2014/6/1 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip1 Consumer and Firm Behavior: The Work-Leisure Decision and Profit Maximization

112/04/10ECO 2021 Intermediate Macroeconomic Theory

Professor C. K. Yip 1

Consumer and Firm Behavior:

The Work-Leisure Decision and Profit

MaximizationChapter 4



Static vs Dynamic Decision Making

• In this and the next chapters, we are considering static decision making, i.e., planning over a single period.

• From chapter 6 to 9, we are going to discuss dynamic decision making, i.e., planning over more one period.

• Chapter 4 first recalls what you’ve learnt in the last semester: the micro behavior of a representative consumer and a representative firm.

• Chapter 5 then assembles these in a macro model in order to address some important macro issues. The role of government is also introduced.



Objectives of the Representative Consumer & the Representative

Firm • Representative Consumer: To maximize utility subject to budget (and time) constraint by allocating time between work and leisure;

• Representative Firm: To maximize profits subject to technological constraint by deciding how much labor to be hired.



Assumptions of the Model

1) Two Goods:– Consumption good, which is an aggregation of all

consumer goods in the economy.– Leisure, which is any time spent other than working in

the market.e.g. Recreational activities, sleep and household work.

2) One Consumer:– All consumers are identical in terms of preferences,

ability, time constraint and budget constraint. Then, the economy will behave as if there were only one consumer, one that we refer to as the representative consumer.



Assumptions of the Model

3) Price-Taking:– The representative consumer is a price-taker, i.e., he

takes all market prices as given, and acts as if his actions had no effect on those prices.

4) No Money:– The economy we’re considering is a barter economy, i.e.,

all trade involves barter exchanges of goods for goods in the absence of money.



The Representative Consumer’s Optimization Problem

• Objective: to make himself as well off as possible given the constraints he faces.

• Two Ingredients in this problem:– Consumer’s preferences– Consumer’s budget constraint



Preferences• The preferences of the representative consumer is

captured by the utility function,U(C , l)

where C is the quantity of consumption, l is the quantity of leisure

• Any particular pair of consumption and leisure (C , l) is called a consumption bundle.

• For each consumption bundle, the utility function U assigns a real number so that different bundles can be ranked.



Preferences

• Consider two distinct bundles (C1 , l1) and (C2 , l2)

– (C1 , l1) is strictly preferred to (C2 , l2) if

U(C1 , l1) > U(C2 , l2)

– Consumer is indifferent between the two bundles if

U(C1 , l1) = U(C2 , l2)

• Assumptions on Preferences:

1) More is always preferred to less– A consumer always prefers a consumption bundle that

contains more consumption, more leisure, or both.



Preferences2) Consumer prefers a more diversified consumption bundle.– If the consumer is indifferent between (C1 , l1) and (C2 , l2),

then some mixture of the two will be preferable to either one.

– Example: Consider a new bundle (C3 , l3), where C3 = C1 + (1 – )C2,

l3 = l1 + (1 – )l2 and lies between 0 and 1 (a fraction), then

U(C3 , l3) > U(C1 , l1) = U(C2 , l2)

3) Consumption and leisure are normal goods.– A good is normal (inferior) for a consumer if the quantity of

the good that he/she purchases increases (decreases) when income increases.



Graphical Representation of Preferences

• An indifference curve connects a set of points, with these points representing consumption bundles among which the consumer is indifferent.

• A family of indifference curves is called indifference map.



Properties of Indifference Curves• Consider a consumption

bundle B. Since a consumer prefers more to less, any bundle that is indifferent to B must lie within quadrant II and IV.

• Implication: An indifference curve slopes downward.

B

I

IIIII

IV

Leisure

Consumption

.



Properties of Indifference Curves• Consider any two

bundles A and B, since a consumer prefers a more diversified bundle C to either A or B, the set of bundles that are indifferent to A and B must lie below the straight line AB.

• Implication: An indifference curve is convex, that is bowed-in toward the origin.

