Basic Mathematics for Economists - Rosser - Chapter 13 pptx

The cobweb model takes into account this delayed response on the supply side of a market by assuming that quantity supplied now Qst depends on the ruling price in the previous time perio

13 Dynamics and difference equations

Learning objectives

After completing this chapter students should be able to:

• Demonstrate how a time lag can affect the pattern of adjustment to equilibrium insome basic economic models

• Construct spreadsheets to plot the time path of dependent variables in economicmodels with simple lag structures

• Set up and solve linear first-order difference equations

• Apply the difference equation solution method to the cobweb, Keynesian andBertrand models involving a single lag

• Identify the stability conditions in the above models

13.1 Dynamic economic analysis

In earlier chapters much of the economic analysis used has been comparative statics Thisentails the comparison of different (static) equilibrium situations, with no mention of themechanism by which price and quantity adjust to their new equilibrium values The branch

of economics that looks at how variables adjust between equilibrium values is known as

‘dynamics’, and this chapter gives an introduction to some simple dynamic economic models.The ways in which markets adjust over time vary tremendously In commodity exchanges,prices are changed by the minute and adjustments to new equilibrium prices are almostinstantaneous In other markets the adjustment process may be a slow trial and error processover several years, in some cases so slow that price and quantity hardly ever reach their properequilibrium values because supply and demand schedules shift before equilibrium has beenreached There is therefore no one economic model that can explain the dynamic adjustmentprocess in all markets

The simple dynamic adjustment models explained here will give you an idea of howadjustments can take place between equilibria and how mathematics can be used to calculatethe values of variables at different points in time during the adjustment process They areonly very basic models, however, designed to give you an introduction to this branch ofeconomics The mathematics required to analyse more complex dynamic models goes beyondthat covered in this text

In this chapter, time is considered as a discrete variable and the dynamic adjustment processbetween equilibria is seen as a step-by-step process (The distinction between discrete andcontinuous variables was explained in Section 7.1.) This enables us to calculate differentvalues of the variables that are adjusting to new equilibrium levels:

(i) using a spreadsheet, and

(ii) using the mathematical concept of ‘difference equations’

Models that assume a process of continual adjustment are considered inChapter 14, using

‘differential equations’

13.2 The cobweb: iterative solutions

In some markets, particularly agricultural markets, supply cannot immediately expand tomeet increased demand Crops have to be planted and grown and livestock takes time toraise Some manufactured products can also take a while to produce when orders suddenlyincrease The cobweb model takes into account this delayed response on the supply side of

a market by assuming that quantity supplied now (Qst) depends on the ruling price in the

previous time period (P t−1), i.e.

• the market is perfectly competitive

• supply and demand are both linear schedules

Before we go any further, it must be stressed that this model does not explain how price

adjusts in all competitive markets, or even in all perfectly competitive agricultural markets

It is a simple model with some highly restrictive assumptions that can only explain howprice adjusts in these particular circumstances Some markets may have a more complex lag

structure, e.g Qs

t = f(P t−1, P t−2, P t−3), or may not have linear demand and supply You

should also not forget that intervention in agricultural markets, such as the EU CommonAgricultural Policy, usually means that price is not competitively determined and hence thecobweb assumptions do not apply Having said all this, the cobweb model can still give a fairidea of how price and quantity adjust in many markets with a delayed supply

The assumptions of the cobweb model mean that the demand and supply functions can bespecified in the format

Qdt = a + bP t and Qst = c + dP t−1

where a, b, c and d are parameters specific to individual markets.

Note that, as demand schedules slope down from left to right, the value of b is expected to

be negative As supply schedules usually cut the price axis at a positive value (and therefore

the quantity axis at a negative value if the line were theoretically allowed to continue into

negative quantities), the value of c will also usually be negative Remember that these functions have Q as the dependent variable but in supply and demand analysis Q is usually

measured along the horizontal axis

Although desired quantity demanded only equals desired quantity supplied when a market

is in equilibrium, it is always true that actual quantity bought equals quantity sold In the

cobweb model it is assumed that in any one time period producers supply a given amount Qst.Thus there is effectively a vertical short-run supply schedule at the amount determined bythe previous time period’s price Price then adjusts so that all the produce supplied is bought

by consumers This adjustment means that

This is what is known as a ‘linear first-order difference equation’ A difference equation

expresses the value of a variable in one time period as a function of its value in earlierperiods; in this case

P t = f(P t−1)

It is clearly a linear relationship as the terms (c − a)/b and d/b will each take a single

numerical value in an actual example It is ‘first order’ because only a single lag on the

previous time period is built into the model and the coefficient of P t−1is a simple constant

In the next section we will see how this difference equation can be used to derive an expression

(Note: In this example and in most other examples in this chapter, no specific units of

measurement for P or Q are given in order to keep the analysis as simple as possible In

actual applications, of course, price will usually be in £ and quantity in physical units, e.g.thousands of tonnes.)

