Basic Mathematics for Economists - Rosser - Chapter 13 pptx

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 in some basic economic models. • Construct spreadsheets to plot the time path of dependent variables in economic models with simple lag structures. • Set up and solve linear first-order difference equations. • Apply the difference equation solution method to the cobweb, Keynesian and Bertrand 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. This entails the comparison of different (static) equilibrium situations, with no mention of the mechanism 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 almost instantaneous. In other markets the adjustment process may be a slow trial and error process over several years, in some cases so slow that price and quantity hardly ever reach their proper equilibrium values because supply and demand schedules shift before equilibrium has been reached. There is therefore no one economic model that can explain the dynamic adjustment process in all markets. The simple dynamic adjustment models explained here will give you an idea of how adjustments can take place between equilibria and how mathematics can be used to calculate the values of variables at different points in time during the adjustment process. They are only very basic models, however, designed to give you an introduction to this branch of economics. The mathematics required to analyse more complex dynamic models goes beyond that covered in this text. © 1993, 2003 Mike Rosser In this chapter, time is considered as a discrete variable and the dynamic adjustment process between equilibria is seen as a step-by-step process. (The distinction between discrete and continuous variables was explained in Section 7.1.) This enables us to calculate different values of the variables that are adjusting to new equilibrium levels: (i) using a spreadsheet, and (ii) using the mathematical concept of ‘difference equations’. ModelsthatassumeaprocessofcontinualadjustmentareconsideredinChapter14,using ‘differential equations’. 13.2 The cobweb: iterative solutions In some markets, particularly agricultural markets, supply cannot immediately expand to meet increased demand. Crops have to be planted and grown and livestock takes time to raise. Some manufactured products can also take a while to produce when orders suddenly increase. The cobweb model takes into account this delayed response on the supply side of a market by assuming that quantity supplied now (Q s t ) depends on the ruling price in the previous time period (P t−1 ), i.e. Q s t = f(P t−1 ) where the subscripts denote the time period. Consumer demand for the same product (Q d t ), however, is assumed to depend on the current price, i.e. Q d t = f(P t ) This is a reasonable picture of many agricultural markets. The quantity offered for sale this year depends on what was planted at the start of the growing season, which in turn depends on last year’s price. Consumers look at current prices, though, when deciding what to buy. The cobweb model also assumes that: • 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 how price adjusts in these particular circumstances. Some markets may have a more complex lag structure, e.g. Q s 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 Common Agricultural Policy, usually means that price is not competitively determined and hence the cobweb assumptions do not apply. Having said all this, the cobweb model can still give a fair idea 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 be specified in the format Q d t = a +bP t and Q s t = 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 © 1993, 2003 Mike Rosser 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 Q s t . Thus there is effectively a vertical short-run supply schedule at the amount determined by the previous time period’s price. Price then adjusts so that all the produce supplied is bought by consumers. This adjustment means that Q d t = Q s t Therefore a + bP t = c + dP t−1 bP t = c − a + dP t−1 P t = c −a b + d