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6. An introduction to the use of economics in the planning and evaluation of disease control programmes


6.1 Introduction
6.2 Prices appropriate for use in economic analyses
6.3 Compound interest, discounting, annual rates of growth and annual loan repayments


6.1 Introduction


6.1.1 Basic philosophy
6.1.2 Application of economics disease control policy


6.1.1 Basic philosophy

Economics is a social science dealing with the production and distribution of goods and hence of wealth. It analyses how scarce resources are allocated between different uses and groups within the economy. Originally, economic thought was developed under the name "political economy" and examined the production and distribution of wealth in a society composed of landlords, peasants and artisans With the advent of industrialization, thinkers looked at the economic relationship between capitalists, workers and landlords. This approach was the one taken by Marx and underlies Marxist economics. Modern economies in the "capitalist" societies looks at the economic interactions between producers and consumers, who meet in the market place and try to satisfy their needs. Its aim is to analyst objectively the "positive" i.e. the verifiable or factual aspects of the economic relationships in society, and thus to derive generally applicable theories. It does not concern itself directly with the "normative" aspects which relate to value judgements about how the economic process ought to function.

The study of economics is conventionally divided into two areas. Micro-economics analyses the behaviour of individual producers and consumers, focussing on the factors influencing their levels of production and consumption and the mix of goods involved. Macro-economics analyses the economy as a whole, and deals with such topics as national income, balance of payments, overall savings and investment.

Development economics has emerged as a branch dealing with the specific problems of the less developed countries. It tries to analyse and explain the particular situation of these countries and to examine economic policies, such as price control, subsidies and taxes, and the channelling of investment funds into certain areas, which can help overcome their problems and improve their people's standard of living. The topics covered include an analysis of the causes and symptoms of poverty, of the dichotomy between the agricultural and the industrial sector in Third World countries, and of the extent of the bias in actual development towards urban areas. Development economics examines the questions of choice of technology, unemployment and underemployment, migration and land reform, from an economic point of view and also studies the roles of trade and commodity markets.

Project appraisal, the economic analysis of projects before they are undertaken (ex-ante analysis), and project evaluation, the assessment of projects after they have been undertaken (ex-post analysis), are practical applications of economic principles to decision-making based on a social benefit-cost analysis. This consists of setting out costs and benefits over a number of years and comparing them according to certain prescribed conventions so as to determine whether the project would be profitable. Budgeting and accounting are also techniques of applied economics.

6.1.2 Application of economics disease control policy

Economics contributes to the improvement of policy formulation and decision-making for animal health projects and programmes at four levels:

· Economic theory explains the behaviour of producers and consumers, and the effect of this on the price structure and on the output of the economy as a whole. In the livestock sector, it explains how economic factors influence producers, how they decide what and how much to produce, what prices are acceptable to them, why production is expanded or contracted, how much they invest etc. It also explains the economic factors underlying demand for livestock products, how these affect the amount and mix of products bought, and how prices are fixed in different circumstances (micro-economics). e The economic aspects of the different livestock production systems can be described by collecting relevant information and using it as well as the knowledge derived from economic theory to analyse how producers and consumers interact. A particular livestock production system can be described in economic terms by looking at the value of output, the cost of the inputs, calculating the income received by the producers, butchers, traders and other middlemen, and examining the final price paid by the consumers.

· Having characterized the production systems involved, as well as the interactions between the consumers and producers, it becomes possible to examine and predict the likely economic effects of any changes introduced into the sector. Such changes would include both changes affecting prices of inputs or outputs, which would affect the incomes of consumers and, therefore, demand, and changes in the technical coefficients of output due to introducing improved inputs, changing the animal health picture etc.

· Finally, the techniques of economic analysis make it possible to arrange this information so as to provide the basic yardsticks for ranking and hence comparing different programmes, projects or measures, and assessing their overall economic feasibility.

Thus, for an animal health project, economic theory can help explain producers' behaviour, describe the production systems involved, then help to predict and quantify the effect of the project on output, prices, demand and incomes, and, finally, provide a framework for arranging this information in the form of a benefit-cost analysis. Then, having ranked and compared the alternatives, a decision can be made whether to implement the project or not.

Obviously, decisions cannot be taken on the basis of economic considerations alone. First, the technical feasibility of any proposed measure must be examined by the relevant specialists (veterinarians, animal husbandry experts, sociologists, management experts etc). Second, its overall compatibility with the stated policies and goals of the livestock sector must be ensured, and, third, its feasibility from an organizational and social point of view needs to be verified.

In this manual, the methodology of the benefit-cost analysis is examined in some detail with regard to long-term decisions on animal health programmes. Let us consider some of the basic economic principles before applying them.