In long-run equilibrium, price and quantity will remain unchanged each time period Thismeans that:

the long-run equilibrium price P∗= P t = P t−1

and the long-run equilibrium quantity Q∗= Qd

S0in Figure 13.1 To sell this amount the price has to be reduced to P0, corresponding to the

point A where S0cuts the demand schedule

Producers will then plan production for the next time period on the assumption that P0is the

ruling price The amount supplied will therefore be Q1, corresponding to point B However,

in the next time period when this reduced supply quantity Q1is put onto the market it will sell

Q

160 100

Figure 13.1

for price P1, corresponding to point C Further adjustments in quantity and price are shown

by points D, E, F , etc These trace out a cobweb pattern (hence the ‘cobweb’ name) which

converges on the long-run equilibrium where the supply and demand schedules intersect

In some markets, price will not always return towards its long-run equilibrium level, as weshall see later when some other examples are considered However, first let us concentrate

on finding the actual pattern of price adjustment in this particular example

Approximate values for the first few prices could be read off the graph inFigure 13.1, but

as price converges towards the centre of the cobweb it gets difficult to read values accurately

We shall therefore calculate the first few values of P manually, so that you can become

familiar with the mechanics of the cobweb model, and then set up a spreadsheet that canrapidly calculate patterns of price adjustment over a much longer period

Quantity supplied in each time period is calculated by simply entering the previously rulingprice into the market’s supply function

Qst = −50 + 10P t−1

but how is this price calculated? There are two ways:

(a) from first principles, using the given supply and demand schedules, and

(b) using a difference equation, in the format (1) derived earlier

(a) The demand function

t Thus, P t can be found by inserting the current quantity supplied, Qst,

into the function for P t Assuming that the initial disturbance to the system when Qsrises to

160 occurs in time period 0, the values of P and Q over the next three time periods can be

Q3= −50 + 10P2= −50 + 10(14.25) = −50 + 142.5 = 92.5

P3= 20 − 0.05Q s

3= 20 − 0.05(92.5) = 20 − 4.625 = 15.375

The pattern of price adjustment is therefore 12, 16.5, 14.25, 15.375, etc., corresponding

to the cobweb graph inFigure 13.1 Price initially falls below its long-run equilibrium value

of 15 and then converges back towards this equilibrium, alternating above and below it butwith the magnitude of the difference becoming smaller each period

(b) The same pattern of price adjustment can be obtained by using the difference equation

Note: do not enter for the word “STABLE” incell G10 The stability condition will be deduced

These are the parameter values for this example

A10 to

A20

Enter numbers from 0 to 10

These are the time periods

Given that Qdt = a + bP t then P t =(Qdt–a )/ b.

Note the $ on cells D4 and D5 Format to 2 dp.C11 to

previous time period according to supply

This uses the Excel “IF” logic function to

determine whether d /(–b) is less than 1, greater

than 1, or equals 1 This stability criterion isexplained later

=IF(-F5/D5<1,"STABLE",IF(-F5/D5>1,"UNSTABLE","OSCILLATING"))

A spreadsheet can be set up to calculate price over a large number of time periods tions are given in Table 13.2 for constructing the Excel spreadsheet shown in Table 13.1 Thiscalculates price for each period from first principles, but you can also try to construct yourown spreadsheet based on the difference equation approach

Instruc-This spreadsheet shows a series of prices and quantities converging on the equilibriumvalues of 15 for price and 100 for quantity The first few values can be checked against themanually calculated values and are, as expected, the same To bring home the point that eachprice adjustment is smaller than the previous one, the change in price from the previous timeperiod is also calculated (The price columns are formatted to 2 decimal places so price iscalculated to the nearest penny.)

