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12: Labour and capital

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    Chapter 12: Labour and capital

    In this chapter we will explore:

    12.1

    The demand for labour

    12.2

    Labour supply

    12.3

    Market equilibrium and labour mobility

    12.4

    The concepts of capital

    12.5

    The capital market

    12.6

    Land

    The chapter deals with the markets for the factors of production—labour and capital. The analysis will be presented in terms of the demand, supply and market equilibrium for each. While this is a standard analytical approach in microeconomics, the markets for labour and capital differ from goods and services markets.

    In the first instance, goods and services are purchased and consumed by the buyers. In contrast, labour and capital are used as inputs in producing those 'final' goods and services. So the value of labour and capital to a producer depends in part upon the value of the products that the labour and capital are used to produce. Economists say that the value of the factors of production derives from the value of the products they ultimately produce.

    Secondly, labour and capital offer services. When an employer hires a worker, that worker supplies her time and skills and energy to the employer. When a piece of equipment is rented to a producer, or purchased by a producer, that equipment provides a stream of productive services also. The employer does not purchase the worker, given that we do not live in a society where slavery is legal. In contrast, she may decide to purchase the capital, or else rent it.

    The third characteristic of these markets is the time dimension associated with labour and capital. Specifically, once built, a machine will customarily have a lifetime of several years, during which it depreciates in value. Furthermore, it may become obsolete on account of technological change before the end of its anticipated life. Labour too may become obsolete, or at least lose some of its value with the passage of time, if the skills embodied in the labour cease to be required in the economy.

    12.1 Labour – a derived demand

    The value of labour springs from the value of its use, that is the value placed upon goods and services that it produces – product prices. The wage is the price that equilibrates the supply and demand for a given type of labour, and it reflects the value of that labour in production. Formally, the demand for labour (and capital) is thus a derived demand, in contrast to being a 'final' demand.

    Demand for labour: a derived demand, reflecting the value of the output it produces.

    We must distinguish between the long run and the short run in our analysis of factor markets. On the supply side certain factors of production are fixed in the short run. For example, the supply of radiologists can be increased only over a period of years. While one hospital may be able to attract radiologists from another hospital to meet a shortage, this does not increase the supply in the economy as a whole.

    On the demand side there is the conventional difference between the short and long run: In the short run some of a firm's factors of production, such as capital, are fixed, and therefore the demand for labour differs from when all factors are variable – the long run.

    Demand in the short run

    Table 12.1 contains information from the example developed in Chapter 8. It can be used to illustrate how a firm reacts in the short run to a change in an input price, or to a change in the output price. The response of a producer to a change in the wage rate constitutes a demand function for labour – a schedule relating the quantity of the input demanded to different input prices. The output produced by the various numbers of workers yields a marginal product curve, whose values are stated in column 3. The marginal product of labour, img256.png, as developed in Chapter 8, is the additional output resulting from one more worker being employed, while holding constant the other (fixed) factors. But what is the dollar value to the firm of an additional worker? It is the additional value of output resulting from the additional employee – the price of the output times the worker's marginal contribution to output, his MP. We term this the value of the marginal product.

    The value of the marginal product is the marginal product multiplied by the price of the good produced.

    Table 12.1 Short-run production and labour demand
    Workers Output MPL img406.png Marginal profit = (VMPL–wage)
    (1) (2) (3) (4) (5)
    0 0
    1 15 15 1050 150
    2 40 25 1750 750
    3 70 30 2100 1100
    4 110 40 2800 1800
    5 145 35 2450 1450
    6 175 30 2100 1100
    7 200 25 1750 750
    8 220 20 1400 400
    9 235 15 1050 50
    10 240 5 350 negative
    Each unit of labour costs $1,000; output sells at a fixed price of $70 per unit.

    In this example the img256.png first rises as more labour is employed, and then falls. With each unit of output selling for $70 the value of the marginal product of labour (img407.png) is given in column 4. The first worker produces 15 units each week, and since each unit sells for a price of $70, his production value to the firm is $1,050 img408.png. A second worker produces 25 units, so his value to the firm is $1,750, and so forth. If the weekly wage of each worker is $1,000 then the firm can estimate its marginal profit from hiring each additional worker. This is the difference between the value of the marginal product and the wage paid, and is given in the final column of the table.

    It is profitable to hire more workers as long as the cost of an extra worker is less than the img407.png. The equilibrium amount of labour to employ is therefore 9 units in this example. If the firm were to hire one more worker the contribution of that worker to its profit would be negative img409.png, and if it hired one worker less it would forego the opportunity to make an additional profit of $50 on the 9th unit img410.png.

    Profit maximizing hiring rule:

    • If the VMPL of next worker > wage, hire more labour.

    • If the VMPL < wage, hire less labour.