6.2 Prices appropriate for use in economic analyses


6.2.1 Theoretical aspects
6.2.2 Opportunity cost and the choice of prices in economic analysis
6.2.3 Adjusting for inflation - price conversions and price indexes


6.2.1 Theoretical aspects

Supply and demand

Prices are the "labels" or weights used in economic decision-making. As such, an understanding of how they are derived and what they represent is crucial. Money is the "unit" in terms of which prices of goods are given in a cash economy, although barter can fix their relative values. For example, if a kilogram of meat costs US$ 3 and a yard of cloth US$ 1.50, 2 yards of cloth could be exchanged for 1 kg of meat in the absence of money, or both could be paid for in cowries, manillas or some other acceptable currency.

Historically, price theory began with the concept of goods having either a scarcity value or a value because of the labour needed to produce them. Modern economies sees prices as being determined by the interaction of supply and demand, reflecting both the balance of the price producers are willing to accept, taking into account their production costs, and the price consumers are willing to pay for a certain quantity of goods. For most goods, the quantity offered increases with increasing price, but the quantity demanded decreases. This process is illustrated in Figure 9.

Figure 9. The equilibrium of supply and demand.

If supply equals demand, the market is said to be "in equilibrium" at price P0. This price is also referred to as the market-clearing price, and it represents the point at which all that is offered is bought. At a higher price, supply exceeds demand, since producers are willing to offer more and consumers are reluctant to purchase. The converse is true if the price is lower than the market-clearing price, in which case consumers are eager to buy but producers are reluctant to sell or produce, and, consequently, the quantity demanded exceeds that supplied. If the individuals were bargaining in a real market place, they would continue to offer each other prices until they arrived at a mutually agreeable price, or else the consumer would decide not to buy or the producer not to sell.

Example: Suppose that a government fixes a maximum price for meat with the objective of ensuring that low-income consumers can afford the commodity. If this price is below the market-clearing price, producers would like to charge more, demand outstrips supply, and a black market develops where meat is sold at prices nearer to, or even exceeding, the market-clearing price to those consumers who can afford it. Conversely, if a government fixes a minimum price which is above the market-clearing price, supply will tend to outstrip demand at that price and suppliers will be forced to sell off their goods cheaply, avoiding the government regulations. This commonly happens when there is a fixed minimum wage for labour: if many people are looking for employment, a large number will end up accepting jobs below the minimum wage.

In fact, if a government wants its price-setting policies to be effective, it will often need to pay a subsidy to compensate producers, if the price is too low, or consumers, if it is too high. The government would need sufficient knowledge of the supply and demand curves for the product, i.e. the lines illustrating what quantity is demanded or supplied at which price, in order to work out at what price (P1) the quantity supplied would be equal to that demanded at a minimum price (P2) and representing the amount the government would like people to consume. The government can then pay producers a subsidy equivalent to the difference between P1 and P2, so that the supply rises to the level equal to the quantity demanded at the minimum price, and the market clears.

The discussion of price theory has raised several points which need to be considered when deciding which prices to use in various economic studies. These can be summarised as follows:

· Since for most goods the quantity demanded falls as the price rises, governments can stimulate demand for an item by setting a low price. Conversely, they can lower demand by setting a high price. A low price can be supported by a subsidy, a high price may be enforced by a purchase tax. For example, the consumption of milk may be encouraged by setting a low price for consumers, backed up by a subsidy to producers. Similarly, new inputs into production systems, such as fertilizers, improved breeds of liverstock, ploughs etc, may be encouraged by subsidizing their cost to whoever is prepared to use them. In the absence of a support for artificially high or low prices, black markets tend to emerge.

· Different consumers may pay different prices for the same goods. For example, because of the costs of transport, goods may cost more in isolated rural areas or if they are imported from another region or country. Products may be more expensive when bought in retail outlets with high overheads, while items sold in large quantities are usually cheaper. If a good passes through many hands before it is sold to the final consumer, it will be more expensive since every middleman on the way expects to make some profit. These are all concrete reasons for price variations.

· A more subtle effect is that of the individual consumer's bargaining power. In the market, one person may be better or worse at negotiating a price than another. On a wider scale, the price an individual will pay may depend on such things as his or her influence in society, whether the seller wishes to gain favour, or considers the purchaser rich and capable of paying a good price. All these effects are intensified in a black market.

· A variety of prices exists for each item affected by a government subsidy or tax. These include:

- The price paid by the consumer, which may include a purchase tax or is the portion of the cost after the subsidy has been removed.

- The price received by the producer, which is the price before purchase tax is added or, in the case of a subsidy, the equivalent to the price paid by the consumer plus the government subsidy.

- The cost to the government of the subsidy or the revenue brought in by the tax.

- The cost to the nation, which is roughly equivalent to the price paid to the producer. A government tax or subsidy is a transfer between tax payers who pay the subsidy or tax and those who benefit from it, either by receiving the subsidy or using the facilities financed with the money collected from the tax.

