ECO101 Lecture 08

ECO101 - Week 08 Household Supply.pdf

Theory of Labour Supply

$f = A + w h$
$h + z = 24$
Goals:
- Price up, Quantity down
- Wage up, H up
- R up, S up
List the conditions under which decrease in price will mean an increase in quantity demanded
- Normal good, or an inferior good, with dominant substitution over income effect
List the conditions under which, an increase in wage, results in an increase in hours worked
- When the subtitution effect dominates the income effect
Higher interest rates will make you save more
- When the subtitution effect dominates the income effect
How to find the budget line?
- $w = 1$
- $A = 16$
- $f = 16 + h$
- $f - 16 = h$
- $f - 16 + z = 24$
- $f = 40 - z$
- All budget lines say the same thing, expenditure cannot exceed income
- $f$ is amount of food purchased per day
- So $f ≱ w h + A$
  - $A$ is non labour income
  - Asset income
- Constraint curve is $h + Z = 24$
  - $Z$ is leisure
  - $24$ hours in a day
  - Sleeping, monkeying around
- $h = 24 - Z$
- $f = 24 w + A - w Z$
- slope is $w$
- OCOST of leisure is $w$
- $O C O S T_{Z}$
- Every hour in bed is sacrificing an hour of wages
- Vertical intercept is max purchasing power
- $Z = 0$
- no leisure
- $w = 1$
- $A = 16$
- Max purchasing power for leisure
- If you sleep for 40 hours a day?
- So we have a constraint curve of $z \leq 24$
- $M R S = \frac{M U_{x}}{M U_{y}}$
- $M R S = \frac{M U_{z}}{M U_{f}}$
- $M R S = \frac{f}{z}$
- Tangency condition
  - $M R S = O C O S T_{Z}$
  - $\frac{f}{z} = \frac{1}{1}$
  - $f = z$
- $2 Z = 40$
  - #tk where did this come from?
- So then $Z = 20$
- So hours worked are $4$ .
- 4 hour work week ahh
- $A^{'} = 24$ ?
- $f = 24 (1) + 24 - Z$
- $f = 48 - Z$
- Tangency Condition
  - $f = z$
- $z = 24$ so then $h = 0$
- If you get $24$ dollars a day, you work nothing in this case.
- This shows that when asset income is increased, then hours worked go down.
  - Eventually, you retire?
Why are indifference curves convex?
- Diminishing marginal benefit / marginal utility
- When you've got lots of marginal utility, the benefit isn't that much.
- When you've got no marginal utility, the benefit is very large
- Sleep is very utile when you have none.
- #tk - When your non-labour income goes up

Comparing Employment Alternatives

Rent a van
20 cents per km, daily fee is 20
$\begin{matrix} \frac{Cents}{k m} & Daily Fee \\ 0.2 & 20 \\ 0.1 & 40 \end{matrix}$
Choose that alternative that gives you the highest level of utility
$f = 24 w + A - w z$
Tangency condition is:
- $f = w z$
- Job 1:
  - $f = 4 z$
  - $f = 96 + 0 - 4 z$
  - $f + 4 z = 96$
  - $z = \frac{96}{8} = 12$
  - $f = 96 + 0 - 4 (12) = 48$
- Job 2:
  - a dollar an hour, but 16 dollar a day flat rate
  - $w_{2} = 1$
  - $A_{2} = 16$
  - Utility curve
  - $u = 20$

Income and Substitution Effects of a Wage Change

- #tk - label c1,c2,c3
$w_{1} = 2.25$
$A_{1} = 36$
$w_{2} = 4$
When wages go up, $S E (a \to b)$ and $O C O S T_{Z}$ goes up, then leisure goes down, so hours worked goes up
The Substitution Effect
- When wage goes up, you should work more.
The Income Effect
- $I E (b \to c)$ means Real Income goes up
- Income goes up
- Depends on if the good is normal or inferior
- If it's an inferior good, leisure will go down, more worked hours.
  - Substitution Effect is reinforced by Income Effect
  - Overall hours worked goes up.
- If it's a normal good, leisure will go up, less worked hours.
  - When Substitution Effect dominates the Income Effect
  - Hours worked will go up
  - But here the Income Effect dominates, so the person will work less when they're offered a wage increase
  - Not a giffen good
  - Overall the person will work more in response to a wage increase

Derive The Labour Supply Curve

Demand treats $M, p_{y}, p_{x}$
Maximize utility, treating the $A, w$ as variables
$f = w z$
$\frac{f}{z} = O C O S T$
$\frac{f}{z} = w$
$f = 24 w + A - w z$
$w z = 24 w + A - w z$
$0 = 24 w + A - 2 w z$
$2 w z = 24 w + A$
$z = 12 + \frac{A}{2 w}$
- Workers demand for leisure
$h = 24 - z$
$h = 24 - (12 + \frac{A}{2 w})$
$h = 12 - \frac{A}{2 w}$
$A = 36$
So sub in
$h = 12 - \frac{18}{w}$
Reservation Wage is when $h = 0$
$12 = \frac{18}{w}$
$w = 1.50$
If asset income goes up, then labour supply curve shifts up. Because people need to be paid more because their assets are paying more.
- A up
- Rw up
Explain why the labour supply curve might bend backwards