    To this point we have determined the profit maximizing amount of labour to employ when the output price and the wage are given. However, a demand function for labour reflects the demand for labour at many different wage rates, just as the demand function for any product reflects the quantity demanded at various prices. Accordingly, suppose the wage rate is $1,500 per week rather than $1,000. The optimal amount of labour to employ in this case is determined in exactly the same manner: Employ the amount of labour where its contribution is marginally profitable. Clearly the optimal amount to employ is 7 units: The value of the seventh worker to the firm is $1,750 and the value of the eighth worker is $1,400. Hence it would not be profitable to employ the eighth, because his marginal contribution to profit would be negative. Following the same procedure we could determine the optimal amount of labour to employ at different wages. This implies that the img407.png function is the demand for labour function because it determines the most profitable amount of labour to employ at any wage.

    The optimal amount of labour to hire is illustrated in Figure 12.1. The wage and VMPL curves come from Table 12.1. The VMPL curve has an upward sloping segment, reflecting increasing productivity, and then a regular downward slope as developed in Chapter 8. At employment levels where the VMPL is greater than the wage additional labour should be employed. But when the VMPL falls below the wage rate employment should stop. If labour is divisible into very small units, the optimal employment decision is where the MPL function intersects the wage line.

    Figure 12.1 The demand for labour
    img411.png
    The optimal hiring decision is defined by the condition that the value of the MPL is greater than or equal to the wage paid. 9 workers are employed when the wage is $1,000 and the price of output is $70; 6 workers are employed when the wage is $1,500 and the price of output is $50.

    Figure 12.1 also illustrates what happens to hiring when the output price changes. Consider a reduction in its price to $50 from $70. The profit impact of such a change is negative because the value of each worker's output has declined. Accordingly, the demand curve must reflect this by shifting inward (down), as in the figure. At various wage rates, less labour is now demanded. The new img407.png schedule can be derived in Table 12.1 as before: It is the img256.png schedule multiplied by the lower value ($50) of the final good.

    In this example the firm is a perfect competitor in the output market, because the price of the good it produces is fixed. It can produce and sell more of the good without this having an impact on the price of the good in the marketplace. Where the firm is not a perfect competitor it faces a declining MR function. In this case the value of the img256.png is the product of MR and img256.png rather than P and img256.png. To distinguish the different output markets we use the term marginal revenue product of labour (img412.png) when the demand for the output slopes downward. But the optimizing principle remains the same: The firm should calculate the value of each additional unit of labour, and hire up to the point where the additional revenue produced by the worker exceeds or equals the additional cost of that worker.

    The marginal revenue product of labour is the additional revenue generated by hiring one more unit of labour where the marginal revenue declines.

    Demand in the long run

    In Chapter 8 we proposed that firms choose their factors of production in accordance with cost-minimizing principles. In producing a specific output, firms choose the least-cost combination of labour and plant size. But how is this choice affected when the price of labour or capital changes? While adjustment to price changes may require a long period of time, we know that if one factor becomes more (less) expensive, the firm will likely change the mix of capital and labour away from (towards) that factor. For example, when the accuracy and prices of production robots began to fall in the nineteen nineties, auto assemblers reduced their labour and used robots instead. When computers and computer software improved and declined in price, clerical workers were replaced by computers that were operated by accountants. But such adjustments and responses do not occur overnight.

    In the short run a higher wage increases costs, but the firm is constrained in its choice of inputs by a fixed plant size. In the long run, a wage increase will induce the firm to use relatively more capital than when labour was less expensive in producing a given output. But despite the new choice of inputs, a rise in the cost of any input must increase the total cost of producing any output.

    A change in the price of any factor has two impacts on firms: In the first place producers will substitute away from the factor whose price increases; second, there will be an impact on output and a change in the price of the final good it produces. Since the cost structure increases when the price of an input rises, the supply curve in the market for the good must reflect this – any given output will now be supplied at a higher price. With a downward sloping demand, this shift in supply must increase the price of the good and reduce the amount sold. This second effect can be called an output effect.

    Monopsony

    Some firms may have to pay a higher wage in order to employ more workers. Think of Hydro Quebec building a dam in Northern Quebec. Not every hydraulic engineer would be equally happy working there as in Montreal. Some engineers may demand only a small wage premium to work in the North, but others will demand a high premium. If so, Hydro Quebec must pay a higher wage to attract more workers – it faces an upward sloping supply of labour curve. Hydro Quebec is the sole buyer in this particular market and is called a monopsonist – a single buyer. Our general optimizing principle governing the employment of labour still holds, even if we have different names for the various functions: Hire any factor of production up to the point where the cost of an additional unit equals the value generated for the firm by that extra worker. The essential difference here is that when a firm faces an upward sloping labour supply it will have to pay more to attract additional workers and also pay more to its existing workers. This will impact the firm's willingness to hire additional workers.

    A monopsonist is the sole buyer of a good or service and faces an upward-sloping supply curve.

    Application Box 12.1 Monopsonies

    Monopsonies are more than a curiosity; they exist in the real world. An excellent example is the cannabis market in Canada. Virtually every province has set up a trading agency that has the sole right to purchase cannabis from growers; growers and processors are not permitted to sell directly to retailers; they may only sell to the monopsony by law. In turn, these provincial cannabis monopsonies are frequently retail monopolists in that the agency owns all of the retail outlets in the province.

