Pf(Z)-W·z=0 These two, especially, when combined with demand give you implications about prices
PfZ W Z 0 These two, especially, when combined with demand give you implications about prices
Now back to the mwg set theoretic approach A production plan is a vectory E R L This includes both inputs and outputs, and an input is a negative element in this vector An output is a positive element in this vector Total profits are p·y
Now back to the MWG set theoretic approach. A production plan is a vector y L. This includes both inputs and outputs, and an input is a negative element in this vector. An output is a positive element in this vector. Total profits are p y
The set of all production plans is y, which is analogous to X in the consumer chapters We generally assume that E r, so that firms can shut down Generally, we assume that y is (1) nonempty(even beyond including O) (2) closed, i.e. includes its limit points (3 no free lunch-there is no vectoryE Y Where y/>0, for all I and yk>0 for at least one factor =k 4) free disposal-if a vector yL) where yk>0, for then all other vectors (I,.xk,.ylE Y Where y you can always get rid of something)
The set of all production plans is Y, which is analogous to X, in the consumer chapters. We generally assume that 0 Y , so that firms can shut down. Generally, we assume that Y is (1) nonempty (even beyond including 0) (2) closed, i.e. includes its limit points. (3) no free lunch– there is no vector y Y where yl 0, for all l and yk 0 for at least one factor lk. (4) free disposal– if a vector y1 ,.. yk ,.. yL Y where yk 0, for then all other vectors y1 ,.. xk ,.. yL Y where xk yk (you can always get rid of something)
(5)irreversibility if yE y then -yE Y
(5) irreversibility: if y Y then y Y
These properties are more particular: (6)Nonincreasing returns to scale. If y∈Y, then ay∈ Y for all a e[0,1-you can always scale down (7 Nondecreasing returns to scale. If y∈Y, then ay∈ Y for all a≥1- you can always scale up 8)Constant returns to scale Ify e y, then ay e y for all a>0you can always scale up or down 9) Additi vity(also free entry)if yE Y and y∈ Y then y+y∈Y (10) Convexity if y∈ Y and y∈ y then ay+(1-a)∈ Y for all a∈[0,1 (11Yis a convex conde. If for any yy∈ y and a≥0,β≥0,ay+By∈Y
These properties are more particular: (6) Nonincreasing returns to scale. If y Y, then y Y for all 0, 1 – you can always scale down. (7) Nondecreasing returns to scale. If y Y, then y Y for all 1– you can always scale up. (8) Constant returns to scale If y Y, then y Y for all 0– you can always scale up or down. (9) Additivity (also free entry) if y Y and y Y then y y Y (10) Convexity if y Y and y Y then y 1 y Y for all 0, 1 (11) Y is a convex conde. If for any y, y Y and 0, 0, y y Y