A

B

C

Consumption

Leisure

.

..



Graphical Representation of Preferences

A

Leisure, l

Consumption, C

.

.

.B

DI1

l2l1

C2

C1

I2



Marginal Rate of Substitution• Marginal rate of

substitution of leisure for consumption (MRSl,C) is the rate at which the consumer is just willing to substitute leisure for consumption good.

• It is also minus the slope of the indifference curve.

• Convexity of indifference curve is equivalent to– Diminishing marginal rate

of substitution. (compared slope at A and slope at B)

A

Leisure, l

Consumption, C

. BD

I1

l2l1

C2

C1

I2

..

.Slope = MRSl,C



Marginal Rate of Substitution• The MRS at A is larger (in

terms of absolute magnitude) than the MRS at B.

• As we increase l and reduce C, i.e. moving from A to B along I1, the consumer needs to be compensated more in terms of l to give up another unit of C.

• The consumer requires this extra consumption because of a preference for diversity.

A

Leisure, l

Consumption, C

. BD

I1

l2l1

C2

C1

I2

..

.Slope = MRSl,C



Marginal Rate of Substitution

Mathematical Derivations:• Suppose indifference curve

I1 represents the utility level ,

• Totally differentiate this with respect to C and l gives

UUlCU ),(

c

lCl

lc

U

U

dldC

MRS

dlUdCU

,

0

A

Leisure, l

Consumption, C

. BD

I1

l2l1

C2

C1

I2

..

.Slope = MRSl,C



Constraints faced by The Representative Consumer

• Two constraints:– Time constraint for l– Budget constraint for C

• The time constraint for the consumer is given by

l + Ns = hwhere h is the total number of hours available (e.g., 24 hours a day), l is the leisure time and Ns is the time spent working (or labor supply).



Budget Constraint• Sources of income:

1) Real wage income, wNs

– w is the real wage, i.e., the price of one unit of labor time in terms of consumption goods (the numeraire).

2) Real dividend income, – Since the firms are owned by the representative

consumer, any profits made by firms are distributed to him as dividends.



Budget Constraint• Taxation T: A lump-sum tax, i.e. a tax that does not

depend on the actions of the economic agent who is being taxed.

• Real Disposable Income = wNs + – T• The consumer first receives income and pays taxes in

terms of consumption goods, and then decides on how much to consume out of the disposable income.

• All disposable income is consumed, i.e.

C = wNs + – T = w(h – l) + – T



Budget Constraint• Reasons:

– Since the consumer only lives for one period, there is no incentive to save anything.

– Since more is preferred to less, any wastage is not optimal.

• The consumer’s budget constraint can be written asC + wl = wh + – T

• RHS = Total implicit real disposable income• LHS = Implicit real expenditure on consumption goods

and leisure• Note: w can also be interpreted as the market price, or

the opportunity cost, of leisure time.



Graphical Representation of the Budget Constraint

• Budget constraint:C = –wl + (wh + – T)

thus slope = –w.• The vertical intercept, wh

+ – T, is the maximum consumption that can be achieved when the consumer consumes no leisure.

• Case 1: < TLeisure, l

Consumption, C

h + ( – T)/w

h

wh + – T

A

B

C = –wl + wh + – T

.

.




• Case 2: > T– The consumer can still

enjoy C = – T > 0 even if he chooses not to work.

– When C = 0, l = h + ( – T)/w, but it is not feasible as the maximum time can only be h

– When l = h, C = – T D

Leisure, l

Consumption, C

h

A.

. – T

.B

C = –wl + wh + – T




• The budget constraint tells us what consumption bundles are feasible to consume given the market real wage (w), dividend income () and taxes (T).

• The consumption bundles within the shaded regions and on the budget constraint, are feasible.

• Thus the shaded region together with the budget constraint is called the feasible set.

Not Feasible

D

Leisure, l

Consumption, C

h

A.

.

