1. Distributed Lag models
Is a model for the time series data in which a regression equation is used to predict current value of a dependent variable based on both the current value of an explanatory variable and lagged (past period) values of this explanatory variable.
The starting point for a distributed lag model is an assumed structure of the form
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2. Distributed Lag models
where yt is the value at time period t of the dependent variable y
a is the intercept term to be estimated, and
wi is called the lag weight (also to be estimated) placed on the value i periods previously of the explanatory variable x
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3. Distributed Lag models
In the first equation
the dependent variable is assumed to be affected by values of the independent variable arbitrarily far in the past
so the number of lag weights is infinite and the model is called an infinite distributed lag model.
In the second, equation,
There are only a finite number of lag weights, indicating an assumption that there is a maximum lag beyond which values of the independent variable do not affect the dependent variable
A model based on this assumption is called a finite distributed lag model.
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4. Distributed Lag models
The role of time or Lags
One type of dynamic model is the Distributed Lag Model.
Finite Distributed Lag Model:
Yt = α + β0Xt + β1Xt-1 + β2Xt-2+ εt
General Form of Finite Distributed Lag Model:
Yt = α + β0Xt + β1Xt-1 + β2Xt-2 +….. + βk Xt-k
The Standard Multiplier is defined as:
βi*= βi / ∑βi
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5. Distributed Lag models
Significance of lags
1. Psychological
2. Imperfect Knowledge
3. Institutional
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6. Distributed Lag models
Psychological
A change in the independent variable does not necessarily lead to an immediate change in the dependent variable.
The reaction to each change may be different depending on psychological reasons
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7. Distributed Lag models
Imperfect knowledge
May lead to lags in a decision making process such as consumption and/or investment.
Institutional
Contractual obligations may prevent agents from changing their economic behaviour
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8. Distributed Lag models
Types of lagged variable
There are two types of lagged variable
1. Endogenous lag variable
Are those variables whose value is estimated within the model
They are dependent variables.
Yt-1 Yt-2 are endogenous lagged variable
2. Exogenous lag variable
Are those variables whose value is estimated outside the model.
Xt-1,Xt-2..are exogenous lagged variable. i.
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9. Distributed Lag models
The general form of a distributed lag mode with only lagged exogenous variable is written as
Xt =Ao + BoXt + B1Xt-1 + B2Xt-2 + ……..+B5Xt-6 + U
The number s may be finite or infinite ,but it is supposed to be finite .
The coefficient Bo is known as short run or impact or multiplier since Yt and Xt are related in same period.
Similarly Σ Bi=Bo + B1 +……+Bs = B is called long run distributed lag multiplier. B1,B2…..are delay or interim multipliers
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10. Simulation of an Inventory Problem
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11. Simulation of an Inventory Problem
let in a retail store, to keep replenishing a certain item in the store by ordering item from wholesaler.
we want to adopt a simple policy for ordering new supplies
When stock of the item goes down to P( called reorder point)
We will order Q(called reorder quantity) from wholesaler
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12. Simulation of an Inventory Problem
If the demand on any day exceeds the amount of inventory on hand the excess represents lost sale and goodwill.
If Overstocking  increase carrying cost (cost of storage insurance..)
If Ordering too few  excessive reorder cost
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13. Simulation of an Inventory Problem
Assume the following conditions
1) there is 3 days lag b/w the order and arrival.
Order at the end of the day and is received at the beginning of 4th day  (i+3)rd day
2) for each unit of inventory the carrying cost for each night is Re 0.75
3) each unit out of stock when ordered  result loss of goodwill Rs loss of Rs net income total loss Rs 18.00/ unit
4) placement of each order cost Rs 75.00
5) the demand in a day can be of any nos of units b/w 0 and 99
6) there is never more then one replenishment order outstanding
7) initially we have 115 units on hand and no order outstanding
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14. Simulation of an Inventory Problem
With the conditions, We have to compare the following policies and select the one that has the minimum total cost( reorder cost + carrying + lost sales cost) P Q
reorder point
reorder quantity
Policy 1
125
150
Policy 2
250
Policy 3
Policy 4
175
Policy 5
300
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15. Simulation of an Inventory Problem
This problem can be solved by simulation
Let as simulate the running of the store for about 6 months (180 days)
Simulation model of this inventory sys can be easily constructed by stepping time forward in fixed increment of days
Starting with day 1 can continue to day 180
Day i first we check to see if merchandise is due to arrive today.
If yes then the existing stock S is increased by Q (the quantity was ordered)
If DEM is the demand for today And DEM<=S
Our new stock at the end of today will be S-DEM units
If DEM> S then our new stock will be 0
In two case we calculate total cost from todays transactions and add to the total cost C incurred till yesterday
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16. Simulation of an Inventory Problem
Then determine if the inventory on hand + units on order > P
If not place an order, by stating the amount ordered and the day it is due to be received.
