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 single-commodity 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.

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 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

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 Psb

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.

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)

= 1.5 * 10000/4

= 3750

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

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.

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 =

Given a demand function Qs = aPe where e is the supply elasticity

Change in consumer surplus =

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

DEPARTMENT OF ECONOMICS2002/2003

Senior Sophister

ECONOMICS OF FOOD MARKETS

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.
  1. 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.

  1. 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.
  1. 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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