ECO101 Lecture 10

ECO101 - Week 10 Long Run Firms.pdf

• Study: #tk
  • $u(x,y)$
  • $u(f,z)$
  • $u(f_1,f_2)$
  • Iso profit analysis
    • $f(\overline{k},l)$
  • Short run cost analysis
    • $sc(q)$
  • Long run
    • $f(k,l)$
    • $c(q)$

Isoquant Analysis

• ISO mean equal
• Isoquant
  • Combinations of labour and capital that give equal quantity
• $q=\sqrt{k}\sqrt{l}$
• $q=2\to 4$
  • $q=2$
    • $2=\sqrt{k}\sqrt{l}$
    • $4=kl$
• $MRTS=$
  • Marginal rate of technical substitution
  • $=\frac{M{P}_l}{M{P}_k}$
  • When we have a symmetric function of $\sqrt{k}\sqrt{l}$
  • Then we get $\frac{k}{l}$ is MRTS
  • $MRTS$ how much capital you're able to give up for another unit of production
  • Diminishing MRTS caused by diminishing MP
• Isocost map
  • $w=4$
  • $r=1$
  • $c=8$
  • $c=12$
  • $c=16$
• $OCOS{T}_L=4$ means firing $4$ units of capital to acquire one more unit of Labour.
• If $r$ goes up, so does $OCOS{T}_L$ because $\frac{w}{r}$ gets smaller.
• $w=4$ and $r=1$ and $q=4$
• $q=\sqrt{k}\sqrt{l}$
• Tangency and Sufficiency conditions
• $q=4$
• Cost minimization
• $4=\sqrt{k}\sqrt{l}$
• Lowest sufficient isocost line
• Tangency Condition
  • $MRTS=OCOS{T}_L$
  • Slope of isoquant
  • Slope of isocost
  • $\frac{k}{l}=\frac{4}{1}$
  • $k=4l$
• Sufficiency Condition
  • $k,l$ have to be enough to get $q=4$
  • $16=kl$
• Lowest Sufficiency Isocost Line
  • Sub $k$ from tangency condition
  • $4l^2=16$
  • $l^2=4$
  • $l=2$
  • $k=4(2)=8$
  • Cost
    • $c=16$

Cost Curve Analysis

• How does the output vary change capital and labour are changed
• Now we have constant returns to scale
  • $CRTS:q=\sqrt{k}\sqrt{l}$, double $k$ and $l$ will double $q$
• $DRTS:q=\sqrt[4]{x}\sqrt[4]{y}$
  • Doubling inputs gives you less than twice the inputs
• $IRTS:q=kl$
  • Doubling inputs means more than twice the outputs
• Costs per unit
  • $IRTS$
    • $AC=\frac{c}{q}$
  • $CRTS$
    • $2l,2k$
    • $⟹$
    • $2c$
    • $2q$
    • Constant line for $AC$
  • $DRTS$
    • Increase costs and quantity
    • But output goes up a little
    • $y=e^x$
  • $IRTS$
    • Average costs decline
    • $y=e^{-x}$
  • - Output levels less than $MES$, firm has $IRTS$ - Output levels higher than $MES$, firm has $DRTS$ - At the $MES$ the firm has $CRTS$ - When $MC=AC$ we have $CRTS$ - The shape of the cost curves comes from the production function

Changes in Scale (Long Run Vs Short Run)

• $q=\sqrt{k}\sqrt{l}$
• $q_0=2$
• $q_1=4$
• $w=1$
• $r=0.25$
• They want to increase production.
• In the short run, you can't change your capital
• All you can do is go to Equilibrium $B$
• Sufficiency condition
  • $kl=4$
• Tangency condition
  • $k=4l$
• A
  • $\frac{k}{l}=\frac{1}{0.25}$
  • $4k=l$
  • $4l^2=4$
  • $l=1$
  • $k=4$
  • $C=2$
• B
  • Stuck with $k=4$
  • Sufficiency condition
    • $4l=4^2$
    • $l=4$
    • $k=4$
  • Short run costs
  • $sc=1\cdot 4+\frac{1}{4}\cdot 4$
  • $sc=5$
• C
  • Long run
  • $kl=4^2$
  • $k=4l$
  • $4l^2=4^2$
  • $l=2$
  • $k=8$
  • Costs
  • $LC=wl+rk$
  • $LC=1(2)+\frac{1}{4}8=4$
• UTM:
  • $2004\sim 2000$ students
  • $2005\sim 8000$ students
  • Go from bundle A to bundle B
  • So hire a lot of labour
    • $SR⟹L++$
  • Still had only 4 rooms, had a lot of TAs
  • In the long run, build more buildings, etc, go up the isoquant and go to bundle C
  • Costs should go down in the long run, in the short run, they go higher.
    • More expensive to expand in the short run.
  • If you try to reduce your costs, it's more effective to do so in the long run than the short run.

