Both x1 and x2 are variable. Firm’s problem is Maximize py – w1x1 – w2x2 Subject to y = f(x1, x2)
Long-Run Profit-Maximization
Firm’s problem now becomes: Maximize pf(x1, x2)– w1x1 – w2x2 The input levels of the long-run profitmaximizing plan satisfy p × MP − w1 = 0 p × MP − w2 = 0. 1 2 That is, marginal revenue equals to input price Important: we have assumed decreasing Marginal Products!
/ Chapter Nineteen Profit-Maximization
What Do We Do in this Chapter?
After working our “producer’s budget sets” (production sets), We are working on “producer’s choices” Please pay attention to the similarity and differences between “producer’s choices” and “consumer’s choices”
y
y*
~ y = f(x1, x2)
w1 Slopes = + p
x* 1
x1
A Useful Math: The Envelope Theorem
Suppose x* maximizes g (x; t), where t is a parameter; Then x* varies with t, i.e, x*=x*(t); We have gx [x*(t), t]=0; Let g*(t) = Max g(x,t) = g[x*(t), t] Then, the dg*(t)/dt=gx [x*(t), t]x*’(t) + gt [x*(t), t] = gt [x*(t), t] This is called the Envelope Theorem.
y
Π ≡ Π′′′ Π ≡ Π′′
Π ≡ Π′
~Байду номын сангаасy = f(x1, x2)
y*
w1 Slopes = + p
x* 1
x1
Comparative Statics of Short-Run Profit-Maximization
An increase in w1 causes decreases in – optimal input level; – optimal output; – optimized profit.
~ . Π ≡ py − w1x1 − w2x2 I.e. ~ w1 Π + w2x2 y= x1 + . p p
Short-Run Iso-Profit Lines
y
Π ≡ Π′′′ Π ≡ Π′′
Π ≡ Π′
w1 Slopes = + p
x1
Short-Run Profit-Maximization
y
Π ≡ Π′′′ Π ≡ Π′′
Π ≡ Π′
~ y = f(x1, x2)
w1 Slopes = + p
x1
Short-Run Profit-Maximization
At the short-run profit-maximizing plan, y the slopes of the short-run production function and the maximal Π ≡ Π′′ iso-profit line are equal. w1 * Slopes = + y p w1 MP = 1 p ~ at (x*, x , y*)
Short-run Economic Profit
Suppose the firm is in a short-run ~ circumstance in which x2 ≡ x2. Its short-run production function is
~ The firm’s fixed cost is FC = w2x2 and its profit function is ~ . Π = py − w1x1 − w2x2
1 2
x* 1
x1
Short-Run Profit-Maximization
w1 MP = ⇔ p × MP = w1 1 1 p
p × MP is the marginal revenue product of 1
input 1, the rate at which revenue increases with the amount used of input 1. If p × MP > w1 then profit increases with x1. 1 If p × MP < w1 then profit decreases with x1. 1
Returns-to Scale and ProfitMaximization
y
y = f(x)
y” y’
Increasing returns-to-scale
x’ x” x
Long-Term Profit in the Case of Constant Returns-to-Scale
It’s long-term profit is either 0 or infinity --- depending on prices; Only the 0 profit case is consistent with our perfect competitive firm assumption.
Comparative Statics of Short-Run Profit-Maximization
What happens to the short-run profit-maximizing production plan as the output price p changes?
Comparative Statics of Short-Run Profit-Maximization
Long-Run Profit-Maximization
Now allow the firm to vary both input levels. Since no input level is fixed, there are no fixed costs.
Long-Run Profit-Maximization
A Mathematical Approach to ShortRun Profit-Maximization
Mathematically, the firm’s short run problem is: ~ Π = py − w1x1 − w2x2. Maximize ~ y = f(x1, x2). Subject to: This gives us: pMP1=w1 Important: We have assumed that MP1 is decreasing in x1
Applying the Envelope Theorem
~ . Π = py − w1x1 − w2x2
~ Where, we plug in y = f(x1, x2). Let Π* be the optimal profit level; We get dΠ*/dp=y>0; Π dΠ*/dw1=-x1<0. Π
The equation of a short-run iso-profit line ~ is w1 Π + w2x2 y= x1 +
p
p
so an increase in p causes a reduction in the slope of the family of iso-profit lines
Comparative Statics of Short-Run Profit-Maximization Π ≡ Π′′′
y
Π ≡ Π′′ Π ≡ Π′
~ y = f(x1, x2)
y*
w1 Slopes = + p
x* 1
x1
Comparative Statics of Short-Run Profit-Maximization
Economic Profit
Π = p1y1+L pnyn − w1x1−Lwmxm. +
Notes: Π= For the time being, we restrict to the case of a competitive firm, which is a tiny relative to the market size and takes prices p1,…,pn w1,…,wm as given constants; The economic profit generated by (x1,…,xm,y1,…,yn) is
~ ). y = f(x1, x2
Short-Run Iso-Profit Lines
An iso-profit line contains all the production plans that yield the same profit level. The equation of an iso-profit line is