Finite math, K201, calculus, and statistics explained

1.4b Budget constraint problem with fixed prices for 2 items

You have a budget of 500 dollars for books ($x$) and CDs ($y$). The total spent on books is $55x$ and the total spent on CDs is $15y$. Write an equation in the form of $y = mx+b$ then solve for $y$.
Linear equation representing total budget

Supply and demand problems

Supply and demand both relate quantity $q$ of an item to the price $p$.

Supply curve: Increasing function. $q$ increases as $p$ increases

  • The higher the price that sellers can charge for an item, the more quantity they will produce.
  • If sellers can NOT sell an item for more money than it costs to manufacture, then they would choose not to manufacture it.

Supply curve example

Demand Curve: Decreasing function. $q$ decreases as $p$ increases

  • The more an item costs, the less buyers will want to buy it (less quantity will be demanded).
  • If price is too high, NO items will be sold.

Demand curve example

Equilibrium point: It is assumed that the market settles to an equilibrium point when $S(p) = D(p)$.

  • The supply and demand curves intersect.
  • Corresponds to a specific price and quantity.

Equilibrium point where supply equals demand


EXAMPLE

Find the equilibrium $p$ and $q$ for the following supply and demand functions:
$S(p) = 10p - 5000$
$D(p) = 2500 - 20p$

  1. Set $S(p)$ equal to $D(p)$.

    $10p - 500 = 2500 - 20p$
  2. Solve for $p$.

    $30p = 3000$
    $p = 100$
  3. To solve for $q$, notice the $S(p)$ or the supply at any given price is actually $q$ or the quantity.

    $q = S(p) = 10p - 500$
  4. Plug in the value that you got for $p$ into that equation.

    $q = 10*100 - 500 q= 500$

Taxes

  • Two kinds:
    • Tax on consumers: sales tax, affects demand curve.
    • Tax on producers: affects supply curve.
  • How taxes affect supply and demand curves:
    • For a tax of $x$ dollars, supply curve is $S(p-x)$.
    • For a tax of $x$ dollars, demand curve is $D(p+x)$.
    • For a tax of $x\%$, supply curve is $S(p-px)$.
    • For a tax of $x\%$, demand curve is $D(p+px)$.
  • Taxes will affect equilibrium.

How do taxes affect suppliers and producers?

The price of an item was 100 dollars and a tax of 6 dollars is imposed. The price is now 102 dollars. How much tax is paid by the consumer? How much is paid by the producers?