Although the stability of this example is obvious from the way that price converges onits equilibrium value of 15, a stability check is entered which may be useful when this

spreadsheet is used for other examples Assuming that b is always negative and d is positive, the market will be stable if d/ − b < 1 and unstable (i.e price will not converge back to its equilibrium) if d/ − b > 1 (The reasons for this rule are explained later in Section 13.3.)

When you have constructed this spreadsheet yourself, save it so that it can be used forother examples.

To understand why price may not always return to its long-run equilibrium level in marketswhere the cobweb model applies, consider Example 13.2

Example 13.2

In a market where the assumptions of the cobweb model apply, the demand and supplyfunctions are

Qdt = 120 − 4P t and Qst = −80 + 16P t−1

If in one time period the long-run equilibrium is disturbed by output unexpectedly rising to

a level of 90, explain how price will adjust over the next few time periods

Q∗= 120 − 4P∗= 120 − 4(10) = 80

You could use the spreadsheet developed for Example 13.1 above to trace out the subsequentpattern of price adjustment but if a few values are calculated manually it can be seen thatcalculations after period 2 are irrelevant

Using the standard cobweb model difference equation

happened is that price has followed the path ABCD traced out in Figure 13.2.

The initial quantity 90 put onto the market causes price to drop to 7.5 Suppliers thenreduce supply for the next period to

These slopes are inversely related to parameters b and d, since the vertical axis measures p rather than q Thus the stability conditions are

Stable:|d/b| < 1 Unstable:|d/b| > 1

A formal proof of these conditions, based on the difference equation solution method, plus

an explanation of what happens when|d/b| = 1, is given in Section 13.3.

Although in theoretical models of unstable markets (such as Example 13.2) price ‘explodes’and the market collapses, this may not happen in reality if:

• producers learn from experience and do not simply base production plans for the nextperiod on the current price,

• supply and demand schedules are not linear along their entire length,

• government intervention takes place to support production

Another example of an exploding market is Example 13.3 below, which is solved using thespreadsheet developed for Example 13.1

of price adjustments show According to these figures, the market will continue to operateuntil the eighth time period following the initial shock In period 9 nothing will be produced(mathematically the model gives a negative quantity) and the market collapses.

Test Yourself, Exercise 13.1

(In all these questions, assume that the assumptions of the cobweb model apply toeach market.)

1 The agricultural market whose demand and supply schedules are

Qdt = 240 − 20P t and Qst = −331

3+ 162

3P t−1

is initially in long-run equilibrium Quantity then falls to 50% of its previous level

as a result of an unexpectedly poor harvest How many time periods will it takefor price to return to within 1% of its long-run equilibrium level?

2 In an unstable market, the demand and supply schedules are

(a) Qdt = 150 − 1.5P t and Qst = −30 + 3P t−1

(b) Qdt = 180 − 125P t and Qst = −20 + P t−1

13.3 The cobweb: difference equation solutions

Solving the cobweb difference equation

There are two parts to the solution of this cobweb difference equation:

(i) the new long-run equilibrium price, and

(ii) the complementary function that tells us how much price diverges from this equilibrium

level at different points in time

A similar format applies to the solution of any linear first-order difference equation The

equilibrium solution (i) is also known as the particular solution (PS) In general, the

par-ticular solution is a constant value about which adjustments in the variable in question takeplace over time

The complementary function (CF) tells us how the variable in question, i.e price in the

cobweb model, varies from the equilibrium solution as time changes

These two elements together give what is called the general solution (GS) to a difference

equation, which is the full solution Thus we can write

GS= PS + CF

Finding the particular solution is straightforward In the long run the equilibrium price P∗

holds in each time period and so

Substituting the formulations (4) and (5) for P t and P t−1back into equation (3) above we get

The value of A cannot be ascertained unless the actual value of P t is known for a specific

value of t (See the following numerical examples.)

The general solution to the cobweb difference equation therefore becomes

P t = particular solution + complementary function = (2) plus (6), giving

P t = a − c

d − b + A



d b

t

Stability

From this solution we can see that the stability of the model depends on the value of d/b If

Ais a non-zero constant, then there are three possibilities

(i) If

d b < 1 thend bt → 0 as t → ∞

This occurs in a stable market Whatever value the constant A takes the value of the

com-plementary function gets smaller over time Therefore the divergence of price from itsequilibrium also approaches zero (Note that it is the absolute value of|d/b| that we consider because b will usually be a negative number.)

diverge from its equilibrium level by greater and greater amounts

−1 depending on whether or not t is an even or odd number Price will continually fluctuate

between two levels (see Example 13.6 below)