b P t−1 (1) 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 earlier periods; 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−1 is a simple constant. In the next section we will see how this difference equation can be used to derive an expression for P t in terms of t. Before doing this, let us first get a picture of how the cobweb price adjustment mechanism operates using a numerical example. Example 13.1 In an agricultural market where the assumptions of the cobweb model apply, the demand and supply schedules are Q d t = 400 − 20P t and Q s t =−50 +10P t−1 A long-run equilibrium has been established for several years but then one year there is an unexpectedly good crop and output rises to 160. Explain how price will behave over the next few years following this one-off ‘shock’ to the market. (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.) © 1993, 2003 Mike Rosser Solution In long-run equilibrium, price and quantity will remain unchanged each time period. This means that: the long-run equilibrium price P ∗ = P t = P t−1 and the long-run equilibrium quantity Q ∗ = Q d t = Q s t Therefore, when the market is in equilibrium Q ∗ = 400 − 20P ∗ and Q ∗ =−50 +10P ∗ Equating to solve for P ∗ and Q ∗ gives 400 −20P ∗ =−50 +10P ∗ 450 = 30P ∗ 15 = P ∗ Q ∗ = 400 − 20P ∗ = 400 − 300 = 100 These values correspond to the point where the supply and demand schedules intersect, as illustrated in Figure 13.1. If an unexpectedly good crop causes an amount of 160 to be supplied onto the market one year, then this means that the short-run supply schedule effectively becomes the vertical line S 0 in Figure 13.1. To sell this amount the price has to be reduced to P 0 , corresponding to the point A where S 0 cuts the demand schedule. Producers will then plan production for the next time period on the assumption that P 0 is the ruling price. The amount supplied will therefore be Q 1 , corresponding to point B. However, in the next time period when this reduced supply quantity Q 1 is put onto the market it will sell Supply Deman d 0 B C F Q160100 Q 1 P 0 A P 5 15 20 P 1 S 0 E D Figure 13.1 © 1993, 2003 Mike Rosser for price P 1 , 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 we shall 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. ApproximatevaluesforthefirstfewpricescouldbereadoffthegraphinFigure13.1,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 can rapidly calculate patterns of price adjustment over a much longer period. Quantity supplied in each time period is calculated by simply entering the previously ruling price into the market’s supply function Q s t =−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 Q d t = 400 − 20P t can be rearranged to give the inverse demand function P t = 20 − 0.05Q d t The model assumes that a fixed quantity arrives on the market each time period and then price adjusts until Q d t = Q s t . Thus, P t can be found by inserting the current quantity supplied, Q s t , into the function for P t . Assuming that the initial disturbance to the system when Q s rises to 160 occurs in time period 0, the values of P and Q over the next three time periods can be calculated as follows: Q s 0 = 160 (initial given value, inserted into inverse demand function) P 0 = 20 − 0.05Q s 0 = 20 − 0.05(160) = 20 − 8 = 12 This price in period 0 then determines quantity supplied in period 1, which is Q s 1 =−50 +10P 0 =−50 +10(12) =−50 + 120 = 70 This quantity then determines the market-clearing price, which is P 1 = 20 − 0.05Q s 1 = 20 − 0.05(70) = 20 − 3.5 = 16.5 The same adjustment process then continues for future time periods, giving Q s 2 =−50 +10P 1 =−50 +10(16.5) =−50 + 165 = 115 P 2 = 20 − 0.05Q s 2 = 20 − 0.05(115) = 20 − 5.75 = 14.25 © 1993, 2003 Mike Rosser Q s 3 =−50 +10P 2 =−50 +10(14.25) =−50 + 142.5 = 92.5 P 3 = 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 tothecobwebgraphinFigure13.1.Priceinitiallyfallsbelowitslong-runequilibriumvalue of 15 and then converges back towards this equilibrium, alternating above and below it but with