    Firm versus industry demand

    The demand for labour within an industry, or sector of the economy, is obtained from the sum of the demands by each individual firm. It is analogous to the goods market, but with a subtle difference. In Chapter 3 we obtained a market demand by summing individual demands horizontally. The same could be done here: At lower (or higher) wages, each firm will demand more (or less) labour. However, if all firms employ more labour in order to increase their output, the price of the output will likely decline. This in turn will moderate the demand for labour – it is slightly less valuable now that the price of the output it produces has fallen. This is a subtle point, and we can reasonably think of the demand for labour in a given sector of the economy as the sum of the demands on the part of the employers in that sector.

    12.2 The supply of labour

    Most prime-age individuals work, but some do not. The decision to join the labour force is called the participation decision. Of those who do participate in the labour force, some individuals work full time, others work part time, and yet others cannot find a job. The unemployment rate is the fraction of the labour force actively seeking employment that is not employed.

    The participation rate for the economy is the fraction of the population in the working age group that joins the labour force.

    The labour force is that part of the population either employed or seeking employment.

    The unemployment rate is the fraction of the labour force actively seeking employment that is not employed.

    Data on participation rates in Canada are given in Table 12.2 below for specific years in the modern era. The overall participation for men and women combined has increased since 1977 from 60.8% to 65.8% This aggregated rate camouflages different patterns for men and women. The rates for women have been rising while the rates for men have fallen. Women today are more highly educated, and their role in society and the economy is viewed very differently than in the earlier period. Female participation has increased both because of changing social norms, a rise in household productivity, the development of service industries designed to support home life, and the development of the institution of daycare for young children.

    In contrast, male participation rates declined over the period, largely offsetting the increase in female participation. Fewer individuals in total are retiring before the age of 55 in the most recent decades. This reflects both the greater number of females in the market place, and perhaps also a recognition that many households have not saved enough to fund a retirement period that has become longer as a result of increased longevity.

    Table 12.2 Labour force participation rate, Canada 1977-2015
    Year Total Men Women All > 55 Unemployment
    1977 60.8 80.2 42.1 30.5 5.9
    1990 66.6 77.1 56.8 25.9 7.4
    1994 65.4 74.9 56.8 24.5 9.2
    2001 66.1 73.4 59.2 26.4 6.2
    2008 67.5 73.6 61.6 34.2 5.1
    2015 66.2 72.0 60.6 37.3 6.0
    2019 65.8 71.0 60.8 38.0 4.4
    Source: Statistics Canada, CANSIM 14-10-0287-02
    September of each year, for individuals aged img413.png 25, unless stated.

    At the micro level, the participation rate of individuals depends upon several factors. First, the wage rate that an individual can earn in the market is crucial. If that wage is low, then the individual may be more efficient in producing home services directly, rather than going into the labour market, earning a modest income and having to pay for home services. Second, there are fixed costs associated with working. A decision to work means that the individual must have work clothing, must undertake the costs of travel to work, and pay for daycare if there are children in the family. Third, the participation decision depends upon non-labour income. If the individual in question has a partner who earns a substantial amount, or if she has investment income, she will have less incentive to participate. Fourth, it depends inversely upon the tax rate.

    The supply curve relates the supply decision to the price of labour – the wage rate. Economists who have studied the labour market tell us that the individual supply curve is upward sloping: As the wage increases, the individual wishes to supply more labour. From the point img414.png on the supply function in Figure 12.2, let the wage increase from img415.png to img416.png.

    Figure 12.2 Individual labour supply
    img417.png
    A wage increase from W0 to W1 induces the individual to substitute away from leisure, which is now more expensive, and work more. But the higher wage also means the individual can work fewer hours for a given standard of living; therefore the income effect induces fewer hours. On balance the substitution effect tends to dominate and the supply curve therefore slopes upward.

    The individual offers more labour, img418.png, at the higher wage. What is the economic intuition behind the higher amount of labour supplied? Like much of choice theory there are two impacts associated with a higher price. First, the higher wage makes leisure more expensive relative to working. That midweek game of golf has become more expensive in terms of what the individual could earn. So the individual should substitute away from the more expensive 'good', leisure, towards labour. But at the same time, in order to generate a given income target the individual can work fewer hours at the higher wage. This is a type of income effect, indicating that income is greater at a higher wage regardless of the amount worked, and this induces the individual to work less. The fact that we draw the labour supply curve with a positive slope means that the substitution effect is the more important of the two. That is what statistical research has revealed.

    Elasticity of the supply of labour

    The value of the supply elasticity depends upon how the market in question is defined. In particular, it depends upon how large or small a given sector of the economy is, and whether we are considering the short run or the long run.

    Suppose an industry is small relative to the whole economy and employs workers with common skills. These industries tend to pay the 'going wage'. For example, very many students are willing to work at the going rate for telemarketing firms, which compose a small sector of the economy. This means that the supply curve of such labour, as far as that sector is concerned, is in effect horizontal – infinitely elastic.