Households Are Savers in Capital Markets

$m_{1} = 30$
$m_{2} = 30$
$r = 50 %$
Why
- Rates go up savings go up?
Another model is $(0 \to 65)$ then $(65 +)$
Suppose you have two periods
- period 1 $(0 \to 21)$ : $f_{1}$ on food
  - No income
- period 2 $(21 +)$ : $f_{2}$ on food
  - Higher income
- $r = 50 %$
- This is to smooth out your spending
- $B L : f_{2} \leq m_{2} + (1.5) [\underset{savings}{\underset{⏟}{m_{1} - f_{1}}}]$
- Spending will not exceed income
- Rearrange
- $\underset{Future Value}{\underset{⏟}{F V}} : f_{2} + (1.5) f_{1} = m_{2} + 1.5 m_{1} = 30 + 1.5 (30) = 75$
- Vertical Intercept
  - Future value of lifetime earnings is 75 dollars
- Horizontal Intercept
  - $\underset{Present Value}{\underset{⏟}{P V}} : f_{1} + \frac{f_{2}}{1.5} = m_{1} + \frac{m_{2}}{1.5} = 30 + \frac{30}{1.5} = 50$
  - If you save everything in period 2
  - We'll give you a bank loan of 20 dollars
  - Future income is 30
  - If they loan you max lending, 20, you will need to pay back 30 after the period of loan.
- Slope is OCOST=1.5
  - Every dollar spent today, sacrifices a dollar 50 tomorrow.
- To draw the savings graph
  - Shift it towards the savings side
Why Save?
- Why you should save or borrow, $r =$
  - Interest Rates
  - Example: $2 %$ interest, shit savings
  - If the interest rate on borrowing is high, no one would borrow
- $M R S$
  - Marginal rate of substitution is probably very low for us $M R S = 0.0001$
  - Are you willing to sacrifice today for tomorrow
- 3rd
  - Small current income, anticipate larger future income, then borrow now to create the future income
Lifetime savings and borrowing
- $25 %$
  - You need to save $25 %$ of your income
  - If you save that 25, 25 percent tax
  - So you have 50 percent left
  - These numbers work, if you work from 25-65
  - Spend half income today, save a quarter, so then spending remains half of income in second period of life too.
- Now we have studies that half of these people will be centennials
  - If deadly diseases at around 70 within 50 years are cured.
  - Then you need to save more from working to finance 40 years of spending when retired
  - 40 years of work, save then 40 years of spending.
  - So 50 percent of stuff saved
If rates are up
- Saver
  - Substitution effect: OCOST of spending today goes up, so current spending goes down. Saving more
  - Income effect: Wealth goes up
    - When interest rates go up, rich get richer compounded
    - Normal good
      - Spend more
      - Saving less
    - Inferior good
      - Spend less
      - Saving more
  - So IE may be the same or opposite as SE
  - TE
    - C1
      - Inferior good, SE reinforces IE, spend less save more
    - C2
      - Normal good, SE dominates IE, save more
    - C3
      - Normal good, IE dominates SE, interest rates goes up, spend more, save less
- Borrower
  - Rates go up
  - SE OCOST Spending down, Borrow less
  - IE Wealth down,
    - Normal Good, spend less, borrow less
    - Inferior Good, spend more, borrow more
  - TE
    - C1
      - Normal good, SE reinforces IE, borrow less
    - C2
      - Inferior good, SE dominates IE, borrow less
    - C3
      - Inferior good, IE dominates SE, borrow more
      - Example:
        
        Guy is loaded
        
        Credit cards up the wazoo
        
        Bro's crazy about borrowing
        
        Interest rates go up, buy more, because everything you buy sucks
        
        #tk

Studying

$u = \sqrt{g} \sqrt{z}$
$w = 4$
- Increase grade by $4$ percent per hour of study
$g = A + w s$
$s$ is hours studied
$A = 20$
- Non labour grade
- If a course was all multiple choice and there were 5 multiple choice per question
- Random chance would get you 20 percent
$s = 10$ 10 hours of studying
$g (10) = 20 + 4 (10) = 60$
$g (15) = 20 + 4 (15) = 80$
Consider that MGT101 was $A = 25$ and $w = 4$ , you would study more in ECO101
If leisure is a normal good
- Then having more non-labour income will make you study less
If leisure is an inferior good
- Having more non-labour income will make you study more
MAT999
- $A = 20$
- $w = 0.0000004$
- Study more or less?
- Depends on if leisure is a normal good
- SE
  - lower OCOST, means more leisure, less studying
- IE
  - whether leisure is a normal good or inferior
- See which effect dominates which
Diminishing $M U_{g}$ ?
- Number of hours you spend studying is a utility maximizing choice
- This might be a case where we don't have diminishing marginal utility for grades
- If you got 1/15, one point will get less marginal utility rather than 14/15 and say 'why'd I miss that point'?
- So there's increasing marginal utility of grades