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Exercises for Class 4 (week 4)
In having a go at the following questions you should be seeking to ensure that
you are able to do the following:
1.
2.
3.
4.
apply formulae for growth
create and use parameter boxes
build models with a dynamic structure
plot graphs of time paths
Ex. 4.1
Set up a worksheet with time in the first column running from 2000 to 2025
and GDP in the second column. In a parameter box away to the side put
entries for the initial value of GDP and for the growth rate. In the parameter
cells set these values initially to 10000 and 0.02 respectively. In the table put
a formula for GDP in 2001 that refers to these initial values: GDP in 2001 will
be (1 + 0.02) times its value in 2000, although you will be referencing the
parameter cell which contains 0.02.
Document the time path for GDP for the period 2000-2025. Then change the
growth rate to 3% per annum via the parameter box. Plot a chart of GDP for
the two growth rates specified.
Ex. 4.2
A simple macro model with many similarities to the cobweb model covered in
the session notes
In this example we will follow the convention of referring to time with a
subscript t rather than a t in brackets: it makes no difference which way you
do it.
(i) Assume that an economy is characterised as follows.
Consumption: Ct = g + h*Yt-1
Investment: It = I0
Aggregate supply equals aggregate demand: Yt = Ct + It
Set up a worksheet with time, consumption, investment and income as your
column headings.
The time variable should run from 0 to 30 in steps of 1.
Put the parameters (g and h) away to the top or side in a parameter box,
giving them values of 50 and 0.7 to start.
The initial conditions, which should also go into a parameter box, are a
starting value for income Y = 500 (=aggregate demand = aggregate supply)
and a static level of investment that remains at 150 throughout.
(ii) Plot the time path of income. You should find it converging to an
equilibrium value (with income taking the same value time after time and thus
Yt =Yt-1. Check by solving algebraically the equation Y=(50+0. 7*Y)+150 that
this value is the correct one.
Ex 4.3
The Multiplier-Accelerator Model is an example of dynamic models used in
growth theory and macroeconomics. It is formulated as follows:
Investment in period t (denoted I(t)) is proportional to the lagged rate of
change of income, where income is denoted Y(t):
I(t)=k{Y(t-1)- Y(t-2)} where k is a constant. Consumption in period t (denoted
C(t)) depends on lagged income: C(t) = g + h Y(t-1) where g is a constant
and h is the marginal propensity to consume, where 0 < h < 1.
Income is the sum of consumption and investment so that: C(t) + I(t) = Y(t)
In technical terms this is a second-order difference equation, since Y(t)
depends on income during the previous 2 time periods. The time-path of
income Y(t) to which it gives rise depends on the relationship between the
parameters.
You are asked to set up a worksheet to track the behaviour of income through
time. Use a parameter box into which you put values for g, h, k and the initial
values of income for the first two periods Y(0) and Y(1). Your worksheet
should have a time measure in column B running from 0 in steps of 1 to, say,
50. In column C will be the time path for C(t). In column D will be the time path
for I(t), referring to values of income for the previous 2 time periods. The first
two entries in the consumption and investment columns will be empty: only
time and income are given.
As starting values for the parameters you might try g = 50, h = 0.8, k=0.70,
Y(0) = 425 and Y(1) = 450.
(i) Plot a graph of the time path of income.
(ii) Experiment with various combinations of the parameters to see whether
you can deduce anything about what determines the characteristics of the
time path of income (hint: is any parameter likely to key-influence the Y level?)
Ex 4.4
A market has demand (D) and supply (S) functions given by: xd(t) = 15 - 5p(t)
and xs(t) = -1 + 3p(t-1).
Set up a worksheet with time (0,1,2,..,30) in the first column, price in the
second and quantity in the third.
In a parameter box to the side have entries for the intercept and slope for
each of the demand and supply functions in turn.
Set the demand and supply parameters to 15, -5, -1, +3 as the demand and
supply functions specify.
You will also need a parameter box for the initial value of price p(0). Set the
initial price in this parameter box to 3.
In the table, go to the cell which requires the Price for time period 0, and insert
the parameter box reference for in the initial value of price. You should now
have a table showing Time=0, Price=3, Quantity still blank.
Remember that the quantity S depends on the price in the previous period.
Therefore it is not possible to calculate quantity for period 0.
So the next move is to go to the quantity cell for period 1 and calculate xs(t) =
-1 + 3p(t-1)
You should find that quantity supplied in Period 1 is 8. But in order to sell this
chosen level of output, price must be set to equate the quantity demanded xd
with the quantity supplied xs., so that xd = xs
First we have to solve the Demand equation for Price, respecifying xd(t) = 15 5p(t) in terms of p.
This gives p = (15-xd)/5
[Note of course that you will refer to the values 15 and 5 by reference to the
parameter box.]
As xd = xs in equilibrium, we need to solve this for the quantity supplied, thus
price p = (15-xs)/5
But we already know the quantity supplied, so we can solve this directly. You
should get Price for Period 1 = 1.4.
Once that has been solved, the rest of the table can be completed rapidly by
copying the formulae down.
(i) Plot the time path of price for the periods t = 0,1,2,..30. You should find that
by about Period 11 or 12, price should be converging on 2 and quantity on 5.
(ii) Comment on the sensitivity of the model to parameter changes by
experimenting with different values for the demand and supply slopes. Just for
fun, try changing the supply slope from 3 to 5 to 7. If the model doesn’t
change, you haven’t linked your table to the parameter boxes correctly. See
how although our original parameters gave a convergent solution, different
values can cause the model to oscillate or diverge.
Suggested solutions for all the 4 exercises can be visualised here