清华大学中级微观经济学讲义

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 MP1 w1 0 p MP2 w2 0. That is, marginal revenue equals to input price Important: we have assumed decreasing Marginal Products!
What happens to the short-run profitmaximizing production plan as the variable input price w1 changes?
Comparative Statics of Short-Run Profit-Maximization
with the amount used of input 1.
If p MP1 w1 then profit increases with x1. If p MP1 w1 then profit decreases with x1.
A Mathematical Approach to Short-Run Profit-Maximization
x”
x
Revealed Profitability
Consider a competitive firm with a technology that exhibits decreasing returns-to-scale. For a variety of output and input prices we observe the firm’s choices of production plans. What can we learn from our observations?
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
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
f(x1,
x~2
)
Slopes w1 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 iso-profit line are
Economic Profit
The economic profit generated by (x1,…,xm,y1,…,yn) is
p1y1 pnyn w1x1 wmxm.
Notes:
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 equation of a short-run iso-profit line
is
y
w1 p
x1
w 2x~ 2 p
so an increase in w1 causes an increase in the slope, and
Comparative Statics of Short-Run
Short-run Economic Profit
Suppose the firm is in a short-run
circumstance in which x2 x~2.
Its short-run production function is
The
firm’s
y f(x1, x~2 ).
Returns-to Scale and Profit-
Maximization
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.
Profit-Maximization
y
y f(x1, x~2 )
y* Slopes w1 p
x*1
x1
Comparative Statics of Short-Run
Profit-Maximization
y
y*
y f(x1, x~2 )
Slopes w1 p
x*1
x1
A Useful Math: The Envelope Theorem
Long-Term Profit in the Case of Decreasing Returns-to-Scale
In this case, the firm has a single long-run profit-maximizing production plan; See the graph next page; The optimal profit might be 0.
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
equal. y*
MP1
w1 p
Slopes w1 p
at (x*1, x~2, y* )
x*1
x1
Short-Run Profit-Maximization
MP1
w1 p
p MP1 w1
p MP1 is the marginal revenue product of input 1, the rate at which revenue increases
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.
Returns-to Scale and Profit-
Maximization
y
y f(x)
y*
Decreasing returns-to-scale
x*
x
Long-Term Profit in the Case of Increasing Returns-to-Scale
In this case, the long-term profit is infinite (see the graph next page); But, this is not consistent with our perfect competitive firm assumption, since the firm is infinite in size.
fixed cost is FC
w 2x~ 2
and its profit function is py w1x1 w2x~2.
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
py w1x1 w2x~2.