    But some industries may not be small relative to the total labour supply. And in order to get more labour to work in such large sectors it may be necessary to provide the inducement of a higher wage: Additional workers may have to be attracted from another sector by means of higher wages. To illustrate: Consider the behaviour of two related sectors in housing – new construction and home restoration. In order to employ more plumbers and carpenters, new home builders may have to offer higher wages to induce them to move from the renovation sector. In this case the new housing industry's labour supply curve slopes upwards.

    In the time dimension, a longer period is always associated with more flexibility. In this context, the supply of labour to any sector is more elastic, because it may take time for workers to move from one sector to another. Or, in cases where skills must be built up: When a sectoral expansion bids up the wages of information technology (IT) workers, more school leavers are likely to develop IT skills. Time will be required before additional graduates are produced, but in the long run, such additional supply will moderate the short-run wage increases.

    Wages can be defined as being before-tax or after-tax. The after-tax, or take-home, wage is more important than the gross wage in determining the quantity of labour to be supplied. If taxes on additional hours of work are very high, workers are more likely to supply less hours than if tax rates are lower.

    12.3 Labour market equilibrium and mobility

    The fact that labour is a derived demand differentiates the labour market's equilibrium from the goods-market equilibrium. Let us investigate this with the help of Figure 12.3; it contains supply and demand functions for one particular industry – the cement industry, let us assume.

    In Figure 12.1 we illustrated the impact on the demand for labour of a decline in the price of the output produced – a decline in the output price reduced the value of the marginal product of labour. In the current example, suppose that a slowdown in construction results in a decline in the price of cement. The impact of this price fall is to reduce the output value of each worker in the cement producing industry, because their output now yields a lower price. This decline in the img407.png is represented in Figure 12.3 as a shift from img419.png to img420.png, which results in the new equilibrium img34.png.

    Figure 12.3 Equilibrium in an industry labour market
    img421.png
    A fall in the price of the good produced in a particular industry reduces the value of the MPL. Demand for labour thus falls from D0 to D1 and a new equilibrium E1 results. Alternatively, from E0, an increase in wages in another sector of the economy induces some labour to move to that sector. This is represented by the shift of S0 to S1 and the new equilibrium E2.

    As a second example: Suppose that wages in some other sectors of the economy increase. The impact of this on the cement sector is that the supply of labour to the cement sector is reduced. In Chapter 3 we showed that a change in other prices may shift the demand or supply curve of interest. In Figure 12.3 supply shifts from img422.png to img423.png and the equilibrium goes from img145.png to img213.png.

    How large are these impacts likely to be? That will depend upon how mobile labour is between sectors: Spillover effects will be smaller if labour is less mobile. This brings us naturally to the concepts of transfer earnings and rent.

    Transfer earnings and rent

    Consider the case of a performing violinist whose wage is $80,000. If, as a best alternative, she can earn $60,000 as a music teacher then her rent is $20,000 and her transfer earnings $60,000: Her rent is the excess she currently earns above the best alternative. Another violinist in the same orchestra, earning the same amount, who could earn $55,000 as a teacher has rent of $25,000. The alternative is also called the reservation wage. The violinists should not work in the orchestra unless they earn at least what they can earn in the next best alternative.

    Transfer earnings are the amount that an individual can earn in the next highest paying alternative job.

    Rent is the excess remuneration an individual currently receives above the next best alternative. This alternative is the reservation wage.

    These concepts are illustrated in Figure 12.4. In this illustration, different individuals are willing to work for different amounts, but all are paid the same wage img415.png. The market labour supply curve by definition defines the wage for which each individual is willing to work. Thus the rent earned by labour in this market is the sum of the excess of the wage over each individual's transfer earnings – the area img424.png. This area is also what we called producer or supplier surplus in Chapter 5.

    Figure 12.4 Transfer earnings and rent
    img425.png
    Rent is the excess of earnings over reservation wages. Each individual earns W0 and is willing to work for the amount defined by the labour supply curve. Hence rent is W0E0A and transfer earnings OAE0L0. Rent is thus the term for supplier surplus in this market.

    Free labour markets?

    Real-world labour markets are characterized by trade unions, minimum wage laws, benefit regulations, severance packages, parental leave, sick-day allowances and so forth. So can we really claim that markets work in the way we have described them – essentially as involving individual agents demanding and supplying labour? While labour markets are not completely 'free' in the conventional sense, the important issue is whether these interventions, that are largely designed to protect workers, have a large or small impact on the market. One reason why unemployment rates are generally higher in European economies than in Canada and the US is that labour markets are less subject to controls, and workers have a less supportive social safety net in North America.

    Application Box 12.2 Are high salaries killing professional sports?

    It is often said that the agents of professional players are killing their sport by demanding unreasonable salaries. On occasion, the major leagues are threatened with strikes, even though players are paid millions each year. In fact, wages are high because the derived demand is high. Fans are willing to pay high ticket prices, and television rights generate huge revenues. Combined, these revenues not only make ownership profitable, but increase the demand for the top players.

