Vertex42 The Excel Nexus

Consider the supply and demand equations

(a) Assuming that the market is in equilibrium, write down a difference equation for price.

(b) Given that P0 = 1, find the values of the price, Pt for t = 1, 2,..., 10 and plot a graph of Pt against t. Describe the qualitative behaviour of the time path.

Solution

12 - Pt = P?4 which rearranges to give

Notice that this difference equation is not of the form considered in this section, so we cannot obtain an explicit formula for Pt in terms of t.

(b) We are given that P0 = 1, so we can compute the values of Pv P2,... in turn. Setting t = 1 in the difference equation gives

This number can now be substituted into the difference equation, with t = 2, to get

Excel provides an easy way to perform the calculations. We type in the values, 0, 1, 2, ..., 10 for each time period down the first column, type in the value of the initial price, P0, in the second column, and then copy the relevant formula down the second column to generate successive values of Pt. Once this has been done, Chart Wizard can be used to draw the time path.

The most appropriate way of representing the results graphically is to use a bar chart. We would like the numbers in the first column of the spreadsheet to act as labels for the bars on the horizontal axis. Unfortunately, unless we tell Excel that we want to do this, it will actually produce two sets of bars on the same diagram, using the numbers in column A as heights for the first set of bars, and the numbers in column B as heights for the second set. This can be avoided by entering the values down the first column as text. This is done by first highlighting column A and then selecting Format: Cells from the menu bar. We choose the Number tab, and click on Text and OK. We can now finally enter the values of 0, 1, 2,..., 10 in the first column, together with the headings and numerical value of P0 in column 2 as shown in Figure 9.3. The remaining entries in column B are worked out using the difference equation

For example, the entry in cell B6 is Pl which is calculated from P1 = 12- P0 8

The value of P0 is located in cell B5, so we need to type the formula =12-B5A0.8

into B6. Subsequent values are worked out in the same way, so all we need do is drag the formula down to B15 to complete the table.

Chart Wizard can now be used to plot this time path. We first highlight both columns and then click on Chart Wizard. The bar chart that we want is the one that is automatically displayed, so we just press the Finish button. To close the gaps between the bars, click on any one bar. A square dot will appear on each bar to indicate that all bars have been selected. We then choose Format: Selected data series for the menu bar. Finally, click on the Options tab, reduce the Gap Width to zero and click OK. The final spreadsheet is shown in Figure 9.4. It illustrates the oscillatory convergence and shows that the price eventually settles down to a value just greater than 7.

Complementary function of a difference equation The solution of the difference equation, Yt = bYt-l + c when the constant c is replaced by zero.

Difference equation An equation that relates consecutive terms of a sequence of numbers. Dynamics Analysis of how equilibrium values vary over time.

Equilibrium value of a difference equation A solution of a difference equation that does not vary over time; it is the limiting value of Yn as n tends to infinity.

General solution of a difference equation The solution of a difference equation that contains an arbitary constant. It is the sum of the complementary function and particular solution.

Initial condition The value of Y0 that needs to be specified to obtain a unique solution of a difference equation.

Particular solution of a difference equation Any one solution of a difference equation such as Yt = bYt-l + c.

Recurrence relation An alternative phrase for a difference equation. It is an expression for Yn in terms of Yn-1 (and possibly Yn-2, Yn-3 etc).

Stable (unstable) equilibrium An economic model in which the solution of the associated difference equation converges (diverges).

Uniformly convergent sequence A sequence of numbers which progressively increases (or decreases) to a finite limit.

Uniformly divergent sequence A sequence of numbers which progressively increases (or decreases) without a finite limit.

6 Calculate the first four terms of the sequences defined by the following difference equations. Hence write down a formula for Yt in terms of t. Comment on the qualitative behaviour of the solution in each case.

(a) Yt = Yt_x + 2; Y0 = 0 (b) Yt = -Yt_x + 6; Y0 = 4 (c) Yt = OY,- + 3; Y0 = 3

7 Solve the following difference equations with the specified initial conditions:

Comment on the qualitative behaviour of the solution as t increases.

8 Show, by substituting into the difference equation, that c

9 Consider the two-sector model:

Given that Y0 = 3000, find an expression for Yt. Is this system stable or unstable?

10 Consider the supply and demand equations

Assuming that equilibrium conditions prevail, find an expression for Pt when P0 = 70. Is the system stable or unstable?

11 The Harrod-Domar model of the growth of an economy is based on three assumptions.

(1) Savings, St, in any time period are proportional to income, Yt, in that period, so that

(2) Investment, It, in any time period is proportional to the change in income from the previous period to the current period so that

(3) Investment and savings are equal in any period so that

Use these assumptions to show that

and hence write down a formula for Yt in terms of Y0. Comment on the stability of the system in the case when a = 0.1 and p = 1.4, and write down expressions for St and It in terms of Y0. __

12 Consider the difference equation

(a) Write down the complementary function.

(b) By substituting Yt = D(0.6) into this equation, find a particular solution.

(c) Use your answers to parts (a) and (b) to write down the general solution and hence find the specific solution that satisfies the initial condition, Y0 = 9.

(d) Is the solution in part (c) stable or unstable?

13 Consider the difference equation

(a) Write down the complementary function.

(b) By substituting Yt = Dt + E into this equation, find a particular solution.

(c) Use your answers to parts (a) and (b) to write down the general solution and hence find the specific solution that satisfies the initial condition, Y0 = 10.

(d) Is the solution in part (c) stable or unstable?

14 (Excel) Consider the two-sector model

(a) Write down a difference equation for Yt.

(b) Given that Y0 = 10, calculate the values of Yt for t = 1, 2, ... 8 and plot these values on a diagram. Is this system stable or unstable?

(c) Does the qualitative behaviour of the system depend on the initial value of Y0?

15 (Excel) An economic growth model is based on three assumptions:

(1) Aggregate output, yt, in time period t depends on capital stock, kt, according to

(2) Capital stock in time period t + 1 is given by k,+i = 0.99k, + s, where the first term reflects the fact that capital stock has depreciated by 1%, and the second term denotes the output that is saved during period t.

(3) Savings during period t are one-fifth of income so that s, = 0.2*,

(a) Use these assumptions to write down a difference equation for kt.

(b) Given that k0 = 7000, find the equilibrium level of capital stock and state whether kt displays uniform or oscillatory convergence. Do you get the same behaviour for other initial values of capital stock? •-•

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