Productivity, Output and Employment

Prof Mike Kennedy

A bit of a review

• What have we covered so far:– A broad brush picture of what some key variables in

the macro economy look like.– A discussion of what lies behind the concepts in the

national accounts and what they mean. • Important here is the derivation of the relationship between

saving and investment.

– How to convert these measures in something we call “real”. This is important.

– Measures of inflation and real interest rates – also important.

Where are we headed to now?

• The first two chapters have been about Description and Measurement.

• We now turn to Analysis• In this lecture we will discuss:– Production Functions– Demand for Labour– Introduce Labour Supply

Macroeconomic Analysis

• Goal: Build a Macroeconomic model; that is, a general framework to study economic questions already raised from an aggregate perspective.

• Then we will put it all together in Chap. 9 IS-LM Framework– we want to see how various markets interact which

we hope will tell us how the macro-economy works.

• We will be assuming that the economy is at full employment – but we will be touching on unemployment as well.

We Have to Understand the Economy Before We Can Improve It

The Production Function• How do we describe production in the economy?• Factors of production are inputs to the production

process. We will be concerned with:– capital (factories, machines) and labour (workers);– and how effectively they are used – e.g., productivity.

• Here we are going to measure the physical capacity of the economy to produce goods and services.

• A production function is a mathematical expression relating the amount of output produced to quantities of capital and labour utilized and to how efficiently capital and labour are combined.

The Production Function (continued)

In its general form, it looks like this:

Y = AF(K,N)

Y is real output produced A is a number measuring overall productivityK is the quantity of capital used (capital stock)N is the number of workers employedF is a function relating Y to K and N; it tells us how changes

in K and N change Y, for a given value of A

The production function (continued)

Y = AF(K,N) = AKαNβ = AKαN(1-α)

• This is the most well-known and used (Cobb-Douglas).• The exponents (α and β) represent shares in income – in

fact knowing income shares allows us to estimate these parameters.

• Because their values are less than unity, we get diminishing returns to factors.

• The “A” represents total factor productivity.

The production function (continued)

• There are three types of returns to scale (RS): Increasing (IRS), decreasing (DRS), constant (CRS).

• We will focus on CRS, i.e. β = 1 – α, with 0 < α < 1.

• With CRS we say that the function is homogeneous of degree one. This means that if we double all inputs (K and N) we double the output.

The production function (continued)

• A formal demonstration of constant returns to scale. Given a Cobb-Douglas production function, multiply each input by λ, a scalar.

Y = A(λK)α(λN)1-α

• The scalar is just a number which scales up or down the input. We can factor out this scalar:

Y = λA(K)α(N)1-α (since α and 1-α sum to one)

• Thus output rises by the same amount as the increase in inputs.

• But what about A? Do we need to double it as well?

The Production Function (continued)• Total factor productivity (productivity) is a measure of

overall effectiveness with which capital and labour are used.– An improvement in production technology or just using

existing factors more efficiently, allows capital and labour to be utilized more effectively.

– We measure this as a residual, using the production function; it is what we can’t explain that well – a measure of our ignorance according to Solow.

• Properties of production functions when one factor is changed holding the others constant:– they slope upward from left to right;– the slope becomes flatter from left to right.

The Marginal Product of Capital

• The marginal product of capital (MPK) is the increase in output produced resulting from a one-unit increase in the capital stock (other factors held constant).

• It can be considered as a demand function since it measures the benefits of employing/buying an additional unit of capital.

The Marginal Product of Capital (continued)

• The MPK equals the slope of the line tangent to the production function at a given point.

MPK = ∆Y/∆K

• It is the return or reward to capital.

The Marginal Product of Capital (continued)

• For those taking calculus, it is the first derivative of Y with respect to (wrt) to K.

MPK = ΔY⁄ΔK = dY/dK = αAK(α–1)N(1–α) > 0

• The properties of the production function:– the MPK is positive; – the MPK declines as the capital stock increases, holding the

other factors constant.

• Diminishing marginal productivity is the tendency for the marginal product of capital to decline as the amount of capital increases.

The Marginal Product of Labour

• The marginal product of labour (MPN) is the increase in output produced by each additional unit of labour (other factors held constant). The concept is the same as MPK.

• The marginal productivity of labour is diminishing for similar reasons as with capital.

• It is also a first derivative, this time of Y wrt N.

MPN = ΔY⁄ΔN = dY/dN = (1-α)AKαN(-α) > 0

• The properties of the MPN are the same as those of the MPK.

The Production FunctionSupply Shocks

• A supply shock (productivity shock) is a change in an economy’s production function.

