Consumer and Firm Behaviour - Southampton · The consumer is indifferent between bundle (C1,ℓ1) and bundle (C2,ℓ2) if U(C1,ℓ1) = U(C2,ℓ2) NOTE: Utility levels are irrelevant—what

Consumer and Firm BehaviourChapter 4

Topics in Macroeconomics 2

Economics DivisionUniversity of Southampton

February 2009

Chapter 4 1/41 Topics in Macroeconomics

Representative ConsumersThe Representative Firm

Introduction

◮ One-period model◮ Consumers and firms make static decisions (no dynamics)◮ There are many consumers and firms, but we assume that

all firms are identical and all consumers are identical◮ We will study a representative consumer and a

representative firm◮ Consumers face a tradeoff between consuming and

working (work/leisure)◮ Firms maximize profits given the state of technology◮ Next Chapter: Market clearing



PreferencesBudget ConstraintOptimization

Consumers Care about Two Goods

1. Good 1: Physical good◮ You can think of this ‘good’ as a aggregation of all

consumer goods (composite consumption good)◮ Measurement: aggregate consumption

2. Good 2: Leisure◮ Time spent not working in the market◮ Includes: recreational activities, sleep, and work at home◮ Measurement: total number of hours available minus total

working hours

3. Note: Representative consumer




Preference Representation

◮ Preferences can be represented by a utility function◮ Individuals get utility from a consumption bundle (C, ℓ)

◮ The utility derived from C1 units of consumption goods andℓ1 quantity of leisure is

U(C1, ℓ1)

◮ The utility function is meant to measure a happiness index:

how happy you are to consume particular bundles of goods




Preference Representation

◮ The bundle (C1, ℓ1) is strictly preferred to bundle (C2, ℓ2) if

U(C1, ℓ1) > U(C2, ℓ2)

◮ The bundle (C2, ℓ2) is strictly preferred to bundle (C1, ℓ1) if

U(C1, ℓ1) < U(C2, ℓ2)

◮ The consumer is indifferent between bundle (C1, ℓ1) andbundle (C2, ℓ2) if

U(C1, ℓ1) = U(C2, ℓ2)

◮ NOTE: Utility levels are irrelevant—what matters is thelevel of utility from one bundle relative to the level of utilityof another bundle




Assumptions on Preference

1. More is always preferred to less◮ Keeping leisure constant, more consumption is preferred to

less, and vice versa◮ We won’t worry about over eating and/or drinking

2. Diversity is a good thing◮ If two consumption bundles give the same level of utility,

then a mixture of the two bundles will be preferred to eitherbundle alone

3. Consumption goods and leisure are normal goods◮ As their income increases, consumers want to consume

more of both goods◮ For eg., after winning the lottery you want to consume more

goods and take more vacation time (leisure)◮ This behaviour seems to be supported by the data




Indifference Curves

Indifference CurveAn indifference curve connects a set of points, with these pointsrepresenting bundles among which the consumer is indifferent




Property 1: Indifference Curves slope downward

This is because more is preferred to less




Property 2: Indifference Curves are convex

This is because diversity is a good thing




Marginal Rate of Substitution

Marginal Rate of SubstitutionThe marginal rate of substitution of leisure for consumption(MRSℓ,C) is the rate at which the consumer is just willing tosubstitute leisure for consumption goods(the negative of the slope of an indifference curve)




Representative Consumer’s Budget Constraint

◮ The time constraint:

ℓ + Ns = h

h Total amount of time availableNs Amount of time spent workingℓ Leisure time

◮ The budget constraint:

C = wNs + π − T

w real wage rate per unit of time (in terms of C)π Firm profits redistributed to consumers (dividend income)T Lump-sum tax




