ECO101 Lecture 09

ECO101 - Week 09 Short Run Firms.pdf

Production Functions

• Short run, capital is fixed.
  • Capital is anything not labour.
• $L=2$
• $K=2$
• $Q=2^2{2}^2=16$
  • Maximum quantity produced
• TE
  • Technological efficiency
  • All capital is functioning properly
  • All labour is functioning properly
  • If we only have $1$ unit of labour, we need $4$ units of capital to get 16 units produced
• $L=10$
• $K=8$
• $Q=2KL-(0.2)L^2$
• $Q=2(80)-(0.2)8^2$
• $Q=2(80)-(0.2)64$
  • $Q=140$

Product Curves

• $T{P}_L=$ total production of labour
  • $f(\overline{k},l)$
  • Output produced by the firm
• $A{P}_L=$ total product per unit of labour
  • $\frac{f(\overrightarrow{k},l)}{l}$
  • Output per worker
• $M{P}_L=\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }L}$
  • Output per worker of the last worker
• Example:
  • 10 tas
  • 900 students
  • $A{P}_L=\frac{900}{10}=9$
  • $M{P}_L=\frac{940}{11}=$
    • One more tutorial of 40 students
  • If $MP<AP$
    • Pulls the average down
  • If $MP>AP$
    • Average gets pulled up
  • $CGPA=AP$
    • All courses
  • $SGPA=MP$
    • This semester
  • $IQ=AP$
    • Iq test corresponds to average product and average $IQ$ over all students
    • But $MP$ would be the $IQ$ of the last person
• • 1:
    • $T{P}_L=16L^2$
    • $A{P}_L=\frac{T{P}_L}{l}=16l$
    • $M{P}_L=\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }L}$
    • • If a total product curve is increasing at an increasing rate, then marginal product is increasing
    • Average product is the ray
  • 2:
    • $T{P}_L=2\sqrt{L}$
• Example:
  • 1:
    • See that $AP$ decreases until $L=8$ until it increases again
  • 2:
    • $T{P}_L=6l^2-l^3$
    • $\overline{k}=1$
    • If $MP=AP$ we don't change anything on $AP$
• • If $AP$ is $2$ and $MP$ is $8$
  • We know that $AP$ is increasing

Isoprofit Analysis

• • $Q=\sqrt{k}\sqrt{l}$
  • $k=4$
  • $p=8$
  • $w=1$
  • $r=4$
  • $\pi =R-C$
  • Profit is Revenue - Costs
  • $\pi =(PQ)-(wl+rk)$
  • $PQ=\pi +rk+wl$
  • $q=\frac{\pi +rk}{p}+\frac{w}{p}l$
  • $\frac{\pi +rk}{p}$ is the vertical intercept of the isoprofit line
  • Slope is $\frac{w}{p}$
  • Tangency condition:
    • $L^{\ast }:M{P}_L=\frac{w}{p}$
  • $P(M{P}_L)=W$
  • Suppose $M{P}_L=2$
    • Last guy hired produced $2$ units of output
  • If $P=8$
  • $16=w$
    • $16\mathrm{\$}$ of value produced by the last worker
    • $w$ is the marginal cost of hiring that person
    • Here if $w$ costs $16$
    • Then $MB=MC$
      • Maximizes profits
    • If $MB>MC$ then hire more
    • If $MB<MC$ then fire more
  • $T{P}_L=2\sqrt{l}$
    • Sub in fixed capital into $q(k,l)$
  • $M{P}_L=\frac{1}{\sqrt{l}}$
  • $\frac{w}{p}=\frac{1}{8}$
  • Tangency Condition
    • $\frac{1}{8}=\frac{1}{\sqrt{l}}$
    • $8=\sqrt{l}$
    • $8^2=l$
    • $64=l$
    • $64$ more workers to achieve the max profit
  • $T{P}_L=2\sqrt{64}=16$
  • So $\pi =PQ-wl-rk$
    • $=8(16)-(1)(64)-(4)(4)$
    • $=48$ dollars of profits
  • $\frac{\pi +rk}{p}=\frac{48+16}{8}=8$
  • Conclusions
    • Hire 64 workers
    • Produce 16 units
    • Profits will be 48 dollars
  • Price Shock
    • $\frac{w}{p}=\frac{1}{2}$
    • $M{P}_l=\frac{1}{\sqrt{l}}$
    • $\frac{1}{2}=\frac{1}{\sqrt{l}}$
    • $l=4$
    • $\pi =(2)(8)-(1)(4)-(4)(4)$
    • Lower profits
  • Wage Shock
    • $w=1\to 4$
    • $M{P}_L=\frac{w}{p}$
    • $\frac{1}{\sqrt{l}}=\frac{4}{8}$
    • $L=4$
    • $Q=16$
    • Same thing as changing $p=8\to 2$
  • Scale Shock
    • $T{P}_0=2\sqrt{l}$
    • $T{P}_1=\sqrt{l}$
    • $L=$
    • $Q=$
    • $\pi =$