• It can be represented as a change in “A” or the factors of production.

• A positive (beneficial) shock (new technology, innovation) raises the amount of output which can be produced with each capital-labour combination.

• We can also have a positive shock to labour supply.

The Production FunctionSupply Shocks (continued)

• A negative (adverse) shock (drought, oil price hike, earth quake, …) lowers the amount of output which can be produced with each capital-labour combination.– Positive shocks shift the production function upward.– Negative shocks shift the production function

downward.

• Often times we don’t really understand the source of the shock.

The Demand for Labour• Assume that: – the capital stock is fixed and the amount of

labour is variable in the short run – not a bad assumption in reality.

– Firms may employ and lay off workers without notice.

• These assumptions may not seem reasonable – real world – but we can test them later.

The Demand for Labour (continued)

• Also let’s assume (heroically) that:– Workers are all alike.– The wage is determined in a competitive market.– A firm employs workers to earn the highest

possible level of profit – up to the point where MPN equals wage, in real terms.

– The real wage is the cost of using labour.

The MPN and the Labour Demand

• The MPN measures the benefit of employing an additional worker in terms of the extra output produced – remember the graph on the production function.

• This is the source of the demand curve for labour.• It is a first derivative of the production function wrt

to N.• The demand for labour is embedded in the

production function.

The MPN and the Labour Demand (continued)

• The marginal revenue product of labour (MRPN) measures the benefit to the firm of employing an additional worker in terms of the extra revenue produced.

MRPN = P×MPN

• P is the price of output.

The MPN and the Labour Demand (continued)

• To an employer the benefit is MRPN and the cost is the nominal wage (W).

• In real terms the benefit is MPN and the cost is the real wage (w) – the nominal wage (W) divided by the price of output (P).

• The w line is horizontal; that is, the wage is constant in a competitive labour market.

• The profit-maximizing amount of labour input is the point where the MPN curve and w line intersect, MPN = w.

The Labour Demand Curve Shifters

The Labour Demand Curve Shifters (continued)

• A positive supply (productivity) shock increases the MPN and increases the quantity of labour demanded at each real wage level.

• Higher capital stock increases the MPN and increases the quantity of labour demanded at each real wage level.

• All of this is represented by a shift in the labour demand curve.

Deriving the demand for labour (Nd)

• Re-organise the above with w on the LHS

• The above is the demand for labour€

w =[(1− α )AKα ]

Nα →Nα =[(1 − α )AKα ]

w

Aggregate Labour Demand

• Aggregate labour demand is the sum of the labour demands of all the firms in the economy.

• The aggregate labour demand curve looks the same and behaves the same as a labour demand curve for an individual firm.

The Supply of Labour

• Demand is determined by firms but supply by individuals.

• Aggregate supply of labour is the sum of labour supplied by everyone in the economy.

• Each person must decide how much time to work for income (the principal benefit) versus how much time to allocate for leisure (off-work activities, the cost of working).

The Supply of Labour (continued)

• When talking about individuals making choices the concept of utility is helpful.

• The utility (happiness) from income for one more hour at work is compared to the cost (lost utility) of one less hour of leisure – there is a trade-off.

• Utility is maximized when these values are the same.

Real Wages and Labour Supply

• The real wage is real income received in exchange for giving up leisure.

• There are two effects at work here.– The substitution effect of a higher real wage is the tendency

of workers to supply more labour and reduce leisure hours in response to a higher real wage.

– The income effect of a higher real wage is the tendency of workers to supply less labour and increase leisure hours as they enjoy higher incomes.

Real Wages and Labour Supply (continued)

• The two effects work in opposite directions.– The substitution effect of a higher real wage leads to an

increase in the quantity of labour supplied.– The income effect leads to a decrease in the quantity of

labour supplied.

Real Wages and Labour Supply (continued)

• We can imagine a pure substitution and a pure income effect.

• Pure substitution effect:– A temporary increase in the wage rate (which doesn’t

affect wealth) like an overtime bonus.• A pure income effect:– A one time increase in wealth (which raises long-term

income).• An important point:– The longer an increase in the real wage is expected to

last, the larger is the income effect.

Real Wages and Labour Supply (continued)

• Because of conflicting effects there is some ambiguity about how labour supply will respond to a real wage change – it becomes an empirical question.

• The empirical evidence suggests that:– when wage increases are perceived as temporary, the

substitution effect dominates; but– when permanent, the income effect dominates.

The Labour Supply Curve• The labour supply curve is the curve which relates

the amount of labour supplied to the current real wage (other factors held constant, including the real wage expected in the future).