    The lay person may be horrified at thirty-million dollar annual salaries. But in reality, many players receiving such salaries may be earning less than their marginal product! If Tom Brady did not play for the New England Patriots the team would have a lower winning record, attract fewer fans and make less profit. If Brady is paid $25m per season, he is being paid less than his marginal product if the team were to lose $40m in revenue as a result of his absence.

    Given this, why do some teams incur financial losses? In fact very few teams make losses: Cries of poverty on the part of owners are more frequently part of the bargaining process, and revenue sharing means that very few teams do not make a profit.

    The impact of 'frictions', such as unionization and minimum wages, in the labour market can be understood with the help of Figure 12.5. The initial 'free market' equilibrium is at img145.png, assuming that the workers are not unionized. In contrast, if the workers in this industry form a union, and negotiate a higher wage, for example img416.png rather than img415.png, then fewer workers will be employed. But how big will this reduction be? Clearly it depends on the elasticities of demand and supply. With the demand curve D, the excess supply at the wage img416.png is the difference img426.png. However, if the demand curve is less elastic, as illustrated by the curve img78.png, the excess supply is img427.png. The excess supply also depends upon the supply elasticity. It is straightforward to see that a less elastic (more vertical) supply curve through img145.png would result is less excess supply.

    Figure 12.5 Market interventions
    img428.png
    E0 is the equilibrium in the absence of a union. If the presence of a union forces the wage to W1 fewer workers are employed. The magnitude of the decline from L0 to L1 depends on the elasticity of demand for labour. The excess supply at the wage W1 is (F-E1). With a less elastic demand curve (img78.png) the excess supply is reduced to (F-img138.png).

    Beyond elasticity, the magnitude of the excess supply will also depend upon the degree to which the minimum wage, or the union-negotiated wage, lies above the equilibrium. That is, a larger value of the difference (img429.png results in more excess supply than a smaller difference.

    While the above discussion pertains to unionization, it could equally well be interpreted in a minimum-wage context. If this figure describes the market for low-skill labour, and the government intervenes by setting a legal minimum at img416.png, then this will induce some degree of excess supply, depending upon the actual value of img416.png and the elasticities of supply and demand.

    Despite the fact that a higher wage may induce some excess supply, it may increase total earnings. In Chapter 4 we saw that the dollar value of expenditure on a good increases when the price rises if the demand is inelastic. In the current example the 'good' is labour. Hence, a union-negotiated wage increase, or a higher minimum wage will each increase total remuneration if the demand for labour is inelastic. A case which has stirred great interest is described in Application Box 12.3.

    Application Box 12.3 David Card on minimum wage

    David Card is a famous Canadian-born labour economist who has worked at Princeton University and University of California, Berkeley. He is a winner of the prestigious Clark medal, an award made annually to an outstanding economist under the age of forty. Among his many contributions to the discipline, is a study of the impact of minimum wage laws on the employment of fast-food workers. With Alan Krueger as his co-researcher, Card examined the impact of the 1992 increase in the minimum wage in New Jersey and contrasted the impact on employment changes with neighbouring Pennsylvania, which did not experience an increase. They found virtually no difference in employment patterns between the two states. This research generated so much interest that it led to a special conference. Most economists now believe that modest changes in the level of the minimum wage have a small impact on employment levels.

    Since about 2015, numerous labor-friendly movements favoring higher wages for low-paid workers have proposed a $15 minimum in both Canada and the US. Some political parties have supported this movement, as have specific cities and municipalities and governments. While any increase in the minimum wage must by definition help those working, care must be exercised in implementing particularly large increases. This is because large increases in particular areas or spheres may induce production units to move outside of the area covered, and thereby shift jobs to lower-wage areas.

    12.4 Capital – concepts

    The share of national income accruing to capital is more substantial than commonly recognized. National income in Canada is divided 60-40, favoring labour. This leaves a very large component going to the owners of capital. The stock of physical capital includes assembly-line machinery, rail lines, dwellings, consumer durables, school buildings and so forth. It is the stock of produced goods used as inputs to the production of other goods and services.

    Physical capital is the stock of produced goods that are inputs in the production of other goods and services.

    Physical capital is distinct from land in that the former is produced, whereas land is not. These in turn differ from financial wealth, which is not an input to production. We add to the capital stock by undertaking investment. But, because capital depreciates, investment in new capital goods is required merely to stand still. Depreciation accounts for the difference between gross and net investment.

    Gross investment is the production of new capital goods and the improvement of existing capital goods.

    Net investment is gross investment minus depreciation of the existing capital stock.

    Depreciation is the annual change in the value of a physical asset.

    Since capital is a stock of productive assets we must distinguish between the value of services that flow from capital and the value of capital assets themselves.

    A stock is the quantity of an asset at a point in time.

    A flow is the stream of services an asset provides during a period of time.

    When a car is rented it provides the driver with a service; the car is the asset, or stock of capital, and the driving, or ability to move from place to place, is the service that flows from the use of the asset. When a photocopier is leased it provides a stream of services to the user. The copier is the asset; it represents a stock of physical capital. The printed products result from the service the copier provides per unit of time.

    The price of an asset is what a purchaser pays for the asset. The owner then obtains the future stream of capital services it provides. Buying a car for $30,000 entitles the owner to a stream of future transport services. The term rental rate defines the cost of the services from capital.

    Capital services are the production inputs generated by capital assets.

    The rental rate is the cost of using capital services.

    The price of an asset is the financial sum for which the asset can be purchased.

    But what determines the price of a productive asset? The price must reflect the value of future services that the capital provides. But we cannot simply add up these future values, because a dollar today is more valuable than a dollar several years from now. The key to valuing an asset lies in understanding how to compute the present value of a future income stream.

    Present values and discounting

    When capital is purchased it generates a stream of dollar values (returns) in the future. A critical question is: How is the price that should be paid for capital today related to the benefits that capital will bring in the future? Consider the simplest of examples: A business is contemplating buying a computer. This business has a two-year horizon. It believes that the purchase of the computer will yield a return of $500 in the first year (today), $500 in the second year (one period into the future), and have a scrap value of $200. What is the maximum price the entrepreneur should pay for the computer? The answer is obtained by discounting the future returns to the present. Since a dollar today is worth more than a dollar tomorrow, we cannot simply add the dollar values from different time periods.

    The value today of $500 received a year from now is less than $500, because if you had this amount today you could invest it at the going rate of interest and end up with more than $500 tomorrow. For example, if the rate of interest is 10% (= 0.1), then $500 today is worth $550 next period. By the same reasoning, $500 tomorrow is worth less than $500 today. Formally, the value next period of any amount is that amount plus the interest earned; in this case the value next period of $500 today is img430.png, where r is the interest rate. It follows that if we multiply a given sum by (1+r) to obtain its value next period, then we must divide a sum received next period to obtain its value today. Hence the value today of $500 next period is simply img431.png. To see that this must be true, note that if you have $454.54 today you can invest it and obtain $500 next period if the interest rate is 10%. In general:

    img432.pngimg433.png
    img434.pngimg435.png

    This rule carries over to any number of future periods. The value of a sum of money today two periods into the future is obtained by multiplying the today value by img436.png twice. Or the value of a sum of money today that will be received two periods from now is that sum divided by img436.png twice. And so on, for any number of time periods. So if the amount is received twenty years into the future, its value today would be obtained by dividing that sum by img436.png twenty times; if received 'n' periods into the future it must be divided by img436.png 'n' times.

    Two features of this discounting are to be noted: First, if the interest rate is high, the value today of future sums is smaller than if the interest rate is low. Second, sums received far in the future are worth much less than sums received in the near future.

    Let us return to our initial example, assuming the interest rate is 0.1 (or 10%). The value of the year 1 return is $500. The value of the year 2 return today is $454.54, and the scrap value in today's terms is $181.81. The value of all returns discounted to today is thus $1,136.35.

    Table 12.3 Present value of an asset (img437.png)
    Year Annual return Scrap value Discounted values
    Year 1 500 500
    Year 2 500 200 454.54 + 181.81
    Asset value today 1,136.35

    The present value of a stream of future earnings is the sum of each year's earnings divided by one plus the interest rate 'n' times, where 'n' is the number of years in the future when the amount will be received.

    We are now in a position to determine how much the buyer should be willing to pay for the computer. Clearly if the value of the computer today, measured in terms of future returns to the entrepreneur's business, is $1,136.35, then the potential buyer should be willing to pay any sum less than that amount. Paying more makes no economic sense.

    Discounting is a technique used in countless applications. It underlies the prices we are willing to pay for corporate stocks: Analysts make estimates of future earnings of corporations; they then discount those earnings back to the present, and suggest that we not pay more for a unit of stock than indicated by the present value of future earnings.

    12.5 The capital market

    Demand

    The analysis of the demand for the services of capital parallels closely that of labour demand: The rental rate for capital replaces the wage rate and capital services replace the hours of labour. It is important to keep in mind the distinction we drew above between capital services on the one hand and the amount of capital on the other. Capital services are produced by capital assets, just as work is produced by humans. Terms that are analogous to the marginal product of labour emerge naturally: The marginal product of capital (MPK) is the output produced by one additional unit of capital services, with other inputs held constant. The value of this marginal product (VMPK) is its value in the market place. It is the MPK multiplied by the price of output.

    The MPK must eventually decline with a fixed amount of other factors of production. So, if the price of output is fixed for the firm, it follows that the VMPK must also decline. We could pursue an analysis of the short-run demand for capital services, assuming labour was fixed, that would completely mirror the short-run demand for labour that we have already developed. But this would not add any new insights, so we move on to the supply side.

    The marginal product of capital is the output produced by one additional unit of capital services, with all other inputs being held constant.

