# Chapter 4: Consumer and Firm Behaviour - The Work-Leisure Decision and Profit Maximization

### The Representative Consumer

• Considers the behaviour of a single consumer who represents the whole economy.
• Preferences - Consumption Good and Leisure.

#### Utility Function

﻿$U(C,l)$﻿ where ﻿$U$﻿ is utility function, ﻿$C$﻿ is quantity of consumption, and ﻿$l$﻿ is quantity of leisure, as a particular consumption bundle.

• ﻿$U(C_{1},l_{1}) > U(C_{2},l_{2})$﻿ then bundle 1 is strictly preferred.
• ﻿$U(C_{1},l_{1}) = U(C_{2},l_{2})$﻿ then consumer is indifferent.

#### Three Assumptions

1. More is preferred to less.
2. The consumer likes diversity in the consumption bundle.
3. Consumption and leisure are normal goods. (purchased more if income is higher.)

#### Indifference Map

A family of indifference curves with points representing consumption bundles where the consumer is indifferent.

• Indifference curves slope downward and are convex.
• Marginal Rate of Substitution – The rate at which the consumer is willing to substitute leisure for consumption goods (or vice versa).
• ﻿$MRS_{l,C}$﻿ = -[The slope of the indifference curve passing through ﻿$(C, I)]$﻿

### Budget Constraint

• Competitive behaviour dictates that the consumer is a price-taker and treats market prices as being given.

Barter Economy – All trade involves exchange of goods for goods. (Just Consumption Goods and Leisure Time)

#### Time Constraint

﻿$(h) = l + N^{s}$﻿

• ﻿$h$﻿ - Time constraint
• ﻿$l$﻿ - Leisure time
• ﻿$N^{s}$﻿ - Time working, also shown as ﻿$(h-l)$﻿

#### Real Disposable Income

Wage income plus dividend income minus taxes.

• w – Price of labour time sold by the consumer, or real wage.
• The consumption good plays the role of numeraire in the economy.
• π (Pi) - Quantity of dividend income.
• T – Lump-sum tax paid by the consumer to the government. (Is not affected by consumer price).

#### The Budget Constraint

﻿$C=wN^{s}+\pi-T$﻿

• Constraint = (wage x hours worked) + dividends - taxes
• Rearranged as ﻿$C+wl=wh+\pi-T$﻿
• Right hand side is implicit disposable income and left hand side is implicit expenditure on consumption and leisure.
• In slope-intercept form - ﻿$C=-wl+wh+\pi-T$﻿
• -w is slope of the budget constraint, ﻿$wh+\pi-T$﻿ is vertical intercept (maximum quantity of consumption attainable)
• To get horizontal intercept, solve for leisure time - ﻿$h +\left(\pi – T \right)/w$﻿

If ﻿$T<\pi$﻿ , then there will be a kink, where the consumer cannot consumer more than ﻿$h$﻿ hours of leisure. This is because there is additional dividend income that could be thrown away at any time.

• Examples will always deal with this case. (graph above

### Consumer Optimization

• Assumes the consumer is rational in preferences and budget constraint
• Assumes that ﻿$T>\pi$﻿, and that the point where the Indifference Curve is tangent to the budget constraint is the optional consumption bundle. (Point ﻿$H$﻿)

• When consumer optimizes, ﻿$MRS_{l,C} = w$﻿
• Marginal rate of substitution of leisure is equal to the real wage.
• Relative Price of a good ﻿$x$﻿ in terms of a good y is the number of units of y that trade for a unit of ﻿$x$﻿.
• In competitive markets, consumer sets the ﻿$MRS$﻿ equal to the relative price of one good in terms of another.
• Points ﻿$A$﻿ and ﻿$B$﻿ are not optimal because a consumer prefers a mix of consumption and leisure.

#### Change in Real Dividends or Taxes

• Changing ﻿$\pi-T$﻿ holds ﻿$w$﻿ constant.
• Pure Income Effect
• An increase in ﻿$\pi$﻿ could be caused by increase in firm productivity, which increases dividends.
• A decrease in ﻿$T$﻿, a tax cut causes an increase in disposable income.
• When ﻿$\pi>T$﻿, an increase in ﻿$\pi-T$﻿ is considered.
• The budget line is shifted out, with a higher vertical intercept and parallel to the original ﻿$BL$﻿.
• Increase in income causes an increase in consumption less than the difference between the vertical intercepts because non-wage income increases, but wage income falls since the consumer works less.
• Reduction in income does not completely offset the increase in non-wage income because consumption is a normal good.

#### Change in Real Wage - Substitution and Income Effects

• Holds ﻿$\pi$﻿ and ﻿$T$﻿ constant to remove the pure income effect.
• An increase in real wage ﻿$w$﻿ causes the budget constraint to shift out, but the new constraint is steeper, and the kink remains fixed.

Substitution Effect – Increase in real wage causes a new budget constraint ﻿$(JKD)$﻿, where the vertical intercept is higher. (Moves from ﻿$F$﻿ to ﻿$O$﻿ on graph)

• Consumption increases and leisure decreases because of a real wage increase.
• Since wage is higher, consumer works more to consumer more.

Income Effect – Real wage stays the same as budget constraint shifts out ﻿$(JKD\; to\; EBD)$﻿ and non-wage income increases. (Moves from ﻿$O$﻿ to ﻿$H$﻿ on graph)

• Both goods are normal, so consumption increases and leisure increases.
• Since wage is higher, consumer consumes more and takes more time off.

An increase in real wage could lead to an increase or a decrease in labour supply ﻿$N^{S}$﻿.

Labour Supply Curve – Shows how much labour the consumer wants to supply for each possible value for the real wage.