Alternative Representations of the Budget Constraint

◮ Replace the time constraint (ℓ + Ns = h) into the budgetconstraint

C = wNs + π − T

C = w(h − ℓ) + π − T

◮ Add wℓ on both sidesC + wℓ︸︷︷︸

Implicit expenditures

= wh + π − T︸︷︷︸

Implicit real disposable income

◮ To graph the budget constraint, it is more convenient towrite

C = −w︸︷︷︸

Slope

ℓ + wh + π − T︸︷︷︸

Intercept




Representative Consumer’s Budget Constraint (T > π)

◮ The horizontal intercept is foundby setting C = 0 and solving theequation−wℓ + wh + π − T = C = 0for ℓ

◮ The solution is ℓ = h + (π − T )/w◮ If T > π, then

π − T < 0 andh + (π − T )/w < h




Representative Consumer’s Budget Constraint (T < π)

◮ The horizontal intercept wasℓ = h + (π − T )/w

◮ If T < π, thenπ − T > 0 andh + (π − T )/w > h

◮ But you cannot consume moreleisure than your endowment oftime

◮ Point along BD all involveworking zero hours

◮ Points below B involve throwingaway income




Consumer Optimization

◮ The consumer will choose thebest bundle that is budgetfeasible

◮ Points below the budget line(e.g. J) would never be chosen(more is preferred to less)

◮ B is preferred to all points alongBD




Consumer Optimization

◮ The market pays w consumptiongoods for each unit of leisure yougive up

◮ Point F : MRSℓ,C > w : willing togive up more than wconsumption goods for an extraunit of leisure

◮ Point E : MRSℓ,C < w : willing togive up less than w consumptiongoods for an extra unit of leisure

◮ Point H: MRSℓ,C = w




Why is point H optimal

◮ The optimal bundle is at a point where the indifferencecurve is tangent to budget line AB

◮ At point H the rate at which the consumer is willing to tradeleisure for consumption (MRSℓ,C) is the same as the rate atwhich leisure trades for consumption on the market (w )

◮ The marginal rate of substitution of leisure for consumptionequals the relative price of leisure in terms of consumptiongoods

◮ More generally:Consumer optimization in competitive markets will implythat the consumer sets the marginal rate of substitution ofany good x for any good y equal to the relative price of x interms of y




The Representative Consumer Chooses not to Work

◮ The consumer could, in principle,choose not to work at all (point B)

◮ This will not be possible whenconsumers and firms interact:There would be no productionand so no consumption

◮ Points like A and B will be ruledout by assuming that theconsumer wishes to consumesome of both goods




Using the Model: 2 Experiments

1. Increase in after-tax dividend income (π − T )◮ Holding wage rate (w) constant◮ As such, produces a pure income effect on the consumer’s

choices

2. Increase in the real wage rate (w )◮ Produces an income effect: the consumer is richer◮ Produces a substitution effect: the price of leisure in terms

of consumption goods is higher




Experiment 1: Increase in Dividend Income

◮ The experiment is to increaseπ − T

◮ ABD was the original budgetconstraint (H was optimal)

◮ FJD is the new budget constraint(higher intercept)New optimum is K

◮ Both C and ℓ increase becauseboth are normal goods




Experiment 2: Increase in Real Wage Rate

◮ ABD was the original budgetconstraint (F was optimal)

◮ EBD is the new budget constraint(higher intercept and higherslope)New optimum is H

◮ We know that C will increase(you are richer)

◮ However, ℓ may increase,decrease, or remain the same

◮ This is because there are twoopposing effects




The Substitution Effect

◮ From the new budget constraint,lower dividend income until theconsumer’s optimal choice is onthe same indifference curve asthe original bundle

◮ i.e. with the budget constraintJKD the consumer choosespoint O which gives him the samelevel of utility as point F

◮ Since leisure is now moreexpensive, bundle O features lessleisure and more consumption




The Income Effect

◮ The movement from O to H is apure income effect

◮ As before, both C and ℓ willincrease since they are normalgoods

◮ Net effect on C must be positive◮ Net effect on ℓ is unclear:

In this examples, both effectscancel each other out




The Labor Supply Curve

◮ The labour supply curve isobtained by finding the optimalbundle under many different realwage rates:(C(w), ℓ(w)

)

◮ For any given w the laboursupply is Ns(w) = h − ℓ(w)

◮ This labour supply curveassumes that the substitutioneffect dominates the incomeeffect:The labour supply curve isupward sloping




Increase in Dividend Income

◮ We know that an increase in(π − T ) increases leisure (pureincome effect)

◮ It follows that the labour supply atany wage rate is lower followingan increase in (π − T )

◮ The labour supply shifts to the leftfollowing an increase in (π − T )



TechnologyProfit Maximization

Firm Behaviour

◮ The firm demands labour and supplies consumption goods◮ We will assume that the firm owns some capital (K ) which

is fixed in this static model◮ The firm uses its capital and hires labour to produce

consumption goods◮ The choice of the firm is determined by the available

technology and profit maximization




Technology and the Production Function

Y = zF (K , Nd)

◮ The production function tells us how much output will beproduced for given quantities of capital and labour

◮ Y is total output produced◮ F is the production function◮ z represents total factor productivity◮ K is capital◮ Nd is the labour input, measured as total hours worked by

employees of the firm




The Marginal Product of Labour

◮ The marginal product of labour(MPN) is the additional outputthat can be produced with oneadditional unit of labour holdingcapital constant

◮ It is equal to the slope of theproduction function (holdingcapital fixed at K ∗)




The Marginal Product of Capital

◮ The marginal product of Capital(MPK ) is the additional outputthat can be produced with oneadditional unit of capital holdinglabour constant

◮ It is equal to the slope of theproduction function (holdinglabour fixed at N∗)




Properties of the Production Function

1. Exhibits constant returns to scale (CRS)◮ If you double the inputs, output doubles as well◮ The size of firms is irrelevant: One big firm is equivalent to

many small ones◮ With CRS, all we need is a representative firm

2. Output increases with inputs◮ You can do more with more inputs◮ Alternatively, marginal products are positive

( MPN > 0 and MPK > 0 )




Properties of the Production Function

3. The marginal product of labour decreases as the quantityof labour increases

◮ The slope of the production function (MPN ) decreases asNd increases

4. The marginal product of capital decreases as the quantityof capital increases

◮ The slope of the production function (MPK ) decreases as Kincreases

5. The marginal product of labour increases as the quantity ofcapital increases (shown on the next slide )




Adding Capital Increases the Marginal Product ofLabor

◮ The curve labeled MP1N

corresponds to a level ofcapital K = K 1

◮ The curve labeled MP2N

corresponds to a level ofcapital K = K 2 > K 1




Effects of an increase in TFP

Shifts the production functionup because more output isproduced with the same inputs

Increases the marginal productof labour (given a level ofcapital)




Revenue, Variable Costs, and Profit Maximization

◮ Profits are equal to productionminus the cost of labour (K isfixed)π = Y − wNd

π = zF (K , Nd ) − wNd

◮ The profit maximizing quantity oflabour is such that the slope ofthe production function is equalto the slope of the labour costcurve (w )

◮ ⇒ MPN = w◮ π∗ is the distance AB




Why is N∗ Optimal?

◮ To the left of N∗, MPN > w , soone more unit of labour at wproduces more than w units ofconsumption goods

◮ To the right of N∗, MPN < w , soone fewer unit of labour wouldsave w but would lower output byless than w

◮ Eventually the firm will setNd = N∗, where MPN = w




Labor Demand Curve of the Profit-Maximizing Firm

◮ The labour demand curve isobtained by finding the optimallevel of labour under manydifferent real wage rates

◮ But firms always set N such thatMPN = w

◮ So the marginal product of labourcurve is the labour demand curve


Documents

Consumer and Firm Behaviour - Southampton · The consumer is indifferent between bundle (C1,ℓ1) and bundle (C2,ℓ2) if U(C1,ℓ1) = U(C2,ℓ2) NOTE: Utility levels are irrelevant—what