.BFeasible



Consumer Optimization• The representative consumer is assumed to be

rational, i.e. he always chooses the best feasible consumption bundle, or the optimal consumption bundle.

• “Best” in the sense that it lies on the highest possible indifference curve.

• “Feasible” in the sense that it lies within the feasible set.



Graphical Solution• Suppose > T.• Claim: H is the optimal

consumption bundle.Reasons:• Any bundle inside the

budget constraint is not optimal (compare J to F).

• B is preferred to any point on BD.

• For any point on AB, the consumer can always improve by moving closer to H.

D

Leisure, l

Consumption, C

h

A.

. – T

.B

F

.H

E

I2I1

...J



Mathematical Solution• The consumer tries to solve the following constrained

optimization problemmax U(C , l)

C , l

subject to C = w(h – l) + – Tand C 0, h l 0.

• Lagrangian

L = U(C , l) + [w(h – l) + – T – C]

where is the Lagrangian multiplier.



Mathematical Solution• We assume that an interior solution can be obtained. This

means choosing C = 0, l = h or l = 0 are not optimal (so that we can ignore the last two constraints).

• Formally, we can impose the restrictions:

Uc(0 , l) = ∞ and Ul(C , 0) = ∞For any C and l, to guarantee an interior solution.

• First-order (Necessary) conditions (FOCs):– Obtained by differentiating the Lagrangian with respect to C, l

and .(Recall: Lagrangian equation L = U(C , l) + [w(h – l) + – T – C]

Uc(C , l) = ,

Ul(C , l) = w,

w(h – l) + – T – C = 0.



Mathematical Solution• From the FOCs, we obtain

• At H, where an indifference curve is just tangent to the budget constraint, the above equality holds.

• If MRS > w (e.g. at F), the consumer would be better off by increasing l and reducing C, thus moving closer to H.

D

Leisure, l

Consumption, C

h

A.

. – T

.B

F

.H

E

I2I1

...J

wlCU

lCUMRS

c

lCl

),(

),(,



Comparative Statics• To determine how C and l changes when any of , T

and w changes.• Recall the FOC of the consumer’s problem,

which can be written asUl(C , l) – wUc(C , l) = 0. (1)

• From the budget constraint,

w(h – l) + – T – C = 0. (2)• The two form a system of equations in terms of C and l

(endogenous variables).

wlCU

lCU

c

l ),(

),(



Comparative Statics• Totally differentiate the two equations

–dC – wdl + (h – l)dw + d – dT = 0 from (2)[Ucl – wUcc]dC + [Ull – wUcl]dl – Ucdw = 0 from (1)

In matrix form,

• Determinant of the bordered Hessian matrix A is = –Ull + 2wUcl – w2Ucc

• Strict quasiconcavity of U > 0.

A

dTddwU

lh

dl

dC

wUUwUU

w

cclllcccl

0

1

0

1)(1



1) Changes in and/or T• Using Cramer’s Rule, we get

• The assumption that consumption and leisure are normal goods is equivalent to the conditions –Ull + wUcl

> 0 and Ucl – wUcc > 0.

.0)(

,0)(

cccl

clll

wUU

dTdl

ddl

Tddl

wUU

dTdC

ddC

TddC



1) Changes in and/or T Graphical Illustration

• Consider a net increase in – T.

• Since w (slope) remains the same, the budget constraint makes a parallel shift (from AB to FJ).

• Since disposable income , while prices remain the same, there is only a pure income effect on the consumer’s choices.

• The new optimal consumption bundle is K, where both C and l (normal goods).

D

Leisure, l

Consumption, C

h

A

.B

K

I2

I1

.H

.

.

.J.

.F

C1

C2

l1 l2



1) Changes in and/or T Graphical Illustration

Remark: • The increase in

consumption (C2 – C1) is less than the

increase in nonwage income (distance AF).

• Since the consumer is working less (leisure ), wage income .

• This will offset part of the consumption increase.

D

Leisure, l

Consumption, C

h

A

.B

K

I2

I1

.H

.