The concept of elasticity

The concept of supply and demand as discussed in the previous section has been much simplified. In practice there are often deviations from the general rule of price increases leading to a fall in demand and a rise in supply. In order to be able to measure precisely how supply and demand respond to changes in prices, the concept of elasticity was developed, which is expressed by the following formula:


Elasticity should be expressed as a positive number. A minus sign is placed before the equation in the case of the price elasticity of demand, since demand falls as price increases, making the overall result positive. Thus, if the demand or supply changes by the same percentage as price, the elasticity is I. If a price increases by 10% and elasticity is 2, supply will increase by 20%. Goods are said to be inelastic if the demand for them changes very little with price, in which case the calculated elasticity is less than 1. Such goods are generally necessities, for which demand is very stable. For luxuries, demand is generally more elastic.

In some cases producers have a target income rather than trying to maximise their profits, and once this income is reached, they cease to supply more goods. Thus, beyond a certain point, price increases may lead to a reduction m supply. This has been alleged to be the case with some nomadic cattle keepers, who only sell their animals to meet their fixed cash needs for such items as school fees, taxes, clothing, veterinary expenses etc.


Changes in income must be taken into account when trying to project how the demand for livestock products will evolve over the years. Generally, the demand for a good increases with increasing incomes. However, as people get wealthier they reduce the consumption of goods that are considered inferior, such as very cheap cuts of meat and/or clothing.

The concept of elasticity thus has the following practical applications in the formulation and assessment of animal disease control policy:

· It assists in the general understanding of the livestock sector, particularly in determining what the future supply and demand are likely to be in response to changes in prices and incomes.

· It is crucial in determining what prices to charge producers for various veterinary treatments. Figure 10 illustrates a hypothetical relationship between the demand for vaccination and its price.

Figure 10. Demand for vaccination at various prices per dose of vaccine.

The elasticity of demand varies, being very elastic as the price of an individual vaccination falls from US$ 0.50 to about US$ 0.10 and relatively inelastic at US$ 0.75 per vaccination. Therefore, to ensure a vaccination coverage of about 80%, it will be necessary to provide the vaccination free of charge. To increase the coverage further, livestock owners would actually have to be paid or coerced. If vaccinations cost more than US$ 0.90 each, less than 5 to 10% of the livestock would be vaccinated. Suppose that a coverage of 755 is thought necessary for a voluntary vaccination campaign to be effective, then the maximum amount that can be charged by the veterinary service is US$ 0.10. If the vaccine costs US$ 0.12 per dose and the average cost of distributing and administering the distribution is US$ 0.27, it will be necessary to subsidise the campaign to the extent of US$ 0.29 per dose. The vaccine might be cheaper if purchased in bulk, and the cost per dose for distribution and administration might go down as more animals are presented at each vaccination session.

However, experience has shown that this analysis of livestock producers' response to opportunities for vaccination may not always correspond to reality. In some cases, producers avoid having their animals vaccinated when the vaccination is free but present them when a fee is imposed. This does not reflect a failure of economic theory to cope with reality, rather the belief of producers that free vaccinations may be inferior to those that are charged for. Their decision is thus quite rational from the economic point of view: it is not worth their while to spend time getting their animals together for a free vaccination of no value, whereas it is worth paying for one that confers a real benefit.

Prices of factors of production and of durable goods

So far we have analysed prices as though they were for consumer goods that were purchased outright. Prices for durable goods and the various inputs of production are slightly more complex. There are three factors of production to be considered:

· Labour, which can be divided into various grades;
· Land, which includes natural resources; and
· Capital, which covers both money itself and production goods such as livestock and machinery.

A fourth factor, entrepreneurship or management, is sometimes added to cover management and risk taking.

The factors of production are subject to the laws of supply and demand in the same way as other goods, but the demand for them is described as derived demand, since it depends on the demand for the products the factors are used to make. Given sufficient information about the production conditions, prices and the demand for final products, input-output models can be constructed for the whole economy to determine the demand for the different factors of production.

The many inputs of production and most durable goods can usually be bought in two ways:

· Outright purchase, which confers on the owner all the incomes that can be earned from using a particular input or all the benefits from a particular durable good.

· Renting or hiring, which enables the purchaser to use the item for a stated period of time.

Thus a durable consumer good, such as a television, can be owned or rented. Machines used for production (tractors, draught oxen, harvesting equipment) can be hired or owned. Labour is usually rented out by an individual by the hour or week against a fixed wage. Capital in the sense of machinery and buildings can be owned or rented. Money in the sense of cash can either be owned, in which case the owner reaps the income it can earn, or rented in return for a payment per unit of the time that it is used. This "rental" is conventionally referred to as borrowing and the payment per unit of time is the interest. Similarly, land or mineral rights can be owned or rented for a period of time.

Underlying all investment or project appraisals is the concept that the various inputs or factors of production at the disposal of an individual or a nation should be used so as to earn that individual or nation the highest possible income. Thus, just as an individual should not borrow money at an interest of 10% per annum to finance an investment from which he expects a profit of 8% per annum, a nation should not invest resources in projects with a return of 8% when alternatives yielding 10% exist.