We can now use this method of obtaining difference equation solutions to answer somespecific numerical cobweb model problems

Substituting the values for this market a = 400, b = −20, c = −50 and d = 10 into the

general cobweb difference equation solution

P t = a − c

d − b + A



d b

To find the value of A we then substitute in the known value of P0

The question tells us that the initial ‘shock’ output level Q0is 160 and so, as price adjusts

until all output is sold, P0 can be calculated by substituting this quantity into the demandschedule Thus

This is usually called the definite solution or the specific solution because it relates to a

specific initial value

We can use this solution to calculate the first few values of P tand compare with those weobtained when answering Example 13.1.

P1= 15 − 3(−0.5)1= 15 + 1.5 = 16.5

P2= 15 − 3(−0.5)2= 15 − 3(0.25) = 14.25

P3= 15 − 3(−0.5)3= 15 − 3(−0.125) = 15.375

As expected, these values are identical to those calculated by the iterative method

In this particular example, price converges fairly quickly towards its long-run equilibriumlevel of 15 By time period 9, price will be

and so the stability condition outlined above is satisfied

Note that, because−0.5 < 0, the direction of the divergence from the equilibrium value

alternates between time periods This is because for any negative quantity−x, it will always

be true that

x < 0, ( −x)2> 0, ( −x)3< 0, ( −x)4> 0, etc.

Thus for odd-numbered time periods (in this example) price will be above its equilibriumvalue, and for even-numbered time periods price will be below its equilibrium value.Although in this example price converges towards its long-run equilibrium value, it wouldnever actually reach it if price and quantity were divisible into infinitesimally small units.Theoretically, this is a bit like the case of the ‘hopping frog’ back inChapter 7when infinitegeometric series were examined The distance from the equilibrium gets smaller and smallereach time period but it never actually reaches zero For practical purposes, a reasonable cut-off point can be decided upon to define when a full return to equilibrium has been reached Inthis numerical example the difference from the equilibrium is less than 0.01 by time period

9, which is for all intents and purposes a full return to equilibrium if P is measured in £.

The above example explained the method of solution of difference equations applied to

a simple problem where the answers could be checked against iterative solutions In othercases, one may need to calculate values for more distant time periods, which are moredifficult to calculate manually The method of solution of difference equations will also

be useful for those of you who go on to study intermediate economic theory where somemodels, particularly in macroeconomics, are based on difference equations in an algebraicformat which cannot be solved using a spreadsheet

We shall now consider another cobweb example which is rather different from Example13.4 in that

(i) price does not return towards its equilibrium level and

(ii) the process of adjustment is more gradual over time

Example 13.5

In a market where the assumptions of the cobweb model hold

Qdt = 200 − 8P t and Qst = −43 + 8.2P t−1

The long-run equilibrium is disturbed when quantity suddenly changes to 90 What happens

to price in the following time periods?

Using the formula derived above, the solution to this difference equation will therefore be

P t = a − c

d − b + A



d b

The first part of this solution is of course the equilibrium value of price which has already

been calculated above To derive the value of A, we need to find price in period 0 The

quantity supplied is 90 in period 0 and so, to find the price that this quantity will sell for, thisvalue is substituted into the demand function Thus

Note that, as in Example 13.1 above, the value of parameter A is the difference between the

equilibrium value of price and the value it initially takes when quantity is disturbed from itsequilibrium level, i.e

A = P0− P∗= 13.75 − 15 = −1.25

Putting this value of A into the general solution (2), the specific solution to the difference

equation in this example now becomes

We can see that, although price is gradually moving away from its long-run equilibrium value

of 15, it is a very slow process By period 10, price is still above 13.00, as

P10= 15 − 1.25(1.025)10= 13.40

and it takes until time period 102 before price becomes negative, as the figures below show:

The following example illustrates what happens when a market is neither stable norunstable

Example 13.6

The cobweb model assumptions hold in a market where

Qdt = 160 − 2P t and Qst = −20 + 2P t−1

If the previously ruling long-run equilibrium is disturbed by an unexpectedly low output of

50 in one time period, what will happen to price in the following time periods?

Solution

Substituting the values a = 160, b = −2, c = −20 and d = 2 for this market into the

cobweb difference equation general solution

P t = a − c

d − b + A



d b

t

(1)gives

P t =160− (−20)

2− (−2) + A

2