Input Price Change ($SE$ and $OE$)

• 4 cases
• $q=\sqrt{k}\sqrt{l}$ and $p=8$
• $w=1,r_0=0.25$ and $r^{\prime }=1$
• #tk
• Case 1:
  • Theory:
    • What happens if the interest rate goes from $r_0\to r^{\prime }$
    • Substitution Effect
    • Income Effect
    • $r++$
    • $SE(a\to b)$
      • When $r++$, we get $OCOS{T}_L=\frac{w}{r}$
      • The opportunity of hiring a worker goes down.
      • If the opportunity cost of anything goes down, do more of it.
      • $l++$ and $k--$
    • $OE(b\to c)$
      • Output Effect
      • Produce less, so $l--$ and $k--$
    • $S{E}_K$ reinforces $O{E}_K$ so then $k--$
    • $S{E}_L$ dominated $O{E}_L$ then $l++$
    • $O{E}_L$ dominates $S{E}_L$ then $l--$
  • Bundle A
    • Sufficiency
      • $kl=q^4$
    • Tangency
      • $k=4l$
      • Then
      • $4l^2=q^4$
      • $l=\frac{1}{2}{q}^2$
      • $k=2q^2$
      • $c=wl+rk$
      • $c=q^2$
      • $AC=q$
      • $MC=2q$
      • If $p=8$ then $q_0=4$
  • Bundle B
    • $SE$
      • $q=q_0=4$
      • Same indifference curve, now we stay on the same isoquant
    • Sufficiency condition
      • $kl=4^4$
    • Tangency condition
      • $MRTS=OCOST$
      • $\frac{k}{l}=\frac{1}{1}$
      • $k=l$
    • Cost
      • $k^2=4^4$
      • $k=4^2=16$
      • $l=16$
      • $C=wl+rk$?
      • $C=32$
  • Bundle C
    • $OE$
    • Sufficiency
      • $kl=q^4$
    • Tangency
      • $k=l$
      • $l^2=q^4$
        $l=q^2$
        $k=q^2$
    • Cost
      • $c=2q^2$
      • $AC=2q$
      • $MC=4q$
      • When $P=MC$
      • $8=4q$
      • $q_1=2$
      • Produce $2$ units now
    • $L=4$
    • $K=4$
    • $C=8$
    • $\frac{w}{r}=\frac{1}{0.25}$
    • $\frac{w}{r}=4$
    • $\frac{w}{r}=1$
    • Two compare the costs:
      • because the denominators are different, you can't compare directly
      • On the bottom, you can see that $\frac{C_c}{w}<\frac{C_a}{w}$
      • This can tell us about costs.
    • If $w++$ or $w--$
      • $\frac{Ca}{w_0},\frac{C_c}{w_1}$ tells us nothing, we'd have to convert the fractions
      • Use $\frac{C_a}{r},\frac{C_c}{r}$
• Case 2:
  • Decrease in interest rates
  • $r--$
• Case 3:
  • $W++$
• Case 4:
  • $W--$

Output Price Change

• $q=\sqrt[4]{k}\sqrt[4]{l}$
• $r=4$
  $w=1$
  $p_0=16$
• $p_1=32$
• Tangency Condition
  • $MRTS=OCOST$
  • $\frac{k}{l}=\frac{1}{4}$
  • $4k=l$
• Sufficiency Condition
  • $kl=q^4$
• Subsitute
  • $4k^2=q^4$
  • $2k=q^2$
  • $k=\frac{1}{2}{q}^2$
  • $l=2q^2$
  • $C=wl+rk$
  • $C=4q^2$
  • $AC=4q$
  • $MC=8Q$
    • Twice as steep
• When $P=MC$
  • $16=8Q$
  • $2=q$
  • $k=\frac{1}{2}{q}^2=2$
  • $l=2q^2=8$
  • $C=4q^2=16$
• $P=32=MC$
  • $32=8q$
  • $q_1=4$
  • $k=8$
  • $l=2q^2$
  • $l=32$
  • $c=64$