.

.J.

.F

C1

C2

l1 l2



2) Changes in w• Using Cramer’s rule,

• C is normal good –Ull + wUcl > 0, together with >

0 and Uc > 0

.))((

,))((

ccclc

clllc

wUUlhU

dwdl

wUUlhwU

dwdC

.0dwdC



2) Changes in w• However, we cannot determine the effect of a change in

w on l.• Reason: It depends on the relative magnitude of the

opposing income and substitution effects.• Substitution effect: w Opportunity cost of leisure (l becomes more expensive

relative to C) Demand for leisure

• Income effect: w Wage income Demand for leisure (normal

good)

0)(

cUsubstdwdl



2) Changes in w Graphical Illustration

• Suppose > T and w .• The budget constraint shifts

from ABD to EBD (with a steeper slope).

• This shows a special case in which leisure remains unaffected.

• Pure substitution effect: Movement from F to O (on the same indifferent curve).

• Pure income effect: Movement from O to H.

• Both income and substitution effects act to C.

D

Leisure, l

Consumption, C

h

A

.

B

H

I2I1

F

K

E

C1

C2

l1

.O..

..

..

.

J



2) Changes in w Graphical Illustration

• Labor supply curve which specifies how much labor the consumer wishes to supply given any real wage.

• Algebraically, the labor supply curve is

Ns(w) = h – l(w),where l(w) is the demand function for leisure.

• Substitution effect > Income effect Upward sloping labor supply curve

• Net in ( – T) Upward shift in labor supply curve

Ns

Real Wage, w

Employment, N



Example: C and l are perfect complements

• Suppose the consumer’s utility function can be represented by

U(C , l) = min{C , al}.(Leontief Function)

where a is a positive constant.• Note that more is not always

preferred to less. The consumer can be better off only if he receives more of both goods.

• Thus, it is always optimal to choose

C = al.

D

Leisure, l

Consumption, C

h

A

.

BF .

E

. .C = al

I2

I1.



Example: C and l are perfect complements

• Combining C = al and the budget constraint gives

• In this case,

• This is because with perfect complements, there are no substitution effects. Thus leisure as real wages .

,wa

Twhl

.)(

waTwha

C

0walh

dwdl



The Representative Firm• The firm owns productive capital and hires labor to

produce consumption goods.• Production technology is captured by the production

function, which describes the technological possibilities for converting factor inputs (capital K and labor Nd) into outputs Y.

Y = zF(K , Nd)where z is total factor productivity.

• z both K and Nd will be more productive.



Assumptions on Production Function

• Production function exhibits constant returns to scale (or homogenous of degree one).– For any x > 0, xY = zF(xK , xNd).– If all factor inputs are changed by a factor x, then output

changes by the same factor x.– In this case, a perfectly competitive economy with

numerous small firms will behave in exactly the same way as one with a single representative firm (same level of efficiency).

– Increasing return to scale: zF(xK , xNd) > xzF(K , Nd).– Decreasing return to scale: zF(xK , xNd) < xzF(K , Nd).




• Positive marginal product of capital (MPK) and marginal product of labor (MPN).– MPK (MPN) is the additional

output that can be produced with one additional unit of capital (labor), holding constant the quantities of labor (capital).

– Fix the quantity of labor at N*, then the MPK at K* is the slope of the production function at point A.

A

Output, Y

Capital Input, K

F(K , N*)

Slope = MPK

K*

.

),(),( d

K

d

K NKzFKNKF

zMP




– Algebraically, we assume that,

FK(K , Nd) > 0 and FNd(K , Nd) > 0.

– Conceptually, this simply means: more inputs yield more output.

• Diminishing Marginal Product– The declining MPK and MPN is

equivalent to the concavity of the production function.

– Algebraically, this means

FKK(K , Nd) < 0,

and FNdNd(K , Nd) < 0.

– Implicitly, we assume that F(. , .) is twice

differentiable.

