BUEC 280 LECTURE 8

Introduction to Labour Demand

Last few weeks …

We developed two simple models of labour supply

Consumption-leisure choice model

Household production model

The goal was to understand the decisions that people

make to determine their labour supply

Now, we’ll do the same for firms

Where does the labour demand curve come from?

Profit maximization

Our basic assumption is that firms maximize profits

They continually ask “what changes can we make to improve profits?”

Firms can only control certain things

Usually assume that firms cannot choose prices (they are determined in a

market)

Firms can choose how much output to produce, and how to produce it

(technology, mix of inputs, etc.)

Focus on marginal changes – small changes in one dimension, holding

other things constant

Inputs to production

Assume there are two factors of production (inputs): labour and capital

Assume that firms expand/contract production by varying the quantity of one or both factors, holding constant their production technology

What do I mean by technology? It is how firms combine labour and capital to produce output.

How are these decisions made?

If income (revenue) generated by using 1 more unit of an input exceedsthe extra expense, then use more of that input

If revenue generated by using 1 more unit of an input is less than the expense, use less of that input

If revenue generated by adding 1 more unit of an input is equal to the expense, no change necessary

Marginal product, marginal revenue, and marginal

revenue product

So the basis of the decision to use more/less of an input is based on the extra revenue generated by using 1 more unit of an input – we call this the marginal revenue product

It is the product of two quantities: the extra output produced by using 1 more unit of the input (the input’s marginal product) and the extra revenue from producing one more unit of output (marginal revenue)

Example: Imagine a tennis tournament. Suppose that if a famous tennis player plays in the tournament, 2000 extra spectators will come. Tickets are $25 each.

The player’s marginal product is 2000 spectators

Marginal revenue is $25 per spectator

Maria’s marginal revenue product is $25 x 2000 = $50,000

Formal definitions

The marginal product of labour (MPL) is the change in physical output (ΔQ) due to a change in the units of labour input (ΔL), holding capital (K) constant:

MPL = ΔQ / ΔL (holding K constant)

Similarly, the marginal product of capital isMPK = ΔQ / ΔK (holding L constant)

The firm’s marginal revenue (MR) is the extra income generated by producing an additional unit of output. In a competitive output market, firms take prices as given, and MR is just the equilibrium price of the good they produce:

MR = p

The marginal revenue product of labour (MRPL) is the extra revenue generated by employing an extra unit of labour:

MRPL = MR x MPL = p x MPL (competitive market)

Similarly, the marginal revenue product of capital (MRPK) is:MRPK = MR x MPK = p x MPK (competitive market)

What about costs?

Hiring an additional unit of labour or capital isn’t free …

For now, assume firms take all prices (including input prices) as given

i.e., decisions made by the firm do not affect prices

The worker’s wage W is the marginal expense of hiring one more unit of labour

The rental rate of capital r is the marginal expense of hiring one more unit of capital

Short-run labour demand with perfectly competitive input

and output markets

We’ll focus first on the short run – a period of time that is short enough so that firms can vary L but notK

Assumption: Declining MPL

We will assume that (eventually) each additional unit of labour hired by the firm is less productive than the previous unit

MPL can rise at first (maybe because of cooperation), but eventually it must fall

Why? Because one factor (capital) is fixed

Example of declining MPL

Consider a hypothetical car dealership

The first worker hired sells 10 cars

MPL=10

When a second worker is hired, total sales are 21 cars

MPL=11>10 because the two salespeople can help one another

# Salespeople # Cars Sold

MPL

0 0

1 10 10

2 21 11

3 26 5

4 29 3

When a third is hired, total sales increase to 26 MPL=5<11 because a fixed building can only contain so many

cars and customers (diminishing marginal product)

The profit maximization conditions

Before, we said that to maximize profits:

If MRPL exceeds the marginal expense of labour, then hire 1 more unit to increase profit

If MRPL is less than the marginal expense, reduce labour input to increase profit

If MRPL is equal to the marginal expense, no change necessary because profit cannot be increased by changing labour input

The marginal expense of labour is W

Profits are maximized when MPRL = W

MPL x p = W

MPL = W/p

What’s W/p? The real wage, in units of output.

The firm’s short run labour demand

L

MPL,Real Wage (W/p)

MPL

W3/p

W2/p

W1/p

L3 L2 L1

W3

W2

W1

L3 L2 L1

Labour Demand

Nominal Wage (W)

L

Where does market labour demand come from?

We just add up all the individual firms’ labour demand

curves

E.g., suppose there are 2 firms in the economy. If firm 1 demands

2 units of labour when W=10 and firm 2 demands 3 units, then

total labour demand is 5 units at this wage

Because MPL is downward sloping for each firm, we know

labour demand is downward sloping for each firm

⇒ market labour demand is also downward sloping

Going a step further …

We can use a graphical model to analyze the firm’s demand for labour Looks like the consumption-leisure choice model

Decision maker is the firm

Decides how many units of labour to hire at given input prices

It is a cost minimization problem

The firm cannot maximize profits without minimizing costs.