    The value of the marginal product of capital is the marginal product of capital multiplied by the price of the output it produces.

    Supply

    We can grasp the key features of the market for capital by recognizing that the flow of capital services is determined by the capital stock: More capital means more services. The analysis of supply is complex because we must distinguish between the long run and the short run, and also between the supply to an industry and the supply in the whole economy.

    In the short run the total supply of capital assets, and therefore services, is fixed to the economy, since new production capacity cannot come on stream overnight: The short-run supply of services is therefore vertical. In contrast, a particular industry in the short run faces a positively sloped supply: By offering a higher rental rate for trucks, one industry can bid them away from others.

    The long run is a period of sufficient length to permit an addition to the capital stock. A supplier of capital, or capital services, must estimate the likely return he will get on the equipment he is contemplating having built. To illustrate: He is analyzing the purchase or construction of an earthmover that will cost $100,000. Assuming that the annual maintenance and depreciation costs are $10,000, and that the interest rate is 5% (implying that annual interest cost is $5,000), it follows that the annual cost of owning such a machine is $15,000. If the entrepreneur is to undertake the investment she must therefore earn at least this amount annually (by renting it to others, or using it herself), and this is what is termed the required rental. We can think of it as the opportunity cost of ownership.

    The required rental covers the sum of maintenance, depreciation and interest costs.

    Prices and returns

    In the long run, capital services in any sector of the economy must earn the required rental. If they earn more, entrepreneurs will be induced to build or purchase additional capital goods; if they earn less, owners of capital will allow machines to depreciate, or move the machines to other sectors of the economy.

    As an example, the price of oil on world markets fell by half during 2015; from about $100US per barrel to $50US. At this price, many oil wells were no longer profitable, and oil drilling equipment was decommissioned. Technically, the value of the marginal product of capital declined, because the price of the good it was producing declined. In the near and medium term, no new investment in capital goods will take place in the oil drilling sector of the economy. If the price of oil should increase in the future, some of the decommissioned capital will be brought back into service. But some of this capital will deteriorate or depreciate and simply 'die', and be sold for scrap metal – particularly the older vintage capital. Only when the stock of oil drilling equipment is reduced by depreciation and decay to the required level will any new investment in this form of capital take place.

    Note that the capital in this example is sector-specific. Drilling equipment cannot be easily redirected for use in other sectors. In contrast, earth movers can move from one sector of the economy to another with greater ease. An earth mover can be used to dig foundations for housing or commercial buildings; it can be used for strip mining; to build roads and bridges; to build tennis courts, golf courses and public parks. Such equipment may thus be moved to other sectors of the economy if in one particular sector the capital no longer can earn the required rental.

    The prices of capital goods in the long run will be determined by the supply and demand for the services they provide. If the value of the services, as determined by supply and demand is high, then the price of assets will reflect this.

    12.6 Land

    Land is an input used in production, though is not a capital good in the way we defined capital goods earlier – production inputs that are themselves produced in the economy. Land is relatively fixed in supply to the economy, even in the long run. While this may not be literally true – the Netherlands reclaimed from the sea a great quantity of low-lying farmland, and fertilizers can turn marginal land into fertile land – it is a good approximation to reality. Figure 12.6 shows the derived demand D0 for land services. With a fixed supply S, the equilibrium rental is R0.

    Figure 12.6 The market for land services
    img438.png
    The supply of land is relatively fixed, and therefore the return to land is primarily demand determined. Shifts in demand give rise to differences in returns.

    In contrast to this economy-wide perspective, consider now a retailer who rents space in a commercial mall. The area around the mall experiences a surge in development and more people are shopping and doing business there. The retailer finds that she sells more, but also finds that her rent increases on account of the additional demand for space by commercial enterprises in the area. Her landlord is able to charge a higher rent because so many potential clients wish to rent space in the area. Consequently, despite the additional commerce in the area, the retailer's profit increase will be moderated by the higher rents she must pay: The demand for retail space is a derived demand. The situation can be explained with reference to Figure 12.6 again. On account of growth in this area, the demand for retail space shifts from img419.png to img420.png. Space in the area is restricted, and thus the vertical supply curve describes the supply side well. So with little or no possibility of higher prices bringing forth additional supply, the additional demand makes for a steep price (rent) increase.

    Land has many uses and the returns to land must reflect this. Land in downtown Vancouver is priced higher than land in rural Saskatchewan. Land cannot be moved from the latter to the former location however, and therefore the rent differences represent an equilibrium. In contrast, land in downtown Winnipeg that is used for a parking lot may not be able to compete with the use of that land for office development. Therefore, for it to remain as a parking lot, the rental must reflect its high opportunity cost. This explains why parking fees in big US cities such as Boston or New York may run to $40 per day. If the parking owners could not obtain this fee, they could profitably sell the land to a developer. Ultimately it is the value in its most productive use that determines the price of land.

    Key Terms

    Demand for labour: a derived demand, reflecting the demand for the output of final goods and services.