Repeat this process to 180 days
initially set day number i =1, stock S=115, no of units due UD=0(because there is no outstanding order) and day that are due DD=0
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17. Simulation of an Inventory Problem
The demand DEM for each day is not a fixed quantity bt random variable
It could assume and integral value from 00 to 99 eavh with equal probalility
Generate Random nos with subroutine
Condition 6  there is no more than one reorder outstanding
Evaluated stock ES
ES compare to P before an order is placed
UD>P if if we already have a replenishment order outstanding another order will not be placed
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18. Simulation of an Inventory Problem
Fortran code
INTEGRAL P,Q,S,ES,UD,DD,DEM
READ P,Q
C=0.0
S=115
UD=0
DD=0
100 IF (DD.NE. 1) GO TO 110
S=S+Q
DEM=RNDY1(DUM)*100.0
IF (DEM.LE. S) GO TO 120
C=C+(FLOAT(DEM)-FLOAT(S))*18
S=0
GOTO 130
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19. Simulation of an Inventory Problem
S=S-DEM
C=C+FLOAT(S)*.75
ES=S+DEM
IF(ES.GT. P) GO TO 140
UD=Q
DD=I+3
C=C+75.0
I=I+1
IF(I .LE. 180) GO TO 100
PRINT P,Q,C
STOP
END
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20. Simulation of an Inventory Problem
The prog yields the following cost
Polocy 4 (p=175, q=250) is the best among the five considered
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21. Agricultural markets there is a lag between planting and harvesting
cobweb model
Model that explains why prices might be subject to periodic fluctuations in certain types of markets.
It describes supply and demand in a market where the amount produced must be chosen before prices are observed.
The cobweb model is generally based on a time lag between supply and demand decisions.
Agricultural markets there is a lag between planting and harvesting
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22. If farmers supply will be high, resulting in low prices.
cobweb model
For example unexpectedly bad weather, farmers go to market with an unusually small crop of strawberries.
results in high prices.
If farmers expect these high price conditions to continue, then in the following year, they will raise their production of strawberries relative to other crops.
If farmers supply will be high, resulting in low prices.
If they then expect low prices to continue, they will decrease their production of strawberries for the next year, resulting in high prices again.
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23. This process is illustrated by the adjacent diagrams.
cobweb model
This process is illustrated by the adjacent diagrams.
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24. cobweb model
The equilibrium price is at the intersection of the supply and demand curves.
A poor harvest in period 1 means supply falls to Q1, so that prices rise to P1.
If producers plan their period 2 production under the expectation that this high price will continue, then the period 2 supply will be higher, at Q2.
Prices therefore fall to P2 when they try to sell all their output.
As this process repeats itself, oscillating between periods of low supply with high prices and then high supply with low prices, the price and quantity trace out a spiral.
In which case the economy converges to the equilibrium where supply and demand cross; or they may spiral outwards, with the fluctuations increasing in magnitude.
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25. The cobweb model can have two types of outcomes:
If the supply curve is steeper than the demand curve,
then the fluctuations decrease in magnitude with each cycle,
so a plot of the prices and quantities over time would look like an inward spiral
This is called the stable or convergent case.
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26. then the fluctuations increase in magnitude with each cycle,
cobweb model
If the slope of the supply curve is less than the absolute value of the slope of the demand curve,
then the fluctuations increase in magnitude with each cycle,
so that prices and quantities spiral outwards.
This is called the unstable or divergent case.
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27. Two other possibilities are:
cobweb model
Two other possibilities are:
Fluctuations may also remain of constant magnitude,
so a plot of the outcomes would produce a simple rectangle,
if the supply and demand curves have exactly the same slope.
If the supply curve is less steep than the demand curve near the point
where the two curves cross, but more steep when we move sufficiently far away, then prices and quantities will spiral away from the equilibrium price
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28. cobweb model
but will not diverge indefinitely; instead, they may converge to a limit cycle.
In either of the first two scenarios, the combination of the spiral and the supply and demand curves often looks like a cobweb, hence the name of the theory.
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29. The outcomes of the cobweb model are stated above in terms of slopes,
but they are more commonly described in terms of elasticities.
In terms of slopes, the convergent case requires that the slope of the supply curve be greater than the absolute value of the slope of the demand curve:
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30. then we see that the convergent case requires
cobweb model
In standard microeconomics terminology, define the elasticity of supply as
If we evaluate these two elasticities at the equilibrium point, that is
then we see that the convergent case requires
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31. cobweb model
In words, the convergent case occurs when the demand curve is more elastic than the supply curve, at the equilibrium point.
The divergent case occurs when the supply curve is more elastic than the demand curve, at the equilibrium point
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32. cobweb model
Model
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