the magnitude of the difference becoming smaller each period. (b) The same pattern of price adjustment can be obtained by using the difference equation P t = c −a b + d b P t−1 (1) and substituting in the given values of a, b,c and d to get P t = (−50) − 400 −20 + 10 −20 P t−1 P t = 22.5 − 0.5P t−1 (2) The original price P 0 still has to be derived by inserting the shock quantity 160 into the demand function, as already explained, which gives P 0 = 20 − 0.05(160) = 12 Then subsequent prices can be determined using the difference equation (2), giving P 1 = 22.5 − 0.5P 0 = 22.5 − 0.5(12) = 16.5 P 2 = 22.5 − 0.5P 1 = 22.5 − 0.5(16.5) = 14.25 P 3 = 22.5 − 0.5P 2 = 22.5 − 0.5(14.25) = 15.375 etc. These prices are the same as those calculated by method (a), as expected. Table 13.1 A B C D E F G H 1 Ex. COBWEB MODEL 2 13.1 Qd=a+bPt Qs=c+dPt 3 4 Parameter a = 400 c = -50 5 values b = -20 d = 10 6 Initial shock Quantity = 160 7 Equilibrium Price = 15 8 Time Quantity Price Change Equilibrium Quantity = 100 9 t Qt Pt in Pt 10 0 160 12.00 Stability => STABLE 11 1 70 16.50 4.50 12 2 115 14.25 -2.25 13 3 92.5 15.38 1.13 14 4 103.75 14.81 -0.56 15 5 98.125 15.09 0.28 16 6 100.9375 14.95 -0.14 17 7 99.53125 15.02 0.07 18 8 100.23438 14.99 -0.04 19 9 99.882813 15.01 0.02 20 10 100.05859 15.00 -0.01 © 1993, 2003 Mike Rosser Table 13.2 CELL Enter Explanation As in Table 13.1 Enter all labels and column headings shown in Table 13.1 Note: do not enter for the word “STABLE” in cell G10. The stability condition will be deduced by the spreadsheet. D4 400 D5 -20 F4 -50 F5 10 These are the parameter values for this example. D6 160 This is initial “shock” quantity in time period 0. A10 to A20 Enter numbers from 0 to 10 These are the time periods. B10 =D6 Quantity in time period 0 is initial “shock” value. C10 =(B10-D$4)/D$5 Calculates P 0 , the initial market clearing price. Given that Q d t = a + bP t then P t =(Q d t – a )/ b. Note the $ on cells D4 and D5. Format to 2 dp. C11 to C20 Copy formula from C10 down column. Will calculate price in each time period (when all quantities in column B are calculated) B11 =F$4+F$5*C10 Calculates quantity in year 1 based on price in previous time period according to supply function Q s t = c + dP t – 1. Format to 2 dp. D11 =C11-C10 Calculates change in price between time periods. B12 to B20 Copy formula from B11 down column. Calculates quantity supplied in each time period. D12 to D20 Copy formula from D12 down column. Calculates price change since previous time period H7 =(F4-D4)/(D5-F5) Calculates equilibrium price using the formula P* = (c – a)/ (b – d ) H8 =F4+F5*H7 Calculates equilibrium quantity Q* = a + bP * G10 Enter the formula below 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 is explained 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. Instruc- tions are given in Table 13.2 for constructing the Excel spreadsheet shown in Table 13.1. This calculates price for each period from first principles, but you can also try to construct your own spreadsheet based on the difference equation approach. This spreadsheet shows a series of prices and quantities converging on the equilibrium values of 15 for price and 100 for quantity. The first few values can be checked against the manually calculated values and are, as expected, the same. To bring home the point that each price adjustment is smaller than the previous one, the change in price from the previous time period is also calculated. (The price columns are formatted to 2 decimal places so price is calculated to the nearest penny.) Although the stability of this example is obvious from the way that price converges on its 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.) © 1993, 2003 Mike Rosser When you have constructed this spreadsheet yourself, save it so that it can be used for other examples. To understand why price may not always return to its long-run equilibrium level in markets wherethecobwebmodelapplies,considerExample13.2. Example 13.2 In a market where the assumptions of the cobweb