哈佛大学：《中级微观经济学》（英文版）Lecture 9 The Producer's Problem

Economics 2010a Fa|2003 Edward L. Glaeser Lecture 9

Economics 2010a Fall 2003 Edward L. Glaeser Lecture 9

9. The Producer's problem Firms and maximization b. Production functions C. Supply and Profit Functions d. Cost Functions e. Duality and Producers f. Application: Urban Systems

9. The Producer’s Problem a. Firms and Maximization b. Production Functions c. Supply and Profit Functions d. Cost Functions e. Duality and Producers f. Application: Urban Systems

1. Technology The more tradition approach is to assume (1)A production correspondence, e.g f(K, L) or more generally f(2, that maps the vector of inputs z which cost w into a vector of outputs, which are then sold at prices P for total revenues Pf(z n many cases, we think of f(z) as a function - i.e. only one output-but is doesnt need to be We assume first that firms treat prices as given-i e they are price takers-ie they dont have market power (2)We assume that firms maximize profits and that they have the option(at least in the long run) to exit, i. e. earn zero profits

1. Technology The more tradition approach is to assume: (1) A production correspondence, e.g. f(K,L) or more generally fZ, that maps the vector of inputs Z which cost W into a vector of outputs, which are then sold at prices P for total revenues PfZ. In many cases, we think of fZ as a function– i.e. only one output– but is doesn’t need to be. We assume first that firms treat prices as given– i.e. they are price takers– i.e. they don’t have market power. (2) We assume that firms maximize profits, and that they have the option (at least in the long run) to exit, i.e. earn zero profits

This is of course a deeply controversial claim

This is of course a deeply controversial claim

(3 )We make some assumption about the number of firms- perhaps free entry of identical firms, perhaps something else This last assumption gives us a great deal of power-this is the equilibrium assumption in action Together, profit maximization and free entry of identical firms gives us the following two sets of conditions(assuming that the production function is continuously differentiable and concave) of() W for each input marginal revenue equals prIce. And given these first order conditions

(3) We make some assumption about the number of firms– perhaps free entry of identical firms, perhaps something else. This last assumption gives us a great deal of power– this is the equilibrium assumption in action. Together, profit maximization and free entry of identical firms gives us the following two sets of conditions (assuming that the production function is continuously differentiable and concave). P fZ Zi  Wi for each input marginal revenue equals price. And given these first order conditions:

Pf(Z)-W·z=0 These two, especially, when combined with demand give you implications about prices

PfZ  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 lk. (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