ECO101 Lecture 11

ECO101 - Week 11 Competition.pdf

Demand Shocks

• $q=\sqrt[4]{k}\sqrt[4]{l}$
• $w=0.5$
• $r=0.5$
• $F=4$
• $D=160-20p$
• Perfect competition is an industry structure
  • This assumes you have: many firms and identical products and perfect information
    • Perfect Info means the firms know your indifference curve between their products and another
  • Meaning every firm in the industry has the same production function
  • Free entry and exit
    • Anyone can open a business in this kind of industry
• Tangency condition is $MRTS=OCOST$
  • $\frac{k}{l}=\frac{w}{r}$
  • $k=l$
• Sufficiency condition
  • $kl=q^4$
  • $l^2=q^4$
  • $l=q^2$
    $k=q^2$
  • $C=\frac{1}{2}{q}^2+\frac{1}{2}{q}^2⟹q^2+4$
  • $AC=q+\frac{4}{q}$
  • $MC=2q$
• Since we assume everyone in the industry has a fixed costs, it's $F=4$ in this case.
• 5 steps to solve:
  • 1
    • Find MES
    • This is when $AC$ is the lowest meaning $MC=AC$
    • $q+\frac{4}{q}=2q$
    • $q^2+4=2q^2$
    • $q=-2,2$
    • So $q_0=2$
  • 2
    • Find the price of industry
    • $p=MC$
    • $p=2(q_0)$
    • $p_0=4$
  • 3
    • $Q_0=160-20p_0=80$
  • 4
    • $n_0=40$
    • $\frac{Q}{q}=n$
  • 5
    • Supply curve
    • $p=MC$
    • $p=2q$
    • $\underset{\text{The firm’s supply curve}}{\underbrace{q^s}}=\frac{1}{2}p$
    • Industry is $n\cdot q$, as we take every firm's supply, multiply it by $n_0$ the amount of players and get $20p$
  • At the initial Equilibrium, profits are $0$, because $p=AC$
  • $\pi =(p-AC)q=0$
  • If $p>AC$ then $\pi >0$
    • Causing new firms to enter the industry
    • When they enter the industry, the supply curve shifts to the right.
    • Price would fall until $\pi =0$ (economic profits).
  • If $p<AC$ and $\pi <0$
    • Causing firms to exit the industry
    • Cause the supply to shift left
    • $p++⟹\pi ++$ then we get to $\pi =0$
  • Theory, how the number of firms $n^{\ast }$ adjust to changes in demand and supply. To create $D=S$
  • Theory:
    • On the pic:
    • $D=320-20p$
    • $A$ is our initial
    • $B$ is our immediate when $n_1=n_0$
    • $C$ is the ultimate
  • Existing firms respond based on their supply curve
  • $S_2$
  • Since $q^s=\frac{1}{2}p$ find $S_2$
  • If $p=4$ then quantity is $320-20(4)=240$
  • $n_2=120$
  • Theory
    • Supply curve we're working with
      • $MRTS=OCOST$
      • It has isoquant analysis built in as the $k=ql$ type function
    • Demand
      • We have budget lines and Utility Maximization
      • That gives us the new demand curve
• Decrease in demand
• Re-presenting the material in a table, they can see if we know how to do the above.
  • a:
    • $n=1000$
    • $p=$ #tk
    • Suppose $p=12$, then $Q=200000$
    • If $p=10⟹Q=300000⟹p=MC⟹10=500q$
    • $n=1000$
    • $Q=500000$
    • Price can't be $10$ dollars
    • $p=9$
    • $Q=350000$
    • $q=450$
    • $n=1000$
    • $Q=450000$
    • So price can't be $9$
    • $p=8$
      • This is when $q\cdot n=Q$
  • b:
    • How many firms will there be in the Equilibrium.
    • 1
      • Find MES
      • When $MC=ATC$
      • At $p=11$ row
    • 2
      • Find the price of industry
      • $p=MC$
      • $10=10$
      • Price is #tk
    • 3
      • $Q_0=160-20p_0=80$
    • 4
      • $n_0=40$
      • $\frac{Q}{q}=n$
      • $\frac{300000}{500}=600=n_1$
    • 5
      • Supply curve
      • $p=MC$