model apply, the demand and supply functions are Q d t = 120 − 4P t and Q s t =−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. Solution The long-run equilibrium price can be determined from the formula P ∗ = c −a b −d = (−80) − 120 −4 − 16 = −200 −20 = 10 Thus, the long-run equilibrium quantity is Q ∗ = 120 − 4P ∗ = 120 − 4(10) = 80 YoucouldusethespreadsheetdevelopedforExample13.1abovetotraceoutthesubsequent pattern of price adjustment but if a few values are calculated manually it can be seen that calculations after period 2 are irrelevant. Using the standard cobweb model difference equation P t = c −a b + d b P t−1 (1) and substituting the known values, we get P t = (−80) − 120 −4 + 16 −4 P t−1 = 50 − 4P t−1 (2) The initial price P 0 can be found by inserting the shock quantity of 90 into the demand function. Thus Q s 0 = 90 = 120 − 4P 0 4P 0 = 30 P 0 = 7.5 Putting this value into the difference equation (2) above we get P 1 = 50 − 4P 0 = 50 − 4(7.5) = 20 P 2 = 50 − 4P 1 = 50 − 4(20) =−30 © 1993, 2003 Mike Rosser Demand Supply A 0 £ 5 20 30 40 10 Q90240120 S 0 D B C 7.5 Figure13.2 Thereisnotmuchpointingoinganyfurtherwiththecalculations.Assumingthatproducers willnotpayconsumerstotakegoodsofftheirhands,negativepricescannotexist.Whathas happenedisthatpricehasfollowedthepathABCDtracedoutinFigure13.2. Theinitialquantity90putontothemarketcausespricetodropto7.5.Suppliersthen reducesupplyforthenextperiodto Q s 1 =−80+16P 0 =−80+16(7.5)=40 ThissellsforpriceP 1 =20andsosupplyforthefollowingperiodisincreasedto Q s 2 =−80+16P 1 =−80+16(20)=240 Consumerswouldonlyconsume120evenifpricewerezero(wherethedemandschedule hitstheaxis)andso,whenthisquantityof240isputontothemarket,pricewillcollapseto zeroandtherewillstillbeunsoldproduce.Producerswillnotwishtosupplyanythingforthe nexttimeperiodiftheyexpectapriceofzeroandsonofurtherproductionwilltakeplace. Thisisclearlyanunstablemarket,butwhyisthereadifferencebetweenthismarketand thestablemarketconsideredinExample13.1?Itdependsontheslopesofthesupplyand demand schedules. If the absolute value of the slope of the demand schedule is less than the absolute value of the slope of the supply schedule then the market is stable, and vice versa. 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. © 1993, 2003 Mike Rosser Althoughintheoreticalmodelsofunstablemarkets(suchasExample13.2)price‘explodes’ andthemarketcollapses,thismaynothappeninrealityif: •producerslearnfromexperienceanddonotsimplybaseproductionplansforthenext periodon the current price, • supply and demand schedules are not linear along their entire length, • government intervention takes place to support production. AnotherexampleofanexplodingmarketisExample13.3below,whichissolvedusingthe spreadsheetdevelopedforExample13.1. Example 13.3 In an agricultural market where the cobweb assumptions hold and Q d t = 360 − 8P t and Q s t =−120 +12P t−1 a long-run equilibrium is disturbed by an unexpectedly good crop of 175 units. Use a spreadsheet to trace out the subsequent path of price adjustment. Solution When the given parameters and shock quantity are entered, your spreadsheet should look like Table 13.3. This is clearly unstable as both the automatic stability check and the pattern Table 13.3 A B C D E F G H 1 Ex. COBWEB MODEL 2 13.3 Qd=a+bPt Qs=c+dPt 3 4 Parameter a = 360 c = -120 5 Values b = -8 d = 12 6 Initial shock quantity = 175 7 Equilibrium Price = 24 8 Time Quantity Price Change Equilibrium Quantity = 168 9 t Qt Pt in Pt 10 0 175 23.13 Stability => UNSTABLE 11 1 157.5 25.31 2.19 12 2 183.75 22.03 -3.28 13 3 144.375 26.95 4.92 14 4 203.4375 19.57 -7.38 15 5 114.84375 30.64 11.07 16 6 247.73438 14.03 -16.61 17 7 48.398438 38.95 24.92 18 8 347.40234 1.57 -37.38 19 9 -101.10352 57.64 56.06 20 10 571.65527 -26.46 -84.09 © 1993, 2003 Mike Rosser [...]... It (3) and can therefore be written as Ct = a + bY ∗ Y ∗ = Ct + It © 1993, 2003 Mike Rosser Table 13. 5 CELL Enter As in Enter all labels and Table 13. 