• The labour supply curve is upward sloping. – An increase in the current real wage leads to an

increase in labour supplied.

• With the exception of the real wage, any factor which changes the amount of labour supply will shift the labour supply curve.

Aggregate Labour Supply

• Aggregate labour supply is the total amount of labour supplied in the economy.

• More things are going on at the aggregate level than at the level of the individual.

• Moving along the curve.– An increase in the current economy-wide real wage raises the

aggregate quantity of labour supplied:– people already working supply more hours (more overtime, second

job);– some people are induced to join the labour force (reservation wage).

• We conclude that the aggregate labour supply curve slopes upward.

Aggregate Labour Supply (continued)

• Shifts in the curve:– An increase in wealth or future income will shift the curve

leftward.– An increase in the working-age population or a change in

the participation rate will shift the curve rightward.The rise in the participation of women is an important example – social attitudes changed – as is the elimination of mandatory retirement.

– The recent reductions in wealth (due to the recession) will cause people to postpone retirement or re-enter the workplace – the curve has shifted rightward.

What do the Data Show on Hours Worked

• Over time, as Canada has become wealthier (a permanent rise in real wages), hours worked and the length of time in work (both measures of labour supply) have declined.

• This pattern shows up in cross section analysis where individual country experiences are compared at a point in time.

Hours worked have declined over time…

… possibly because of gains in productivity which increase wages

Labour Market Equilibrium

• Interpreting equilibrium: – The classical model of the labour market assumes that the

real wage adjusts quickly to equate labour supply and labour demand.

– The Keynesian view is that the relationship holds but only in the long run.

• In equilibrium both sides of the labour market (firms and individuals) are satisfied. If they were not, adjustment would occur.

Labour Market Equilibrium (continued)

• The equilibrium level of employment after the complete adjustment of wages and prices is full-employment level of employment N bar

• The corresponding market clearing real wage is w.

• Both workers and employers are balancing off marginal costs with marginal benefits.

Labour Market Equilibrium(continued)

• Factors that shift aggregate labour demand or supply curve affect:– the equilibrium real wage;– the full-employment level of unemployment.

• A temporary adverse productivity shock shifts the demand curve (by changing the “A” in the production function);

• but, because it is temporary, there is no shift in labour supply – longer-run income is unaffected, so supply remains unchanged.

Labour Market Equilibrium(continued)

• The advantage of the model is that it can say something about how shocks can affect employment and wages.

• It also has some micro underpinnings, which was always a criticism of the simple Keynesian model.

• It is considered a good description of long-run equilibrium.

Labour Market Equilibrium(continued)

• The model, with some modifications, can be used to answer questions about income distribution.– Part of the reason for adverse income distribution can be

attributed to skill differentials.– The data show that the proportion of skilled workers in the

labour force is rising – a characteristic of technical change is that it has been “skill-biased”.

• A complication not shown, supply curves could shift as well.

Labour Market Equilibrium(continued)

• Full-employment output (often called potential output), , is the level of output that firms in the economy supply when wages and prices are fully adjusted.

• We can calculate it by substituting full employment labour into the production function.

Y

€

Y = AF(K,N )

N

Labour Market Equilibrium(continued)

• Effects of an adverse supply shock that affect the labour market now hits output as well:– The output is reduced directly by reduction in

productivity, A.– The MPN falls, employment falls, full

employment output falls.N

Y

Unemployment

• An important drawback of the model is that it implies that there is zero unemployment, which is not realistic, especially in the short run.

• Possible explanations of unemployment:– the real wage could be slow to adjust.– matching people to jobs can be a time consuming

process.

• This can be shown graphically by having supply greater than demand at a given wage rate.

Labour Market Definitions• First some measurement/definitions issues:– an employed person (E) is someone who worked

full-time or part-time during the past week.– an unemployed person (U) is someone who did

not work during the past week, but who had actively sought work in the previous four weeks, and was also available for work.

Labour Market Definitions (continued)

• Someone not in the labour force is a person who did not work during the past week and did not look for work during the past four weeks – they are not participating.

• The rest are in the labour force (LF), which is defined as all employed and unemployed workers (E + U).

• The working age, or adult population (P), sometimes called the source population, is the sum of those in and not in the labour force.

Labour Market Definitions (continued)

• The unemployment rate is the fraction of the labour force that is unemployed [U/(E+U) = U/LF].

• The participation rate is the fraction of the labour force in the working-age population (LF/P).

• The employment ratio is the fraction of the employed in the working-age population (E/P).