Short Run Cost Curve Analysis

• Find $L_{SR}=$
  • Short term labour
  • $Q=2\sqrt{l}$
  • $Q^2=4l$
  • So $L_{SR}=\frac{1}{4}{Q}^2$
• $SVC=$
  • Short variable costs
  • Variable inputs
  • $wL$
  • $1\cdot \frac{1}{4}{q}^2$
  • $\frac{1}{4}{q}^2$
• $SFC=$
  • $4$ units of fixed capital at $4$
  • $kr$ is $SFC$
  • $16$
  • Fixed
• $SC=SVC+SFC$
  • $=\frac{1}{4}{q}^2+16$
• $SAVC=$
  • $\frac{SVC}{Q}=\frac{1}{4}Q$
• $SAFC=$
  • $\frac{16}{q}$
• $SAC=$
  • $\frac{1}{4}q+\frac{16}{q}$
• $SMC=$
  • $\frac{1}{2}q$
  • Twice as big as the slope of $SAVC$
• Why is it $8$?
  • Why do $SAFC$ and $SAVC$ intersect at $8?$
  • Why is $SAC$ minimized at $SMC=SAC$
    • Same example as when $CGPA=SGPA$
  • Do $SAVC$ and $SAC$ ever cross?
    • No, because $SAC>SAVC$ due to $SAC=SAVC+SAFC$
    • Only when $SAFC=0$ could they touch.
      • However $SAFC\ne 0$
    • So clearly $SAC>SAVC,\mathrm{\forall }q$
• Firms maximize profit when $SMC(Q)=P$
  • $SAC=5$
    • $\frac{1}{4}(16)+\frac{16}{16}$
    • $5$
  • $\pi =(p-SAC)q$
  • $\pi =(8-5)(16)=48$
• $L_{SR}=\frac{1}{4}{Q}^2$
  • $L_{SR}=\frac{1}{4}(256)=64$
• $P=8$
• $K=4$
• $R=4$
• $W=1$
  • The firm should hire $64$ workers, produce $16$ and get profits of $48$
  • Another method of maximizing profits.
• If $P=4$
  • $L=16$
  • $0$ profit
  • When price falls profit falls
• If $P=2$
  • Average cost is $5$ dollars
  • $(2-5)4=-12$
  • Shut down the firm?
    • No
    • If you shut down, then $Q=0$
    • $R=0$
    • $R=PQ=0$
    • $L=0$
    • $SVC=0$
    • $SFC=16$
      • Still have to pay the $16$
    • $\pi =0-0-16=-16$
    • You'll lose more money per month rather than before.
  • If losses are less than fixed costs, don't shut down.
• • A firm with $SVC=q^2+1$
  • $SFC=3$
  • $SMC=2q$
  • $STC=q^2+4$
  • Will shut down once the price drops below:
    • $1$
    • $2$
    • $3$
    • $4$
  • $P=SMC=q=\pi =3$
• • $Q^{\ast }:P=SMC$ rule
  • If $P>SMC$
    • Costs $2$ dollars to make something, and you can sell it for more, then produce more
  • If $P<SMC$
    • If costs are more than price, sell less
  • If $P=SMC$
    • We optimize the balance here

  • If the firm tries to maximize output: