# 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

- More is preferred to less.
- The consumer likes
*diversity*in the consumption bundle. - 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.

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