• Chapter 11 Profit Maximization
  • Marginal revenue
  • Uncompensated Factor Demand
    • Short-Run supply by a price-taking firm:
    • Graph
    • Long Run supply by a price taking firm
• Chapter 12 Partial Equilibrium
  • Consumers’ decisions
  • Firms’ decisions
  • Equilibrium price and quantity
    • Short Run market reaction
    • Long run (firms can enter and exit)
      • Perfect competitive market:

Chapter 11 Profit Maximization

choose an appropriate level of output.

• Total revenue $R(q)=p(q)·q$, when $R$ depends on $q$
  (lower the price to sell a larger amount of products)

• total cost of producing $q$ is $C(q)$ profits

• $\pi= R(q) - C(q)$

1. Output choice $~max ~ R(q)-C(q)$, Take derivatives $\frac{d\pi}{dq}=0$

2. Profits are max when $MR=\frac{dR}{dq} = \frac{dC}{dq} = MC$

3. Second derivatives: $\frac{d^2 \pi}{dq^2}<0$, concave, mountain shape at maximum.

Marginal revenue

$MR(q) = \frac{dR(q)}{dq} = \frac{d[q~p(q)]}{dq} = p(q)+q\frac{dp}{dq}$

Example:
Demand function: $q=100-10p$

1. Solve for the Price $p=10 - \frac{q}{10}$

2. Given total cost = $5q$

3. Maximize profit Max $\pi (q)= p*q -TC = (10 - \frac{q}{10})q - 5q$

4. f.o.c: $10-\frac{q}{5}-5=0$

5. $q^* = 25, p^* =7.5, \pi = 62.5$

Uncompensated Factor Demand

When price does not depend on the quantity.

By maximizing profit,
Max $\pi (q)= pq-$ total cost = Max $p*f(k,l)-(vk+wl)$

• take f.o.c to $0:$
  $p*f_l(k,l)-w=0, ~l^* = l(p,v,w)$
  $p*f_k(k,l)-v=0, ~ k^* = l(p,v,w)$

Short-Run supply by a price-taking firm:

(firm takes price as given)
labor is easier to be changed than capital.

Make decision:

1. Positive quantity ($q>0$)
   MR=Short MC, p=Short MC

2. Shut down production at $q^{* }$
   p<SAVC
   price is not lager enough to cover...

Example:

$\pi = pq-C(q)$

$= q (p - \frac{C(q)}{q})$

$= q(p-SAC_q)$

$= q(p-(SAVC_q +SAFC_q))$

$= q*(p-(SAVC_q) - q*SAFC_q$

$= q(p-(SAVC_q) - q\frac{SFC_q}{q}$

$= q(p-(SAVC_q) - SFC$

when choose to Shut down the production, meaning that
$\pi_s = -SFC > q*(p-(SAVC_q) - SFC = \pi_0$

→ $0> q*(p-(SAVC_q))$
→ $p < SAVC_q$

Graph

profits = $pq-SC(q) = q(p-SAC)$

If price rises to $q^{** }$
If price falls to $q^{*** }$, $p=MC$, MC upper-forwarding slope
If price falls to $q^{**** }$, shut down, p < $SAVC_q$

Long Run supply by a price taking firm

Make decisions:

1. produce positive $q$
   $MR=MC$, $p=MC$

2. exit the market at $q^{* }$.
   $\pi < 0$ → $p< AC$
   price is not high enough to cover the average costs.

Chapter 12 Partial Equilibrium

Consumers’ decisions

market demand function

• Only two goods: $x,y$
  Marshallian demand for x is $~x^* (p, p_y, I)$

• Market demand is the sum of all individuals marshallian demand.
  Market demand $Q_D (p, p_y, I) = \Sigma_{i=1} ^n x_i (p, p_y, I)$

Change in $p_x$ result in movement along the curve, quantity demanded.

Change in $p_y$ or $I$, results demand curve for x shift.

Firms’ decisions

market supply function

• Market supply is the sum of all individuals marshallian demand.
  Market demand $Q_S (p,v,w) = \Sigma_{i=1} ^n q_i (p,w,v)$

• • Short run: market supply is the sum of the quantity supplied by existing Firms. (cannot enter ot exist the industry in the short run)

  • Long run un market supply s the sum of the quantity supplied by existing and entering firms.

Equilibrium price and quantity

• Equilibrium price, quantity demanded = quantity supplied.
  Every one is on their optimal options.

• At $p^{* }, Q_S (p,v,w) = Q_D (p, p_y, I)$

Short Run market reaction

Many buyers experience an increase in the demands, the market demand curve will shift to the right.

Long run (firms can enter and exit)

1. (Assume $\pi >0$, positive profit)

• new market will enter.
  • short run, market supply curve shifts outward
  • price goes down, profit decreases.

2. (Assume $\pi < 0$, negative profit)

• existing firms will leave market,
  • short run: market supply curve shifts inward
  • price goes up, profit increase.

Perfect competitive market:

(market is in long-run Equilibrium.)

• all firm maximize profit;
• no firm has intent to enter nor exit.

$p=MC=MR=AC$, operate at minimum $AC$
ZERO profits $= q(p-AC)$
(for different cost functions, marginal firms are make zero profits)

• LRE Constant cost case:
  • entry of new firms has no effect on the cost of inputs;
  • firms enter or leave an industry, the firm’s cost curves will remain unchanged.