A look at the labour force

Over time the participation of women in the workforce has risen

Employment-to-population ratio, which is highly cyclical, is still recovering from the “Great Recession”

A Look at the US

Important definitions: A Summary

• Labor Force = U + E = 19.1 (million)• Unemployment rate = U/LF = 7.1%• Participation rate = LF/P = 66.5%• Employment ratio = E/P = 61.8%• The fraction of those not in the labour

force = (1 – Participation rate) = 38.2%

Changes in Employment StatusSee Figure 3.15 in the text • The labour market is in a constant state of flux – it

is very dynamic. The net numbers reported mask a lot of action.

• Workers lose and find jobs continuously. – 21.8% of unemployed find find work in the next month.– 17.2% will exit the labour force.– The rest 61% of unemployed stay unemployed the

following month.

• Some workers will become discouraged and stop searching.

Changes in employment mask very large gross flows (see slide 54)

How Long are People Unemployed?

• An unemployment spell is the length of time that an individual is constantly unemployed. – Its length is called the duration of the unemployment

spell.

• Most unemployment spells are of short duration, about two month or less.

• Most people who are unemployed on a given date are experiencing unemployment spells with long duration.

• How can both statements be true?

Unemployment duration

How Long are People Unemployed? (continued)

• Duration of unemployment varies across the country and over time.

• Duration of unemployment by 2008 was down to levels last seen in the 1970s.

• It has since risen back in the wake of the “Great Recession”.

• The business cycle has an effect.• But other factors are at play as well

(Chapter 13).

At look at duration in each province in 2012

Why are There Always Unemployed People? Frictional Unemployment

• Frictional unemployment arises as workers search for suitable jobs and firms search for suitable workers.– The search-and-match process takes time.– During that time someone is unemployed.

Structural Unemployment• Structural unemployment is the long-term and

chronic unemployment that exists when the economy is not in a recession.– Unskilled or low skilled workers are unable to find

long-term jobs.– Workers lose their skills.– Reallocation of labour from industries/regions in

decline takes time.– Perhaps unemployment benefits hinder the

process.

The Natural Rate of Unemployment

• The natural rate of unemployment ( ) is the rate of unemployment that prevails when output and employment are at their full-employment levels.

• The natural rate of unemployment exist due to frictional and structural causes.

u

The relationship between the actual and the “natural” unemployment rate for Canada

The relationship between the actual and the “natural” unemployment rate for the US

Cyclical Unemployment

• Cyclical unemployment is the difference between actual and natural unemployment rates

• Cyclical unemployment is positive whenever the actual unemployment rate is above the natural rate (and vice versa).

)u(u

Okun’s Law• Okun’s law states that the gap between an

economy’s full-employment and actual levels of output increases by about 2 percentage points for every 1 percentage point increase in the unemployment rate.

• A rise in cyclical unemployment has a magnified effect on output – the number of people in the work force, hours worked and productivity also fall.

Okun’s Law (continued)

• Okun’s law can also be expressed in growth terms. Taking the total derivative and assuming that:

€

Y −Y

Y = −2(u − u )

€

ΔY × Y

Y 2−

ΔY × Y

Y 2= −2Δu

€

Y = Y

€

ΔY

Y=

ΔY

Y − 2Δu

Okun’s Law: Original version for Canada

A look at Okun’s Law in Canada: The growth version doesn’t work as well

Okun’s Law in the recent Canadian recession: The version with the output and unemployment gap

Okun’s Law in the recent US recession: The version with the output and unemployment gap

Bank of Canada Governor Steve Poloz (BAH, Queen’s ‘78) suggests that college grads who are having a hard time finding jobs should consider

working as interns for free so as to gains work experience.

Addendum: Properties of the production function

• Let’s start with the production function:(1) Y = AKαN(1-α)

• The return to capital is its marginal product, (the slope of the production function, fig. 3.2):(2) MPK = ΔY⁄ΔK = αAK(α-1)N(1-α) > 0

• The return to labour is the real wage (fig. 3.3):(3) MPN = ΔY⁄ΔN = (1-α)AKαN-α > 0

(4) Real income = returns to factors times number of them = MPK×K + MPN×N

The production function and why product divides (next steps)

(5) MPK×K = αAK(α-1)N(1-α)×K

= αAKαN(1-α)

= αY

• In a similar fashion it can be shown that:(6) MPN×N = (1-α)Y

• When we add up total wages and the total return on capital we get:(7) MPK×K + MPN×N = αY + (1-α)Y = Y

• Income = output – pretty cool result!

The evolution of humans and jobs in the economy over time

Canadian Unemployment Rate

Canadian Unemployment Rate

US Unemployment Rate

US Unemployment Rate