6.2.2 Opportunity cost and the choice of prices in economic analysis

In a project appraisal or budget, the main economic input lies in the choice of prices, since it is assumed that the technical inputs which give the main physical components of costs and benefits have been derived by the professionals responsible for ensuring the technical feasibility of the project. In the same way as all the assumptions necessary for deriving the physical parameters must be clearly stated, so the origin or derivation of every price or group of prices chosen must be given as well as the justification for using them. A simple rule determining which prices can be used in a particular analysis is that the prices chosen should approximate, as far as possible, to the opportunity cost of the relevant items to the individual, firm, institution or country from whose point of view the analysis is being made.

Opportunity cost and shadow prices

The opportunity cost of making a particular economic choice is given by the cost of whatever alternative production or consumption had to be foregone as a result of that choice. The allocation of labour in a village production system means that new projects introducing new work patterns need to take into account opportunity costs.

Example: The labour needed to grow fodder crops could be valued at the government's minimum wage rate of, say, US$ 5 a day. After consideration, this rate might be found artificially high, so a black market wage rate of US$ 3 per day might be applied. We may also look at the problem from the point of view of opportunity cost and ask the question, What would the farmer be doing with his time if he were not cultivating his fodder crop? If the answer is that he would be doing nothing but lying in the shade sleeping, the opportunity cost - unless he is very tired - may be nil. If the answer is that he would be drinking beer with his friends, it may be that the opportunity cost is negative - by not drinking he saves money and has fewer hangovers. Alternatively, his drinking may be a way of finding out information on marketing issues, pasture availability, local politics etc. Most often, however, the opportunity cost will be expressed in terms of another crop or of time spent trading or on craftwork or some other remunerative occupation. In order to assess the true cost of transferring the farmer's labour to fodder crop production, the cost of the income foregone from the alternative occupation must be estimated.

The opportunity cost of capital, i.e. of using money or investment funds, is the rate of return or interest rate that can be earned in alternative uses.

From the concept of opportunity cost, the idea of shadow prices can be derived. Shadow prices are used with the broad objective of bringing prices to values nearer their true opportunity cost and thus, in project analysis, they lead to the selection of projects which use up the different resources at rates reflecting the real cost to society. Shadow prices can be defined as artificial prices calculated for certain items in order to ensure that their real opportunity cost is taken into consideration when making decisions. These shadow prices may be different from the money actually received or paid for the items at the time they are used.

Shadow prices are generally used in the following circumstances:

- Where market prices do not reflect real opportunity costs. This is often the case when prices are fixed by the government or are affected by speculators indulging in monopolistic trading.

- To accomplish particular policy objectives by encouraging the use of some items by setting artificially low prices for them and discouraging that of others by setting artificially high prices.

Thus, in project appraisal, shadow prices will present the costs and benefits of the projects at prices that: a) reflect, as far as possible, the real opportunity costs of the choices being made and the policies being proposed; and b) follow government policy by making those projects that use a higher proportion of the inputs whose use or production the government wishes to encourage, seem relatively more profitable. This is because shadow prices give such inputs an artificially low cost and such outputs an artificially high value.

Shadow prices are most commonly used in the case of two commodities:

· Labour, which can be rather difficult to value in monetary terms, as was illustrated by the example given above. Moreover, governments often want to encourage projects that use a high proportion of local labour while maintaining a relatively high minimum wage rate. A low shadow price for labour would make such projects appear relatively cheaper compared to projects substituting other inputs for local labour.

· Foreign exchange. Foreign exchange is a market commodity just like any other. It is accumulated by exporting and receiving aid in hard currencies and spent on imports, foreign debt repayments etc. A low price for foreign exchange means that the value of the local currency is high. This is often felt to give the country prestige and to imply a strong economy. It also makes the repayment of international loans artificially cheap. As with any other market, an artificially low price will lead to demand exceeding supply. Imports are artificially cheap, but exports are artificially expensive and hence not competitive, resulting in a shortage of foreign exchange. So governments end up restricting imports by imposing quotas, licences or banning certain commodities. One way to ensure the selection of a project that saves foreign exchange is to use a high shadow price for it.

Shadow prices can be used for any commodity if the need arises. For instance, if the objective of government policy is to raise the living standard of a particular group of people in a country, shadow prices can be used to give a higher value to incomes gained by that group as compared to those of another group. A comprehensive system of shadow pricing based on world market prices has been devised by Little and Mirrlees (1977).

An example of the application of shadow prices is given in Table 36, which presents a comparison of costs of different techniques used for the control of tsetse in Nigeria. A shadow price for foreign exchange was calculated, based on the prevailing black market rate for the Naira (N). The shadow price calculated for labour was 1 N per day. This was partly based on the actual rate paid locally outside the civil service and on an estimate of alternative earnings in the rural sector. Since the shadow price for labour was lower than the market price of 2 N per day, its effect was to lower costs. The shadow price for foreign exchange was N 2.10 per pound sterling instead of N 1.40, thus increasing costs.

