Download simulating market models

SIMULATING MARKET MODELS
Alan Matthews, October 2002
Introduction
This note explains the steps involved in building an empirical market model to assess
the impact of policy changes on prices and quantities produced and on the welfare of
consumers, producers and taxpayers. For simplicity we work with a singlecommodity model, but the steps are common to more complex multi-commodity
models. A method to solve the model using the Goal Seek command in Excel is
described at the end of the handout.
For example, suppose you are asked to examine the effects of introducing a consumer
subsidy for a product. The basic diagram will look like the following.
Price
S
Ps
Pd
Dplus subsidy
D
Identify the changes in consumer surplus and producer surplus on the diagram.
Identify also the change in government expenditure as a result of the subsidy. Note
the size of the welfare loss as a result of this distortion.
While diagrams can teach us a lot about the directions of change, policy makers
generally want quantification. They want to know how much it will cost the budget to
introduce the subsidy. They want to know what will be the effect on total
consumption of the product, on the incomes of producers, etc. For this, we need to
move from graphs to models.
Model-building steps
1. Assemble the basic data for the initial equilibrium
In a supply-demand model, you would expect to collect information on quantities
marketed (supply equals demand) and the market price. In an open economy model
you will also need information on trade. You will also need information on any initial
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distortions in the market, e.g. taxes, subsidies, tariffs, etc (referred to as policy
wedges).
2. Write down the general algebraic formulation.
At its simplest, a four equation model for a single market can be written as
Qs = f(Ps)
This is the supply function, stating that supply is a function of the producer price Ps.
Qd = g(Pd)
This is the demand function, stating that demand is a function of the consumer price
Pd.
Pd = Ps
This is the price equation which states that the producer and consumer prices are the
same, i.e. no marketing margin or transport costs are assumed.
Qs = Qd
This is the equilibrium condition that the market must clear.
This model can be modified in two ways:
a) to take account of marketing margins.
Pd = Ps + M
This represents a fixed marketing margin M. Its absolute value remains the same
regardless of the level of prices
Pd = Ps (1 + m)
This represents a proportional marketing margin m where m is entered in decimal
form, i.e. 0.25 represents a margin of 25%. Here the absolute value of the margin
changes as the level of prices change.
It is of course possible to combine both types of margin but we usually make one
assumption or the other.
b) to take account of taxes and subsidies
Here we need to distinguish between the original market (demand or supply) curves
and the tax (subsidy)-laden curves. Take the example of VAT, a tax on consumption.
We denote the ex-tax price as Pc, which in turn is either equal to the producer price Ps
or the producer price plus some marketing margin.
However, the price to which consumers respond is the tax-inclusive price. With one
additional price variable, we need another equation to explain it.
Pd = Pc (1 + t) where t is the rate of VAT, e.g. 0.21 or 21%.
In the case of an excise or absolute tax T, then the equation becomes
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Pd = Pc + T where T is the value of the excise.
We often simplify the model and relate the tax-inclusive demand price directly to the
producer price. For example, in the case of a fixed marketing margin and a VAT, the
price equation becomes
Pd = (Ps + M) * (1 + t)
Note that the VAT tax applies to the combined value of the raw material and
marketing margin value added.
3. Choose functional form
linear vs. power (also called Cobb-Douglas)
Linear equations have the advantage that they are easier to work with algebraically.
Cobb-Douglas formulations have the advantage that the exponents are the relevant
elasticities and taking logs converts them to a linear form.
Example of a linear form of the supply equation
Qs = a + b Pp
Example of a Cobb-Douglas form of the supply equation
Qs = a Ps b
Taking natural logs of both sides of this equation gives
ln Qs = ln a + b ln Ps
which can be an easier form to work with in Excel.
4. Choose parameter values
The most important parameter values are the elasticities assumed (other parameters
which might be used include price transmission elasticities in open economy models
or acreage set-aside percentages). Elasticity values might be estimated from
econometric work but, for large-scale models, are often based on a literature search.
5. Calibration.
Calibration is the process of initialising the model to the base year data. The purpose
of calibration is to identify the parameters in each supply and demand equation given
their functional form, estimates of elasticity values and one price-quantity observation,
such that the model reproduces the base period data and is consistent with the
elasticity parameter assumptions.
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In the Cobb-Douglas form of the supply function, for example, given the supply
elasticity b, the problem of calibration is to find the value of the constant term a which
is consistent with the data. This is easily done in the spreadsheet using the LN()
function to convert the equation to a linear form.
If you are assuming linear supply and demand equations in the first place, then
calibration means finding the values of the slope and the constant of the function
given knowledge of the price and quantity values and the elasticity. Here we make
use of the elasticity formula
Elasticity = Slope * Price / Quantity
to first determine the value of the slope. Then substitute this is to the supply or
demand equation to find the value of the constant.
Example of wheat supply function
Supply curve is assumed linear Qs = a + bPs
Current market supply of tickets = 10,000
Current market price = £4
Supply elasticity E = 1.5
The slope b can be found by rewriting the elasticity formula as
b = E . Qs/Ps
= 1.5 * 10000/4
= 3750
(1)
Constant term a is found by substituting current price and quantity values into the
supply equation
10000 = a + 3750 * 4
a = 10000 - 15000 = -5000
Therefore estimated equation is
Qs = -5000 + 3750 Ps
6. Specify policy simulation experiment.
A policy simulation experiment is specified by altering a policy wedge, for example,
changing the value of a tax or subsidy.
Working through the individual supply and demand equations will alter domestic
supply and demand quantities, and market disequilibrium will result. We need to
solve the model to find a new equilibrium.
7. Solution procedure
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For small models, the new equilibrium can be found algebraically but it is much easier
to use Goal Seek in Excel.
Using Goal Seek
When you can specify the answer you want, and you must work backward to find the
input value that gives you that answer, use Tools, Goal Seek in Excel. You use this
command, for example, to determine the needed growth rate to reach a sales goal, or
to determine how many units must be sold to break even. We will use it to find the
price that ensures that market supply equals market demand.
To solve for a specific answer using Goal Seek, follow these steps:
1. Select a goal cell that contains a formula that you want to force to produce a
specific value.
2. Choose Tools, Goal Seek. The Goal Seek dialog box appears (this is also visible
in the figure). Notice that the Set Cell text box contains the cell selected in step 1.
3. In the To Value text box, type the target value you want to reach. If your formula
in the goal cell gives the difference between the values of two equations, then
enter 0 here to find the equilibrium value for the unknown.
4. In the By Changing Cell text box, enter the cell reference of the input cell. In the
example, the cell being changed is B19, so enter this reference. For a system of
equations, this is the cell containing the unknown parameter whose value you want
to find.
5. Choose OK.
6. After a solution has been found, choose OK to replace the values in the original
worksheet with the new values shown on-screen, or choose Cancel to keep the
original values.
7. Advice on preparing solution worksheets
Always document the worksheet clearly. Put in headings and labels, different colours
and backgrounds, comments (Insert, Comment command) and callouts (found on the
Drawing Toolbar) to draw attention to how your worksheet is constructed.
Name crucial cells in the your worksheet to read your formulae easier. Use Insert,
Name, Define command or simply type in the name in the Name box on the extreme
left corner of the Formula Bar.
Calculation of welfare changes
The areas representing welfare changes can be calculated using simple geometry in the
case of linear supply and demand curves, or by using integration (necessary if
constant-elasticity functional forms are assumed).
Linear functions
For example, in the case of the change in producer surplus, the relevant area is shown
in grey in the figure below.
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Price
P1
P0
Qo
Q1 Quantity
This might be measured as Q1* (P1-P0) - 0.5 * (Q1-Q0) * (P1 - P0)
Alternatively, the shaded area is the integral under the supply curve between the prices
P1 and P0 can be measured using integration
If supply curve is Q = a + b.P
then, the change in producer surplus is  (a + b.P) dP which evaluates to
a.P + 0.5 * b.P2
This expression is then evaluated for the price range P0 to P1.
Either approach will give the same answer for the change in producer surplus.
The change in consumer surplus is found in an analogous manner.
The change in government revenue must also be calculated. In the case of border
interventions, it will be the difference between the final expenditure on export
subsidies or revenue from import duties and the original cost of export subsidies or
revenue from import duties. These amounts are simply calculated as the quantity of
export/imports multiplied by the size of the export subsidy or import duty.
A similar procedure is followed in the case of a consumer or input subsidy (or tax).
Constant elasticity functions
The following formulae apply:
Given a supply function Qs = aPd where d is the supply elasticity
Change in producer surplus =  aP d .dP 