• ﻿$N^{S}(w) = h – l(w)$﻿
• Where ﻿$h$﻿ = total hours available, and ﻿$l(w)$﻿ is a function that shows the amount of leisure desired in terms of wage ﻿$w$﻿.
• Assumes that the substitution effect is larger than the income effect, meaning that the supply curve is upward sloping.

### Perfect Complements

• Representative consumers’ goods are desired to be consumed in fixed proportions, so ﻿$C/l$﻿ is always equal to some constant “﻿$a$﻿”.
• ﻿$C=al$﻿, where ﻿$a>0$﻿ is a constant, where the optimal consumption bundle will lie. (The quantities of consumption and leisure must satisfy the budget constraint and this equation)
• Solving these reveals -
• ﻿$L = (wh + \pi – T) \div (a + w)$﻿
• ﻿$C = a(wh + \pi – T) \div (a+w)$﻿
• Leisure and consumption increase with non-wage disposable income ﻿$(\pi – T)$﻿ and they both increase with real wage ﻿$w$﻿ increases.
• If ﻿$a$﻿ increases, the consumer prefers more consumption relative to leisure, so at optimum, there is more ﻿$C$﻿ and less ﻿$l$﻿ .
• There are no substitution effects.

### The Representative Firm

• Firms demand labour and supply consumption goods, shown by a single representative firm.
• Assumes that firms own productive capital and hire labour to produce consumption goods.
• Production technology available is shown as - ﻿$Y = zF(K, N^{d})$﻿
• ﻿$z$﻿ is total factor productivity (increasing will make ﻿$K$﻿ and ﻿$N^{d}$﻿ more productive, causing higher output.
• ﻿$Y$﻿ is output of consumption goods.
• ﻿$K$﻿ is quantity of capital input in the production process.
• ﻿$N^{d}$﻿ is the quantity of labour input measured in total hours worked by employees.
• ﻿$F$﻿ is a function of ﻿$K$﻿ and ﻿$N^{d}$﻿
• In ﻿$SR$﻿ , firms cannot vary quantity of plant and equipment ﻿$K$﻿ but are flexible in hiring and laying off workers ﻿$N^{d}$﻿

#### Marginal Product

Additional output that can be produced with one additional unit of that factor input, holding constant the quantities of other factor inputs.

• Slope of the production function is the ﻿$MP$﻿.
• Varying the quantity of labour (fixed capital) causes the Marginal Product of Labour ﻿$MP_{N}$﻿ to be the slope.
• ﻿$MP_{N}$﻿ declines with quantity of labour.
• Varying the quantity of capital (fixed labour) causes the Marginal Product of Capital ﻿$MP_{K}$﻿ to be the slope.
• ﻿$MPK$﻿ declines with quantity of capital.

### Production Functions have 5 Key Properties

#### 1 Constant Returns to Scale

• At any constant ﻿$x > 0, xzF(K,N^{d}) = zF(xK,xN^{d}).$﻿
• If all factor inputs are changed by a factor “﻿$x$﻿” then output changes by the same factor. (i.e. ﻿$x=2$﻿, output doubles as well)
• Alternatives
• Increasing Returns to Scale – Larger firms are more efficient than small firms.
• Decreasing Returns to Scale – Smaller firms are just as efficient as large firms.

#### 2 Output increases when either capital input or labour input increases

• Marginal products of labour and capital are both positive ﻿$MP_{N}>0$﻿ and ﻿$MP_{K}>0$﻿
• More inputs yield more output.

#### 3 The Marginal Product of Labour decreases as Quantity of Labour increases

• The more labour input added, the less marginal product can be obtained.
• Slope (concavity) decreases in a production function.

#### 4 The Marginal Product of Capital decreases as Quantity of Capital increases

• The more capital input added, the less marginal product can be obtained.

#### 5 The Marginal Product of Labour increases as the Quantity of Capital Input Increases

• Adding more capital increases the Marginal Product of Labour for each quantity of labour – if there is a worker to utilize the capital, then MPN increases.

### The Effect of a Change in Total Factor Productivity on the Production Function

• An increase in total factor productivity ﻿$z$﻿ has 2 effects.
• There is an upward shift in the production function as more output can be produced given capital and labour inputs
• Marginal product of labour shifts to the right, as it increases when ﻿$z$﻿ increases.
• Caused by -
• Anything that permits more output to be produced for given inputs, such as technological innovation (i.e. assembly line) increases ﻿$z$﻿.
• Good weather (for agricultural and construction) increases ﻿$z.$﻿
• Government regulations may decrease ﻿$z$﻿
• Increase in relative price of energy decreases ﻿$z$﻿

### Profit Maximization Problem

• Representative firm behaves competitively – It takes the real wage as given. (Price at which labour trades for consumption goods)
• It wants to maximize its profits, given ﻿$Y – wN^{d}$﻿
• ﻿$Y$﻿ – Total revenue received from selling output, ﻿$wN^{d}$﻿ – total real cost of the labour input (total real variable costs).
• Subbing in the production function ﻿$Y = zF(K,N^{d})$﻿, the firm must choose ﻿$N^{d}$﻿ to maximize ﻿$\pi = zF(K,N^{d}) - wN^{d}$﻿
• ﻿$zF(K,Nd)$﻿ – Firm’s revenue,
• ﻿$wNd$﻿ – Variable cost
• ﻿$K$﻿ is fixed, and ﻿$\pi$﻿ is real profit.
• To maximize profits, the firm will choose ﻿$Nd = N^{\ast}$﻿ to get maximized profits ﻿$\pi^{\ast}$﻿ (which is distance ﻿$AB$﻿ in this graph)
• Located at Marginal Revenue = Marginal Cost, or ﻿$MPN=w$﻿.

A firm’s marginal product of labour schedule is its demand curve for labour.