>

>: R1 0

logxqdq if he works in cottage sector;

R1 0

logx_qdq V if he works in modern sector.

As before, under cottage production:

w = 1;

p_q = 1 for all q.

Under modern production, once again p_q =1 (for the same reason as before). However, the wage rate in the moden sector can no longer be the same as that of the cottage sector. The modern sector must o¤er a wage premium to compensate for the associated disutility.

Model II: Factory Wage Premium

Letwˆ denote the wage rate in the modern sector. What is its value in equilibrium?

Notice that at this wage rate, an agent should be indi¤erent between working in the cottage sector and working in the modern sector.

Now the indirect utility of any agent who is working in cottage production and earning a wage income ofL¯ in given by:

Uc =

Z1

0

logLdq¯ =logL.¯

On the other hand, the indirect utility of any agent who is working in modern production and earning a wage income ofwˆL¯ in given by:

U_c =

Z1

0

logwˆLdq¯ V =logwˆL¯ V.

Model II: Factory Wage Premium (Contd.)

Now an agent will be indi¤erent bewteen working in the modern sector vis-a-vis the cottage sector i¤:

logwˆL¯ V = logL¯

) ^{logwˆL¯ logL¯} =V ) ^log(wˆ) =V ) ^wˆ =exp^V >1 Without any loss of generality, let us write

ˆ

w =1+v

wherev is the wage premium associated with factory production.

Model II: Demand-Pro…t Interlinkage

Once again the demand for any variety q is given by the aggregate income y.(Notice however that now income across agents may di¤er depending on which sector they are engaged in, although their utilities would be the same).

Thus pro…t of a monopolist operating in sectorq is given by:

πq = y (1+v) ¹ αy+F

= 1 (1+v)

α y (1+v)F (4) We shall assume the the modern sector is productive enough so that

α>1+v.

Model II: Pro…t-Demand Interlinkage

Once again the aggregate household income is the given by the sum of the wage bill and the distributed pro…t :

y =W +θΠ

Let n proportion of the sectors be operated by the respective monopolist producers.

Then

W =n(1+v) ¹

αy+F + L^¯ n 1 αy+F while

Π=nπq. Exercise:

Using (4) and simplifying, write y as a function ofn (and other parameters). Also …nd the corresponding pro…t function (π(n)).