Lecture Notes

Profit MaximizationProfits (π) = TR – TC or

π = ΣPiYi - Σ wjXj for i = 1 to n (outputs) and j = 1 to m (inputs).

Types of FirmsProprietorship – single ownerPartnership – two or more ownersCorporation – firm is a separate legal entityAdvantages and Disadvantages

Factors of Production (Inputs) – 3 TypesVariable inputs – varies with y.Fixed inputs – fixed even if y=0.Quasi-fixed inputs (if y=0 then x=0 but if y>0 then x

is fixed).

Short-run π maximizationAssume only 1 output Y and two inputs L & KK is fixed => in short-runFirm problem is to choose L to maximize:

Given that y = f(K,L) => choose L to max: How?

MPL = w/p or p*MPL = w – what does this mean? Graphically

Recall π = py – wL – rK, with K fixed or solve for y to get

y = π/p + wL/p + rK/p Recall that p, w, r, and K are all fixed paramters, if we

also hold π constant then we get a straight line in L, y space:

Kr– – wLpy

KrwLKLPf ),(

L

YSlope = w/p

Intercept = π /p + rK/p

The curve in the graph above is called an iso-profit curve. Why? Along a given iso-π 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.

L

Y

L1

π↑

L

Y

Isoprofit curve

),( KLfY

L*

Y*

Discuss the Graph above:Maximize profit by getting on highest iso-profit

curve.Subject to constraint that must be on the

production function.=> tangency between the two orSlope of production function = slope of iso-profit orMPL = 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 ↓ => what happens to L* and Y*? w ↑ or ↓ => what happens to L* and Y*? Show both graphically. Hint what happens to the slope

of all isoprofit lines?

The Long-RunMaximize profit = pf(L,K) – wL – rKBoth L and K are variable and hence a more complex

problem.Recall that in the SR, profit max required MPL = w/pIn LR is this still true?Also requires that MPK = r/por P*MPK = r; P*MPL = wBoth conditions must occur simultaneouslyUse 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*)

Inverse factor demand curvesw = f(L, K*, P) = P*MPL (L, K*)r = f(L*, K, P) = P*MPK (L*, K)or graphically – why downward sloping?

DL

L

w

Returns to ScaleWhat happens to profit with constant returns to

scale? Inputs double => costs double => output doubles =>

profits double but => original output level was not profit max.

What happens with increasing returns to scale?Problems

If original profit = 0 => 2*0 = 0 => original profit may have been profit max.

Eventually get decreasing returns to scale => what happens to profit?

If firm size increases (Y increases) => under what conditions will P remain constant?

Even if market competitive => if all firms increase Y => P will increase.

Revealed ProfitabilityWhat is it?Suppose at time = t => with pt , wt, rt then the

firm chooses Yt, Lt, and Kt.At a different time = s => 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 π maximization point the π from actual

choices must be ≥ π from other possible choices at that time.

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)≥

0Or (4) ΔpΔY – ΔwΔL – ΔrΔK ≥ 0Suppose Δw = Δr = 0 => equation 4 implies that

ΔpΔY ≥ 0 => if Δp > 0 then it must also be the case that ΔY > 0. What does this mean?Supply of the output is upward sloping

More implications of WAPMSuppose Δp = Δw = 0 => equation 4 implies

that – ΔrΔK ≥ 0 or ΔrΔK ≤ 0 or if Δr ≥ 0 then it

must be true that ΔK ≤ 0. What does this mean?

Demand for Capital is downward slopingDo 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?