That is, to maximize profit, the firm must minimize the cost of producing a given level of output

The production function

We’ll continue to assume there are two inputs to production: labour (L) and capital (K)

We will describe a firm’s technology by a function:

Q = f(L,K)

Here, Q is the output produced by the firm, and f is the name of the function

We call this a production function, and it tells us how many units of output Q the firm can produce if it uses K units of capital and L units of labour

The production function describes the firm’s production technology

Isoquants

We represent the production function graphically with an

isoquant

A curve that tells us all the different combinations of L,K that can

be used to produce a given level of output, Q

Think of it like an indifference curve

Instead of measuring different combinations of leisure &

consumption that yield the same level of U, it measures

combinations of L,K that can be used to produce the same level of

output Q

Properties of Isoquants

Negative slope

L,K are substitutes

Convex

Mixtures of L,K are more productive than extremes

Diminishing marginal returns

Don’t cross

The bundles (L1,K3), (L2,K2), and (L3,K1) all produce the same level of output Q=100.

If we fix capital at K3, changing the quantity of labour changes output: f(L1,K3) = 100f(L2,K3) = 150f(L3,K3) = 200 L

K

Q=100

Q=150

Q=200

L1L2 L3

K1

K2

K3

The Marginal Rate of Technical Substitution

In the leisure-consumption choice model, the slope of the indifference curve was called the marginal rate of substitution (MRS)

Rate at which individual was willing to trade off consumption and leisure to hold utility constant

We call the slope of the firm’s isoquant the marginal rate of technical substitution (MRTS)

Rate at which the firm can trade off capital and labour and hold output constant

It is negative because if we increase L, we must decrease K to hold output constant

QQL

KMRTS

Short-run Labour Demand

In the short run K is fixed, say at K*

Profit max: MPL = W / p

This defines short run labour demand

Recall we assumed MPL declining (need this for the MPL = W/p condition to have meaning)

We can see that MPL declines in the figure at right:L3 - L2 > L2 – L1

L

K

Q=100

Q=150

Q=200

L1L2 L3

K*

The Long Run

In the long run, firms can vary both L and K

Just like before, the firm maximizes profits only if the extra revenue from hiring one more unit of an input equals the extra cost

i.e., marginal revenue = marginal cost

In the short run, this was just MRPL = W

In the long run, this condition must hold for L and K:

MRPL = W p x MPL = W p = W / MPL

MRPK = r p x MPK = r p = r / MPK

W / MPL = r / MPK

Since MPL = ΔQ / ΔL and MPK = ΔQ / ΔK, we see that

WΔL / ΔQ = rΔK / ΔQ

Extra cost of producing 1 more unit of output using L = Extra cost of producing 1 more unit of output using K

Why is this profit maximizing?

Our conditions is: W / MPL = r / MPK

Extra cost of producing 1 more unit of output using L = Extra cost of producing 1 more unit of output using K

Suppose this equality didn’t hold, so that W / MPL > r / MPK. This means Extra cost of producing 1 more unit of output using L >

Extra cost of producing 1 more unit of output using K

The firm could reduce its use of labour and increase its use of capital, save money, and produce the same level of output TO MAXIMIZE PROFITS, THE FIRM MUST MINIMIZE COSTS

A Graphical Treatment

We can show this graphically

Need to introduce the analog to the budget line: called an isocost line

The isocost line represents combinations of L,K that cost the same amount

just like a budget line, except the firm can choose their level of expenditure, i.e., which isocost line they are on

The isocost line

Suppose W = $10 r = $20

Here are three isocost lines. They give combinations of L,K that cost $1000, $1500, and $2000 respectively

In each case,Slope = – W / r

= - 1 / 2L

K

100 150 200

50

75

100

Cost=$1000

Cost=$1500

Cost=$2000

The cost minimization problem

Recall our profit max (or cost min)

condition:

W / MPL = r / MPK

WΔL / ΔQ = rΔK / ΔQ

W / r=(ΔK / ΔQ) / (ΔL / ΔQ)

W / r = ΔK / ΔL

Slope of isocost = MRTS!!

The cost-minimizing way to produce

output level Q* is (L*,K*), which costs

C*

Is this profit maximizing?L

K

Q=Q*

Cost = C*

L*

K*

Scale and Substitution Effects

How do firms respond to changes in input prices?

Suppose W increases

Because labour is now more expensive relative to capital, and because labour and capital are substitutes in production, firms will change their input mix to use less labour and more capital to produce any level of output

This is a substitution effect

Because it is now more expensive to produce any level of output, the firm will reduce output (and hence reduce use of labour (and probably capital too)

This is a scale effect (like an income effect in the leisure consumption choice model)

Scale and Substitution Effects Graphically

L

K

Q=Q0

Cost = C0, slope = -r/W0

L0

K0

Cost > C0, slope = -r/W1

1. Wage is W0, and firmchooses (L0,K0) to produceQ0 units at cost C0

2. Wage increases to W1

Substitution effect:(L0,K0) to (Ls,Ks)[change in input mix due to higher wage, holding outputconstant]

Scale effect:(Ls,Ks) to (L1,K1)[change in inputs due toreduced output Q1, becauseproduction is more expensive]Note: we don’t know Q1

Ls

Ks

Q=Q1

L1

K1