ECO 365

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ECO 365 Intermediate Microeconomics

• Lecture Notes

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Profit Maximization

• Profits (p) TR TC or
• p SPiYi - S wjXj for i 1 to n (outputs) and
  j 1 to m (inputs).
• Types of Firms
• Proprietorship single owner
• Partnership two or more owners
• Corporation firm is a separate legal entity
• Advantages and Disadvantages
• Factors of Production (Inputs) 3 Types
• Variable inputs varies with y.
• Fixed inputs fixed even if y0.
• Quasi-fixed inputs (if y0 then x0 but if ygt0
  then x is fixed).

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• Short-run p maximization
• Assume only 1 output Y and two inputs L K
• K is fixed gt in short-run
• Firm problem is to choose L to maximize
• Given that y f(K,L) gt choose L to max
• How?
• MPL w/p or pMPL w what does this mean?
• Graphically
• Recall p py wL rK, with K fixed or solve
  for y to get
• y p/p wL/p rK/p
• Recall that p, w, r, and K are all fixed
  paramters, if we also hold p constant then we get
  a straight line in L, y space

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• The curve in the graph above is called an
  iso-profit curve. Why?
• Along a given iso-p then profit is fixed.
• If profit ? (?) then the line shifts up (down).
• Thus, have a family of curves, all with same
  slope where profit increases as you move and
  decreases as you move down.

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Y
p?
L
L1
7
Y
L
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• Discuss the Graph above
• Maximize profit by getting on highest iso-profit
  curve.
• Subject to constraint that must be on the
  production function.
• gt tangency between the two or
• Slope of production function slope of
  iso-profit or
• MPL w/p notice that this is the same
  condition as we already derived for profit
  maximization.
• What happens as P or w change?
• P ? or ? gt what happens to L and Y?
• w ? or ? gt what happens to L and Y?
• Show both graphically. Hint what happens to the
  slope of all isoprofit lines?

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• The Long-Run
• Maximize profit pf(L,K) wL rK
• Both L and K are variable and hence a more
  complex problem.
• Recall that in the SR, profit max required MPL
  w/p
• In LR is this still true?
• Also requires that MPK r/p
• or PMPK r PMPL w
• Both conditions must occur simultaneously
• Use these conditions to derive Factor Demand
  curves
• L f(p, w, r) and K f(p, w, r)
• Substitute these into production function to get
  optimal output Y f(L, K)

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• Inverse factor demand curves
• w f(L, K, P) PMPL (L, K)
• r f(L, K, P) PMPK (L, K)
• or graphically why downward sloping?

DL
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• Returns to Scale
• What happens to profit with constant returns to
  scale?
• Inputs double gt costs double gt output doubles
  gt profits double but gt original output level
  was not profit max.
• What happens with increasing returns to scale?
• Problems
• If original profit 0 gt 20 0 gt original
  profit may have been profit max.
• Eventually get decreasing returns to scale gt
  what happens to profit?
• If firm size increases (Y increases) gt under
  what conditions will P remain constant?
• Even if market competitive gt if all firms
  increase Y gt P will increase.

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• Revealed Profitability
• What is it?
• Suppose at time t gt with pt , wt, rt then the
  firm chooses Yt, Lt, and Kt.
• At a different time s gt with ps , ws, rs then
  the firm chooses Ys, Ls, and Ks.
• Must be true that
• (1) pt Yt - wt Lt - rt Kt pt Ys - wt Ls - rt Ks
  - why?
• (2) ps Ys - ws Ls - rs Ks ps Yt - ws Lt - rs Kt
  - why?
• WAPM Weak Axiom of Profit Maximization
• To be p maximization point the p from actual
  choices must be p from other possible choices
  at that time.

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• Implications of WAPM?
• Transform equation 2 to get
• (3) -ps Yt ws Lt rs Kt -ps Ys ws Ls rs
  Ks
• Now add equations 3 and 1 together to get
• (pt ps)Yt (wt ws)Lt (rt rs)Kt (pt
  ps)Ys - (wt ws) Ls - (rt rs)Ks or
• (pt ps)(Yt Ys ) (wt ws)(Lt - Ls) (rt
  rs)(Kt - Ks) 0
• Or (4) ?p?Y ?w?L ?r?K 0
• Suppose ?w ?r 0 gt equation 4 implies that
  ?p?Y 0 gt if ?p gt 0 then it must also be the
  case that ?Y gt 0. What does this mean?
• Supply of the output is upward sloping

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• More implications of WAPM
• Suppose ?p ?w 0 gt equation 4 implies that
  
• ?r?K 0 or ?r?K 0 or if ?r 0 then it
  must be true that ?K 0. What does this mean?
• Demand for Capital is downward sloping
• Do the same thing to show demand for labor is
  also downward sloping.
• Finally, note that profit maximization implies
  cost minimization (producing given Y with lowest
  costs). Why?