a
1 d 
1 d 
P1
 P0
1 d

Given a demand function Qs = aPe where e is the supply elasticity
Change in consumer surplus =  aP e .dP 

a
1 e 
1 e 
P0
 P1
1 e
Note the order of the prices in both equations.
Note that in Excel the exponentiation operator is ‘^’
e.g. to write P2 in Excel, enter =P^2
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
DEPARTMENT OF ECONOMICS
Senior Sophister
ECONOMICS OF FOOD MARKETS
2002/2003
Simulation exercises with the Partial Equilibrium EU Wheat Market
Model using Excel
These exercises should be carried out using the Excel file Wheat.xls
1. Calculate the price and quantity changes following the introduction of a 30 per
cent consumer subsidy on bread (assumed the same as wheat in this model).
Evaluate the changes in consumer surplus(CS), producer surplus (PS) and
government expenditure.
2. How are the results of the previous policy changed if you assume
(a) a supply elasticity of 3 instead of 1.5
(b) a demand elasticity of –0.2 instead of –0.8
Examine the different changes in CS and PS in the three scenarios. What does this
tell you about the role of supply and demand elasticities in determining the incidence
of a tax/subsidy.
3. Calculate the price and quantity changes following the introduction of a 2
euro/tonne direct payment (assumed coupled to production). Evaluate the changes
in consumer surplus, producer surplus and government expenditure.
4. How are the results of the previous policy changed if you assume
(a) a supply elasticity of 3 instead of 1.5
(b) a demand elasticity of –0.2 instead of –0.8
(c) the direct payment is assumed decoupled from production.
Examine the different changes in CS and PS in the three scenarios. What does this
tell you about the role of supply and demand elasticities in determining the incidence
of a tax/subsidy.
Alan Matthews
October 2002
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