4 column headings B5 40 B6 0.6 E5 134 E6 110 D6 160 A10 to Enter numbers A28 from 0 to 18 E7 =(B5+E5)/(1-B6) E8 =(B5+E6)/(1-B6) B10 =B5+B6*E7 C10 =B10+E$6 B11 =B$5+B$6*C10 B12 to B28 C11 to C28 Copy formula from B11 down column Copy formula from C10 down... we mean putting it in the format Yt = f(t) so that the value of Yt can be determined for any given value of t © 1993, 2003 Mike Rosser Table 13. 4 A B 1 Ex LAGGED 2 13. 7 3 4 Parameters 5 a= 40 6 b= 0.6 7 8 Time 9 t C 10 0 301.00 11 1 286.60 12 2 277.96 13 3 272.78 14 4 269.67 15 5 267.80 16 6 266.68 17 7 266.01 18 8 265.60 19 9 265.36 20 10 265.22 21 11 265 .13 22 12 265.08 23 13 265.05 24 14 265.03 25... when the original value of It of 134 is inserted into the accounting identity the model becomes Y ∗ = Ct + 134 Ct = 40 + 0.6Y (1) ∗ By substitution of (2) into (1) Y ∗ = (40 + 0.6Y ∗ ) + 134 Y ∗ (1 − 0.6) = 40 + 134 0.4Y ∗ = 174 Y ∗ = 435 © 1993, 2003 Mike Rosser (2) E E (I = 134 ) A Y* EЈ (I = 110) Y0 B 45° 0 375 Y* Figure 13. 3 This is the initial equilibrium value of Y before the change in I Assume time... log 0.083333 =t log 0.5 3.585 = t Therefore Y will have exceeded 2 ,130 by the end of the fourth time period Only the most basic lagged Keynesian model has been considered so far in this section Other possible formulations have been suggested for the ways in which past income levels can determine current expenditure For example Ct = a + bYt−2 © 1993, 2003 Mike Rosser or Ct = a + b1 Yt−1 + b2 Yt−2 The... 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 equilibrium value, 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 would never... level of Yt which was 435 in time period ‘minus one’ Therefore C0 = a + bYt−1 = 40 + 0.6(435) = 301 Y0 = C0 + I0 = 301 + 110 = 411 (13) This is the same initial value Y0 as that calculated in Example 13. 7 The new equilibrium value of income is Y∗ = 40 + 110 150 a + It = = = 375 1−b 1 − 0.6 0.4 (14) Substituting (13) and (14) into the formula for the general solution to the difference equation derived... 1993, 2003 Mike Rosser Substituting these values into the general solution for the lagged Keynesian macroeconomic model difference equation, we get the general solution for this example, which is Yt = Y ∗ + (Y0 − Y ∗ )bt = 2,140 + (2,020 − 2,140)0.5t = 2,140 − 120(0.5)t Therefore, six time periods after the increase in investment Y6 = 2,140 − 120(0.5)6 = 2,140 − 1.875 = 2 ,138 .125 Example 13. 9 How many... to reach 2 ,130 in the preceding example? Solution We know that Yt = 2 ,130 and we wish to find t Thus substituting this value and the initial value Y0 and the new equilibrium Y ∗ calculated in Example 13. 8, into the Keynesian model general solution formula Yt = Y ∗ + (Y0 − Y ∗ )bt we get 2 ,130 = 2,140 + (2,020 − 2,140)0.5t −10 = −120(0.5)t 0.08333 = (0.5)t To get t, put this into the log form, which... first substitute the given value of 50 for Q0 into the demand function so that 160 − 2P0 = 50 = Q0 110 = 2P0 55 = P0 © 1993, 2003 Mike Rosser Now substitute this value for P0 into the general solution (2) above, so that P0 = 55 = 45 + A(−1)0 55 = 45 + A 10 = A The specific solution to the difference equation for this example is therefore Pt = 45 + 10(−1)t Using this formula to calculate the first few values... 13. 69 P3 = 15 + 1.25(1.025)3 = 16.35 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 © 1993, 2003 Mike Rosser and it takes until time period 102 before price becomes negative, as the figures below show: P100 = 15 − 1.25(1.025)100 = 0.23 P101 = 15 + 1.25(1.025)101 . 115 14.25 -2 .25 13 3 92.5 15.38 1 .13 14 4 103.75 14.81 -0 .56 15 5 98.125 15.09 0.28 16 6 100.9375 14.95 -0 .14 17 7 99.53125 15.02 0.07 18 8 100.23438 14.99 -0 .04 19 9 99.882 813 15.01 0.02. 100.05859 15.00 -0 .01 © 1993, 2003 Mike Rosser Table 13. 2 CELL Enter Explanation As in Table 13. 1 Enter all labels and column headings shown in Table 13. 1 Note: do not enter for the word “STABLE”. 183.75 22.03 -3 .28 13 3 144.375 26.95 4.92 14 4 203.4375 19.57 -7 .38 15 5 114.84375 30.64 11.07 16 6 247.73438 14.03 -1 6.61 17 7 48.398438 38.95 24.92 18 8 347.40234 1.57 -3 7.38 19 9 -1 01.10352