    Value of the marginal product is the marginal product multiplied by the price of the good produced.

    Marginal revenue product of labour is the additional revenue generated by hiring one more unit of labour where the marginal revenue declines.

    Monopsonist is the sole buyer of a good or service and faces an upward-sloping supply curve.

    Participation rate: the fraction of the population in the working age group that joins the labour force.

    The labour force is that part of the population either employed or seeking employment.

    Unemployment rate: the fraction of the labour force actively seeking employment that is not employed.

    Transfer earnings are the amount that an individual can earn in the next highest paying alternative job.

    Rent is the excess remuneration an individual currently receives above the next best alternative. This alternative is the reservation wage.

    Physical capital is the stock of produced goods that are inputs to the production of other goods and services.

    Gross investment is the production of new capital goods and the improvement of existing capital goods.

    Net investment is gross investment minus depreciation of the existing capital stock.

    Depreciation is the annual change in the value of a physical asset.

    Stock is the quantity of an asset at a point in time.

    Flow is the stream of services an asset provides during a period of time.

    Capital services are the production inputs generated by capital assets.

    Rental rate: the cost of using capital services.

    Asset price: the financial sum for which the asset can be purchased.

    Present value of a stream of future earnings: the sum of each year's earnings divided by one plus the interest rate raised to the appropriate power.

    Marginal product of capital is the output produced by one additional unit of capital services, with all other inputs being held constant.

    Value of the marginal product of capital is the marginal product of capital multiplied by the price of the output it produces.

    Required rental covers the sum of maintenance, depreciation and interest costs.

    Exercises for Chapter 12

    EXERCISE 12.1

    Aerodynamics is a company specializing in the production of bicycle shirts. It has a fixed capital stock, and sells its shirts for $20 each. It pays a weekly wage of $400 per worker. Aerodynamics must maximize its profits by determining the optimal number of employees to hire. The marginal product of each worker can be inferred from the table below. Determine the optimal number of employees. [Hint: You must determine the VMPL schedule, having first computed the MPL.]

    Employment 0 1 2 3 4 5 6
    Total output 0 20 50 75 95 110 120
    img439.png
    img440.png
    EXERCISE 12.2

    Suppose that, in Exercise 12.1 above, wages are not fixed. Instead the firm must pay $50 more to employ each individual worker: The first worker is willing to work for $250, the second for $300, the third for $350, etc. But once employed, each worker actually earns the same wage. Determine the optimal number of workers to be employed. [Hint: You must recognize that each worker earns the same wage; so when one additional worker is hired, the wage must increase to all workers employed.]

    EXERCISE 12.3

    Consider the following supply and demand equations for berry pickers. Demand: W=22–0.4L; supply: W=10+0.2L.

    1. For values of img446.png, calculate the corresponding wage in each of the supply and demand functions.

    2. Using the data from part (a), plot and identify the equilibrium wage and quantity of labour.

    3. Illustrate in the diagram the areas defining transfer earnings and rent.

    4. Compute the transfer earnings and rent components of the total wage bill.

    EXERCISE 12.4

    The rows of the following table describe the income stream for three different capital investments. The income flows accrue in years 1 and 2. Only year 2 returns need to be discounted. The rate of interest is the first entry in each row, and the project cost is the final entry.

    Interest rate Year 1 Year 2 Cost
    8% 8,000 9,000 16,000
    6% 0 1,000 900
    10% 4,000 5,000 11,000
    1. For each investment calculate the present value of the stream of services.

    2. Decide whether or not the investment should be undertaken.

    EXERCISE 12.5

    Nihilist Nicotine is a small tobacco farm in south-western Ontario. It has three plots of land, each with a different productivity, in that the annual yield differs across plots. The output from each plot is given in the table below. Each plot is the same size and requires 3 workers and one machine to harvest the leaves. The cost of these inputs is $10,000. If the price of each kilogram of leaves is $4, how many plots should be planted?

    Land plot Leaf yield in kilograms
    One 3,000
    Two 2,500
    Three 2,000
    EXERCISE 12.6

    The timing of wine sales is a frequent problem encountered by vintners. This is because many red wines improve with age. Let us suppose you own a particular vintage and you envisage that each bottle should increase in value by 10% the first year, 9% the second year, 8% the third year, etc.

    1. Suppose the interest rate is 5%, for how many years would you hold the wine if there is no storage cost?

    2. If in addition to interest rate costs, there is a cost of storing the wine that equals 2% of the wine's value each year, for how many years would you hold the wine before selling?

    EXERCISE 12.7

    Optional: The industry demand for plumbers is given by the equation W=50–0.08L, and there is a fixed supply of 300 qualified plumbers.

    1. Draw a diagram illustrating the supply, demand and equilibrium, knowing that the quantity intercept for the demand equation is 625.

    2. Solve the supply and demand equations for the equilibrium wage, W.

    3. If the plumbers now form a union, and supply their labour at a wage of $30 per hour, illustrate the new equilibrium on your diagram and calculate the new level of employment.


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