Given a choice of techniques between insecticidal spraying by ground teams and by helicopter at market prices, the difference in cost per km2, N 357 and N 400 respectively, was not large. However, 90% of the field costs for the helicopter consisted of foreign exchange as compared to 34% for ground spraying. In addition, 43% of ground-spray costs were payments for local labour while only 3% of the costs of helicopter spraying were used for this purpose. Taking the shadow prices into account, the resulting costs were N 354/km2 for ground spraying and N 552/km for helicopter spraying.

Generally, it is not recommended that individuals working within a government framework attempt to use a variety of shadow prices that they have calculated themselves. Ideally, the ministry in charge of planning and appraisal should give clear guidelines as to which shadow prices are acceptable. In the absence of this, individuals should make their initial calculations at market prices, and only if they feel that there is a strong case, should they apply their own shadow prices, stating clearly what these are and how they have been derived. Because the issue of shadow pricing is a complex one, the advice of a professional economist should be sought before attempting to assign shadow prices to goods and resources.

Table 36. Comparison of costs for and helicopter spraying against tsetse flies - Nigeria, 1978.

Component of costs/km2

Ground spraying

Helicopter spraying

Field costs



Breakdown of average field costs (%)



Insecticide*

16.7

35.4

Labour**

43.2

2.7

Flying time*

-

52.0

Junior staff

17.2

3.2

Senior staff

2.5

1.3

Vehicle running and maintenance

3.4

2.6

Depreciation of equipment*

17.0

2.8

Total

100.0

100.0

Average field cost of newly reclaimed area (N)



Without shadow prices

87.0

238.0

With shadow price for labour and foreign exchange

82.0

342.0

Adjustments and overheads to average field costs (N)



Barrier resprays

5.0

0.2

Resprays of reinvasions and residual foci

35.0

109.0

Costs of staff not included above (administrative, headquarters, junior and senior staff outside spraying season)

100.0

24.0

Share of all other costs of running units and headquarters

130.0

29.0

Total



Without shadow prices

270.0

162.2

With shadow prices for labour and foreign exchange used in respray operations

272.0

210.0

Final cost



Without using shadow prices

357.0

400.2

Using shadow prices

354.0

552.0

* Foreign exchange costs.
** Local labour costs.

Choice of prices for financial and economic analyses

In economic studies, a distinction is made between financial and economic analyses. Financial analyses examine the monetary implications of any particular activity by an individual person, enterprise or institution, looking at the actual expenses and receipts from the point of view of the individual or firm concerned. The prices used in these analyses are usually market prices.

Economic analyses study the effect of a particular activity on the whole economy. The prices used should approximate to their opportunity cost, so they may be shadow prices. Since the analysis is undertaken from the point of view of the whole economy, all prices are net of purchase taxes and subsidies.

As a study undertaken from the point of view of an individual person (firm or institution) examines the implications of a particular activity to that individual, the prices used must be those that the individual faces. Thus to a farmer who ends up buying all the supplementary feed for his cattle on the black market, the application of the government's subsidised price makes no sense. Supplying supplementary feed at subsidised prices costs the government the handling and distribution expenses plus the value of the subsidy. Whereas if a trader is involved, the feed brings him a profit if he sells it at a higher price, less his own costs of transport, handling, storage etc. These are all financial viewpoints.

From the nation's (economic) point of view, the cost of the supplementary feed is probably best estimated using the price paid by the livestock producer, if the feed is sold on the open market. In economic evaluations involving most agricultural and livestock products, the so called "farm-gate price", which is the price paid to the producer, should be used. The retail price paid by consumers includes the profits of middlemen, transport and handling charges etc. which do not form part of the real value of the product. Where the farm-gate price is artificially fixed, a shadow price reflecting the black market price may be used. World market prices for particular items should only be applied if these prices are being used throughout and if the government or agency for whom the evaluation is being undertaken desires this.

The distinction between economic and financial analyses will be used throughout the rest of this manual. Up to now, the word "economic" has been used to cover both aspects. Used on its own without contrasting it to the word "financial", it will continue to be the general term covering all studies of this nature.

6.2.3 Adjusting for inflation - price conversions and price indexes

Dealing with inflation falls naturally into a general discussion on prices but the reader is also referred to the relevant sections in Chapter 8 on benefit-cost analysis. The relationship of inflation to interest rates is discussed in Section 6.3, as is the principle of compounding, which will be of use in estimating the effect of an annual rate of inflation on prices over a number of years.

For the purposes of project appraisal, making budgets or other economic or financial activities, it is often necessary to convert prices at current levels (i.e. for the year in which they occur) to constant values i.e. to those in a chosen base year.

Since any cost (C) is obtained by multiplying the quantity (Q) by the price (P) i.e.