Marg

inal

Pro

du

ct o

f L

ab

or,

MP

NLabor Input, Nd

MPN




• MPN as K – Algebraically, this

means

– Increase in the quantity of machinery and equipment enhances the productivity of the workers.

• F(. , .) is quasiconcave.M

arg

inal

Pro

du

ct o

f L

ab

or,

MP

NLabor Input, Nd

MPN1

MPN2

0),(2

d

d

NKNKF



Cobb-Douglas Production Function

• Probably the most commonly used form of production function which satisfies all the above properties

Y = zKa(Nd)b

where 0 < a, b < 1.• a + b = 1 Constant return to scale.

a + b > (<) 1 Increasing (decreasing) return to scale.

• If there are profit-maximizing price-taking firms and a + b = 1, then a will be the share that capital receives of national income.



Changes in Total Factor Productivity (z)

• Changes in z is critical to our understanding of the causes of economic growth and business cycles (real business cycles theory).

• Effects of z :1) Output for given values of K and Nd.

2) MPN for given value of K.• Factors that would affect z:

– Technological innovation– Weather– Government regulations– Price of energy

Labour Input, N

Output, Y

Z1F(K* ,

Nd)

Z2F(K* ,

Nd)

Marginal Product of Labor, MPN

Labor Input, Nd

MPN1

MPN2



Profit Maximization Problem• Assume that capital K is fixed. Then the firm’s problem

is to choose a quantity of Nd in order to maximize its profits.

• The representative firm is assumed to behave competitively, i.e. taking the real wage w as given.

• The problem can be stated as (choosing Nd)max = zF(K , Nd) – wNd

• Similar to the consumer’s problem, we assumeFNd(K , 0) = ∞ and FNd(K , ∞) = 0

to ensure interior solution in the firm’s profit maximization problem.



Profit Maximization Problem• The optimal condition (FOC) is

z[∂F(K , Nd)/∂Nd] = w• This states that it is optimal

for the firm to hire workers up to a level in which the MPN equals the real wage.

• Graphically, the optimal quantity of labor N* is at A, where the slope of total revenue function is equal to the slope of the total variable cost function.

• The maximized profits * is given by the distance AB.

A

Revenue, Variable Costs

Labor Input, Nd

zF(K , Nd)

N*

.B..

.

wNd



Profit Maximization Problem• The FOC of the profit-

maximization can also be interpreted as the firm’s demand curve for labor, for given values of z and K .

• The optimal condition (FOC) is

MPN(K , N) = w.

• Diminishing MPN implies w and N are inversely related.

Real Wage, w

Quantity of Labor Demanded, Nd

MPN or Labor Demand Curve



Comparative Statics• Recall the FOC of the firm’s problem

zFNd(K , Nd) = w

• Totally differentiate this gives

zFNdNddNd – dw + FNddz + zFKNddK = 0.

Thus, we obtain

0

,0,01

dd

d

dd

d

dd

NN

KNd

NN

Nd

NN

d

zF

zF

dkdN

zF

F

dzdN　

zFdwdN



Quasiconcavity• A function f(x) is quasiconcave if

f(x1) f(x2) f[x1 + (1 – )x2] f(x2)for any 1 0.

• f is strictly quasiconcave if

f(x1) f(x2) f[x1 + (1 – )x2] > f(x2)for any 1 > > 0.

• Consider a strictly quasiconcave utility function U(C , l).

Suppose x1 = (C1 , l1), x2 = (C2 , l2), then

U(x1) = U(x2) U[x1 + (1 – )x2] > U(x1) = U(x2)for any 1 > > 0.Thus the indifference curves are strictly convex.



Quasiconcavity• Strict quasiconcavity also implies that the bordered

Hessian matrix of the utility function is negative definite, i.e.,

–Ull + 2wUcl – w2Ucc > 0

01

10

lllc

clcc

UUw

UU

w

Documents

2014/6/1 ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip1 Consumer and Firm Behavior: The Work-Leisure Decision and Profit Maximization