Supply Shock

• Per unit tax
• $D=160-20p$
• $t=2$
• $C_0=q^2=4$
• $M{C}_0=2q$
• $A{C}_0=q+\frac{4}{q}$
• Add a $2$ dollar tax, these taxes shift costs up by $2\mathrm{\$}$
• $C_1=q^2+4+tq$
• $C_1=q^2+2q+4$
• $A{C}_1=q+2+\frac{4}{q}$
• $M{C}_1=2q+2$
• It shifts average and marginal by $2$
• Solve for $S_1$
  • $p=MC$
  • $MC=2q+2=p$
  • $q^s=\frac{1}{2}p-1$
• $A{C}_1$ causes firms to lose money, to exit, to drive up the price.
• Incidence of the tax:
  • You snooze you lose
  • Whoever is most inelastic
  • $P_0=4$, $P_2=6$
  • $t=2$
  • The $2$ dollar tax makes the price go from $4\to 6$, so the incidence is $100\mathrm{\%}$
  • Existing firms have time to exit the industry, it can take a lot of time though.
  • If we give them time to exit and shutdown, while demand hasn't moved.
  • So this drives up the price and consumers pay all of the tax then.
• So tax shifts up the $MC$ and $AC$ by $t$ but also rotates $S$.
• Lump Sum Tax
  • $C_0=q^2+4$
  • $D=160-20p$
  • $T=5$
  • This shifts up $AC$ but not the $MC$, it's because it's a fixed cost
  • $C_1=q^2+4+T=q^2+9$
  • $M{C}_0=2q$
  • $M{C}_1=2q$
  • $A{C}_0=q+4q^{-1}$
  • $A{C}_1=q+9q^{-1}$
  • When fixed costs go up, MES rises. Because you need to be producing more to cover the $5\mathrm{\$}$ fixed costs.
  • If $MC$, then $n\cdot MC$ for each firm doesn't change either, so $S_1=S_0$
  • $P_1=P_0$
  • Explain why ${\pi }_1=-5$
  • If ${\pi }_0=5$, then the govt took away $5$ dollars from you with nothing changing, then clearly ${\pi }_1=-5$
  • Creates losses equal to the value of the tax.
• Per unit subsidy
  • $MC--⟹M{C}_2$
  • $AC--⟸A{C}_2$
  • This shifts down your costs by the subsidy
  • $C=q^2+4$
  • $D=160-20p$
  • $s=2$
  • The incidence is $100\mathrm{\%}$
  • Per unit subsidy shifts the marginal and average down, by the amount of the subsidy
• Lump Sum Subsidy
  • $C=q^2+4$
  • $D=160-20p$
  • $S=3$
  • A lump sum subsidy only affects the costs, but doesn't affect marginal, which doesn't affect quantity output.

Technological Process

• $C_{\text{old}}=q^2+16$
• $D=160-10p$
• Taxi Gasoline
• Improved technology will shift the supply curve to the right.
• 1
  • Find MES
  • $MC=AC$
  • $A{C}_{\text{old}}=q+16q^{-1}$
  • $A{C}_{\text{old}}=M{C}_{\text{old}}$
• 2
  • Find price
• 3
  • Demand
• 4
  • Number of firms
• 5
  • Supply curve
• Supply curve shifts right twice

ICI DCI and CCI

• ICI
  • Increasing cost industry
  • Some inputs are scarce, rivalrous
  • If the number of firms are increased, all $AC$ and $MC$ increase.
• DCI
  • Decreasing cost industry
  • Firms share the costs of some inputs
  • Firms in a city, share the cost of the roads
  • If the firms increase, each firms part of the pie to pay goes down.
  • $MC$ and $AC$ go down
• CCI
  • Constant cost industry
  • This is when you increase the number of firms, but cost curves are constant and unaffected.
• Examples:
  • Demand Shock:
    • $C=q^2-nq+16$
    • Decreasing because the function decreases too
    • DCI
      • $-n$ is in the cost curve
    • $D_0=24-2p$
    • $D_1=28-2p$
    • 1
      • Find MES
      • $MC=AC$
      • $2q-n=q-n+16q^{-1}$
      • $q=4$
    • 2, 3, 4
      • $p=MC$
      • $2q-n=8-n$
      • $q=4$
      • $D=24-2p$
      • $n=\frac{Q}{4}$
      • $Q=4n$
      • $4n=24-2(8-n)$
      • $4n=24-16+2n$
      • $n=4$
      • Find price
        • $p=8-n=4$
      • $D(4)=24-2(4)=16$
    • Solve for $S_0$
      • $p=MC$
        • $p=2q-n$
        • $p=2q-4$
        • $q=\frac{p}{2}+2$
        • $q^s=\frac{1}{2}p+2$
        • $q^s=2p+8$
      • $n_0=4$
    • $p_1$
      • $28-2p=2p+8$
      • $p=5$
    • $p_2$
      • $4n=28-16+2n$
      • $n=6$
      • $8-6=2$
      • $p_2=2$
    • Discuss and illustrate $t=2$ in DCI
      • Because ${\pi }_1<0$ and $n--$, in a DCI. $A{C}_1++A{C}_2$ and $M{C}_1++M{C}_2$
      • CCI
        • $p_0=4$
        • $t_0=2$
        • $p_2=6$
      • DCI
        • $p_2>6$
        • So tax incidence is $>100\mathrm{\%}$
        • A $t=2$ makes it go from $4\to 7$
        • So consumer incidence is also more than $100\mathrm{\%}$
      • $n_2=\frac{Q_2}{q_2}$
      • Denominator doesn't change, but numerator is smaller so $n_2--$
      • $P_2++$
      • $Q_2--$
      • $n_2--$
      • So it's a loss