C = P x Q

it follows that, if for any year two out of the three items (C, P or Q) are known, and the price for the base year is known, costs can be converted to their value in the base year. Most commonly, it will be necessary to convert the cost of a particular item or undertaking in year n to that in the base year 0. Since the item or undertaking is the same, it follows that:

Q0 = Qn

so that

C0 = Cn x P0/Pn

i.e. the costs in the year n are converted to costs in the base year by multiplying them by the ratio obtained when prices in the base year (P0) are divided by those in year n (Pn). Sometimes this ratio is given in the form of a price index for a fixed quantity of goods.

Usually the price level in the base year 0 is assigned the number 100, so that price changes will show up as percentages of prices in year 0. Thus as the price changes, the price ratio for each year n (Pn/P0) is calculated and multiplied by 100. Similarly, to convert costs from year n to a base year, they should be divided by the price index and multiplied by 100.

Example: Suppose that milk cost F 180 per litre in 1981 and F 250 in 1983, then the ratio 250/180 multiplied by 100 will give a price index of 139 if the base year is 1981. To create this index a constant quantity (1 litre) was used. Thus the quantity of milk bought for F 15 000 in 1981 would cost 15 000 x 139/100 or F 20 850 in 1983. Conversely, expenditure on milk of F 25 000 in 1983 would have cost 25 000/39 x 100 or F 17 986 in 1981. Often price indices are presented in a series for a fixed quantity. Thus if the 1982 price was given as F 215, the complete series would be as follows:


Base year 1981

Base year 1983

1981

100

72

1982

119

84

1983

139

100

The base year in this series is given by 100. Using such a series makes it possible to convert costs from any year to those of any other, but most conveniently to the base year. Frequently an economist evaluating a project will be confronted with a series of expenditure figures extending over many years. If detailed information is not available, price indices published by government statistical services can be used in the analysis or else such indices can be- put together from the existing information on prices and quantities. Until costs over a number of years have been converted to constant prices, it is meaningless to compare them, since any decreases or increases could be due to price changes.

Any project manager, planner or individual planning his finances must make it a priority to collect not only information on costs but also on prices. Ideally all quantities, prices and expenses should be recorded. In fact, since the objective is to compare expenditure or receipts at constant prices, a record of total costs and unit prices would be sufficient. Expenditure and receipts could then be converted to the base year by making price indices out of the price series. This is the most practical approach. An alternative approach is to note all quantities purchased or sold. When the moment for comparing expenditure and receipts comes, these can converted to current costs for all items since the quantities and current prices are known.

In many eases price indices actually cover a mixed sample of goods of a particular category. Examples of these include consumer price indices, share indices, construction goods indices, industrial price indices etc. In each case, the same principle applies. As before, the quantity must be fixed, but this quantity is a fixed selection of goods, usually called a "basket".

Table 37 gives an example of a price index created to convert costs to constant prices for the evaluation of a tick control project in Malawi. The last year of the project, 1981/2, was chosen as the base year, with prices increasing to that level.

6.3 Compound interest, discounting, annual rates of growth and annual loan repayments


6.3.1 Simple vs compound growth (or interest) rates
6.3.2 Discounting and compounding tables
6.3.3 Estimating present and future values using annuity tables
6.3.4 Loan repayments
6.3.5 Interest or discount rates and inflation


This section explains the formulae needed for calculating annual rates of growth, inflation and compound interest and for discounting, which is, in effect, deducting compound interest. These are all based on a single, simple formula which is explained below.

6.3.1 Simple vs compound growth (or interest) rates

If a given number (a livestock population, a sum of money, a price) is said to increase at a percentage rate per annum (population growth, interest or inflation rates), this increase could be interpreted as simple or compound growth. Table 38 illustrates these two types of growth for a sum of money (US$ 100) growing at an interest rate of 10% over 5 years.

Simple growth is calculated by applying the percentage rate only to the initial sum, so that the numerical value of annual growth is always the same. Thus simple interest is paid only on the sum initially invested (US$ 100) and is fixed at 10% of this (US$ 10).

Table 37. Price indexes calculated for a tick control project in Malawi (base year 1981/82).

Year

Blantyre low-income consumer price index

Salaries in the veterinary and livestock departments

Construction work: dip tanks and staff housing

Vehicles

68/69

32.6

40

18

12

69/70

33.7

40

19

13

70/71

37.0

40

20

14

71/72

39.4

40

23

15

72/73

40.9

46

27

17

73/74

44.1

46

32

20

74/75

50.9

48

42

24

75/76

57.2

48

51

36

76/77

59.7

48

55

41

77/78

62.9

64

57

48

78/79

68.7

82

65

59

79/80

77.7

84

71

71

80/81

90.0

89

87

87

81/82

100.0

100

100

100

Compound growth is calculated by applying the percentage rate each year to the initial sum plus the previous year's growth, so that the annual growth rate also increases each year. Thus compound interest is paid not only on the principal but also on the interest that has accumulated. In the example given, the interest payments over the 5 years increase from US$ 10 to US$ 15.

In practice, almost all forms of annual increase are calculated on a compound basis. Interest is always paid on the full amount of money in the account, so simple interest would generally only apply if the individual removed the previous year's interest (US$ 10), leaving the original sum (US$ 100) in the bank. Human and livestock population growth rates apply each year to the whole of the population existing in the previous year, so the growth rate is again compound. The same is true of the annual inflation rate.

If the present value (PV) and the annual rate of increase (i) are known, the future value (FV) can be calculated from the formula:

FV = PV (1 + i)n

Table 38. Simple vs compound interest.

Year

Simple interest

Compound interest

Sum at start of year

Interest at 10%

Total at end of year

Sum at start of year

Interest at 10%

Total at end of year

1

100

10

110

100

10

110

2

100

10

120

110

11

121

3

100

10

130

121

12

133

4

100

10

140

133

13

146

5

100

10

150

146

15

161

By manipulating this formula, three further formulae can be derived, enabling the calculation of either the present value (PV), the annual rate of increase (i) or the number of years (n) provided that the three other values are known.

Examples

1) Calculation of future values

The current rate of inflation on housing is estimated at 6% per annum. An individual's house is currently valued at US$ 30 000. How much could he expect to sell it for in 5 year's time?

The three known values are:

n = 5
i = 0.06
PV= 30000

The formula for calculating future values is:
FV = PV(I +i)n

Thus:

FV = 30 000 x (1.06)5 = 40 147

The individual could expect to sell his house for just over US$ 40000.

2) Calculation of present values

In 1983, a country estimates that in order to provide sufficient beef for its population in 1990 at least 300 000 head of cattle must be slaughtered annually. The number of cattle present in the country in 1983 is unknown, but an annual growth rate in the national herd of 3.5% and an offtake of 12% are considered to be reasonable values. What would the minimum cattle population in the country need to be in 1983 to be able to satisfy demand in 1990?

With offtake at 12%,300 000 would have to represent 12% or less of the 1990 cattle population for the demand to be satisfied. Thus:

FV = 300 000/0.12 = 2 500 000
n = 1990- 1983 = 7
i = 0.035

The formula for calculating present values is:

PV = FV / (1+i)n

Thus:

PV = 2 500 000 / (1 + 0.035)7 = 1 964 977

To satisfy demand in 1990, the minimum cattle population in the country in 1983 should be 1.965 million.

3) Calculation of growth rate or rate of increase

In a census carried out in 1980, the human population in a region was given as 5 350 071. In 1970, the result was 3 897 136. What is the annual rate of growth of the population?

PV = 3 897 136
FV = 5 350 071
n = 1980-1970 = 10

The formula for calculating growth rate is:

Thus:

The annual rate of growth is 3.22%.

4) Calculation of n

If the interest rate is 12%, for how long must money be invested to double its face value?

FV/PV = 2
i = 0.12

The formula for calculating n is:

n = Log (FV/PV) / log (1 + i)

Thus: n = log (2) / log (1 + 0.12) = 6 116

It will take 6.12 years to double the face value of money invested at 12%.

6.3.2 Discounting and compounding tables

Discounting is the process of converting future values to present values. It is used in project appraisal, when considering a stream of future costs and benefits in order to determine what their total present value would be. Items for different years are "discounted" separately by calculating their present value and then the total present value of all items is calculated by adding these together. In order to simplify the process, tables exist giving the conversion factors for a range of i's and n's - usually 2% to 50% and I to 50 years - worked out to three decimal places. For the reader's convenience discounting and compounding tables are given in Appendix 1.

Table 39 compares the future values of US$ 1000 invested in year 0 and earning interest from years I to 10, to the present values of the same sum received in each of years 0 to 10.

Table 39. Discounting and compounding present and future values.

Year

Future value of US$ 1000 at 10%

Present value of US$ 1000 at 10%

FV = PV(1 + i)n

PV = FV/(1 + i)n

0

1000

1000

1

1100

909

2

1210

826

3

1331

751

4

1464

683

5

1611

621

6

1772

564

7

1949

513

8

2144

467

9

2358

424

10

2594

386

i = interest rate; r = discount rate.

We can see in the table that US$ 1000 received in 10 years' time has a present value of only US$ 386 at a discount rate of 10%. If, however, the sum of US$ 1000 was invested in year 0, it would be worth US$ 2594 in 10 years' time at 10% interest. The conversion factor is the same:

(1 + 0.10)10 = 2.5937

1 / (1+10)10 = 0.3855

FV = 1000 x (1.10)10 = 2594

PV = 1000 / (1.10)10 = 386

Table 40 shows how discount factors are used to discount the present value of a stream of incomes.

Compounding is the process of converting present values to future values. Compounding tables exist showing the future values of money invested now for different i's and n's.

The different values of i or n can be estimated by looking down the column giving the appropriate ratio for FV/PV in the compounding table or PV/FV in the discounting table. Using Table 3 in Appendix I and applying this to Example 3 we find that for FV/PV = 1.3728 and n = 10, the value in the row for 10 years closest to 1.3728 is 1.344, and this occurs under 3%, so that i can be estimated as being just over 3%.

In Example 4, FV/PV = 2 and i = 12%. Looking down the column for 12%, the closest value to 2 is 1.974 in the row for 6 years, so n can be estimated at just over 6.

Compounding tables can be applied to any form of compound growth, not just interest rates.

Table 40. Discounting a stream of incomes using present value tables.

Year

Undiscounted values of benefits less costs

10% discount factor

Discounted values factor less costs

1

- 12 500

.909

- 11 362

2

- 4 000

.826

- 3 304

3

6 500

.751

4 881

4

6 500

.683

4 440

5

6 500

.621

4 036

6

6 500

.564

3 666

7

5 750

.513

2 950

8

5 750

.467

2685

9

5 750

.424

2438

10

8 750

.386

3 378

Total

35 500

Present value

13 808

6.3.3 Estimating present and future values using annuity tables

So far, the discussion has been in terms of the present value of US$ 1 received at a given future date or of the future value at a given date of US$ I invested today. The present value of an annuity table (see Appendix 1, Table 2) gives the present value of US$ 1 received or spent annually at a given rate of discount i and for a given number of years n. Similarly another annuity table (see Table 4, Appendix 1) gives the future value of US$ 1 invested annually at a rate i for n years.

Such tables are derived by making a year-by-year cumulative total of the compounding or discounting factors, as illustrated in Table 41.

Annuity tables can greatly facilitate the process of discounting if the same figure appears for a number of years in the stream of figures to be discounted. In Table 42 the figure for years 3 to 6 inclusive is identical at 6500. Since the present value of an annuity is a simple cumulative total of the discount factors, we can take the figure for year 6, which gives the total annuity over 6 years, and subtract from it that for year 2, which gives the total for the 2 years not to be included, to obtain a discount factor of 2.619 for years 3 through 6. The same process is applied for years 7 through 9. This considerably reduces the work that was necessary to arrive at the total present value of the same costs in Table 40.

Annuity tables giving present and future values of an annuity can be found in Appendix One.

Table 41. Derivation of tables for calculating present and future values of an annuity of I at 10%.

Year

Discount factor

Present value of an annuity

Compounding factor

Future value of an annuity

1

.909

0.909

1.10

1.10

2

.826

1.736

1.21

2.31

3

.751

2.487

1.33

3.64

4

.683

3.170

1.46

5.11

5

.621

3.791

1.61

6.72

6.3.4 Loan repayments

The average amount that must be repaid annually to repay the interest and principal on a loan at an i rate of interest over n years can be calculated using the average capital recovery or amortization factor, which can be derived as follows:

The lender needs to fix annual repayments at a rate of interest i over n years at a value such that:

PV (all repayments) = amount lent.

These repayments are a form of annuity, being an annual and equal amount. From Table 41 we can see that at an interest rate of 10% over 5 years, an annuity of US$ 1000 would have a present value of US$ 3791. Thus to repay a loan of US$ 3791 at 10% in equal annual installments over 5 years, US$ 1000 would have to be repaid annually. Similarly, to repay a loan of US$ 1000 at 10%, five equal annual installments would be necessary. Each installment would be:

1000 x (1/3.791) = US$ 263

Table 42. Discounting a stream of costs using present-value and present-value-of-an-annuity tables.

Year

Undiscounted values of benefits less costs

10% discount factor

Discounted value of benefits less costs

1

- 12 500

.909

- 11 362

2

- 4 000

.826

- 3 304

3

6500

2.619*


4

6 500

2.619*

17 023

5

6500

2.619*


6

6500

2.619*


7

5 750

1.404**


8

5750

1.404**

8073

9

5 750

1.404**


10

8 750

.386

3378

Total

35 500

Present value

13 808

Present value of an annuity at 10% for:

* Year 6 - Year 2 = 4.355 - 1.736 = 2.619
** Year 9 - Year 6 = 5.759 - 4.355 = 1.404

The factor 1/3.791 is the reciprocal of the present value of an annuity table (Appendix I, Table 2) and is referred to as the capital recovery or amortization factor (Appendix I, Table 5).

6.3.5 Interest or discount rates and inflation

Market interest rates that are actually paid in the economy include inflation since to make money by investing it, the rate of interest being paid must be higher than the rate of inflation. Often this is not the case. If, for example, the rate of inflation is 15% while the rate of interest is only 12% per annum, the real rate of interest is negative (-3%). The real rate of interest is defined as the market rate of interest less the rate of inflation; discount rates should usually reflect the real rate of interest.


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