Technology 1. that describes the (net) output of the L goods of the production process of the firm. By convention,

Size: px

Start display at page:

Download "Technology 1. that describes the (net) output of the L goods of the production process of the firm. By convention,"

Silas Townsend
5 years ago
Views:

1 Production Theory

2 Technology 1 Consider an economy with L goods. A firm is seen as a black box which uses these goods to serve as inputs and/or outputs. The production plan is a vector y = (y 1,..., y L ) R L that describes the (net) output of the L goods of the production process of the firm. By convention, a positive y i indicates a good which is a net output and a negative y i indicates a good which is a net input.

3 Technology 2 The existing technology, taken as primitive datum, defines the production set Y R, which is the set of the feasible production plans Can describe the production set using the transformation function F (.), which is such that: Y = {y R L : F (y) 0}; F (y) = 0 iff there is no y Y such that y y. F (y) = 0 iff y is (technically) efficient, that is there is no way to produce more of a least one output with the same inputs or the same output with less of at least one input. The set {y R L : F (y) = 0} is defined as the transformation frontier.

4 Production set and transformation frontier

5 Marginal rate of transformation Provided that F (.) is differentiable and F (y) = 0, then the marginal rate of transformation (at y) is given by: MRT lk (y) = F (y) y l F (y) y k obtained simply by totally differentiating F (.) and evaluating it at y. how much the (net) output of good k can increase if the firm decreases marginally the (net) output of good l Graphically, this is simply the slope of the transformation frontier.

6 Technology 3 Sometimes we ll use two simplifications separation between inputs and outputs. With L M inputs (always negative) and M outputs (always positive), Y = {( z 1,..., z L M, q 1,..., q M ) : (z 1,..., z L M ), (q 1,..., q M ) 0 and F (.) 0}; single-output technology. With L 1 inputs and 1 output, we can make use of the production function q = f (z 1,..., z L 1 ) defined as Y = {( z 1,..., z L 1, q) : q f (z 1,..., z L 1 ) 0 and (z 1,..., z L 1 ) 0}.

7 Single output technology Useful concepts in the single-output case: input requirement set isoquant marginal rate of technical substitution elasticity of substitution

8 Single output technology Useful concepts in the single-output case: input requirement set isoquant marginal rate of technical substitution elasticity of substitution

9 Input requirement set the input requirement set V (q) the set of all input bundles that produce at least q, given by: V (q) = {(z 1,..., z L 1 ) : ( z 1,..., z L 1, q) Y and (z 1,..., z L 1 ) 0}.

10 Single output technology Useful concepts in the single-output case: input requirement set isoquant marginal rate of technical substitution elasticity of substitution

11 Isoquant the isoquant Q(q) is the set of all input bundles that produce exactly q, given by: Q(q) = {(z 1,..., z L 1 ) : (z 1,..., z L 1 ) 0, (z 1,..., z L 1 ) V (q) and (z 1,..., z L 1 ) / V (q ) if q > q}.

12 Single output technology Useful concepts in the single-output case: input requirement set isoquant marginal rate of technical substitution elasticity of substitution

13 Marginal rate of technical substitution if f (.) is differentiable, the marginal rate of technical substitution of input k for input l (MRTS kl ), (holding output fixed at q = f (z)) is the given by: f (z) z MRTS kl = l f (z) z k simply obtained by totally differentiating f (.). The MRTS kl is simply the slope of the isoquant Q(q) and it is the analogue of the MRT kl (when k and l are inputs).

14 Single output technology Useful concepts in the single-output case: input requirement set isoquant marginal rate of technical substitution elasticity of substitution

15 Elasticity of substitution if the marginal rate of technical substitution gives the slope of an isoquant, the elasticity of substitution measures the curvature of an isoquant. More technically, the elasticity of substitution of input k for input l (with output fixed at q) is given by : σ kl = or, for infinitesimal variations σ kl = MRTS kl (z l /z k ) (z l /z k ) (z l /z k ) MRTS kl MRTS kl d(z l /z k ) = d ln(z l/z k ) dmrts kl d ln MRTS kl Intuitively, the more the factor input ratio changes for a given change in the slope of the isoquant, the larger the elasticity of substitution.

16 Properties of ALL production sets 1 Y is closed. If y n y and y n Y, then y Y : the production set contains its own boundary; Y is no empty: At least one productions plan is always possible;

17 Properties of ALL production sets 2 Y satisfies no free lunch: not possible to produce something from nothing (i.e. if y Y and y 0, then y = 0

18 Properties of ALL production sets 4 Y satisfies free disposal: If y Y and y y, then y Y : always possible to throw away (at no cost) some inputs or outputs

19 (Possible) properties of production sets 1 Not always satisfied, sometimes mutually exclusive Y satisfies possibility of inaction: 0 Y : firm can shut down production. It holds before any production decision is made. Otherwise, sunk costs or fixed factors of production may make invalid.

20 (Possible) properties of production sets 2 Y is convex. If y, y Y and α [0, 1], then αy + (1 α)y Y ; balanced input combinations are more productive than unbalanced ones.

21 (Possible) properties of production sets 3 nonincreasing returns to scale: y Y αy Y, α [0, 1]. Any feasible production plan can be scaled down.

22 (Possible) properties of production sets 4 nondecreasing returns to scale: y Y αy Y, α [1, ]. Any feasible production plan can be scaled up.

23 (Possible) properties of production sets 5 constant returns to scale: y Y αy Y, α [0, ].

24 Scale in the single-output case In the single-output case, t > 1, then f (tz) < tf (z) nonincreasing RS; f (tz) = tf (z) constant RS; f (tz) > tf (z) nondecreasing RS;

25 Profit maximisation 1 Assume now L-dimensional vector of prices p = (p 1,..., p L ) > 0, independent from the choices of the firm: firm is price taker in input and output markets firm maximises profits Y is not empty, closed and satisfies free disposal The firm s problem can be stated as or, equivalently max y p y PMP s.t y Y max y p y s.t F (y) 0

26 Profit maximisation 2 If F (.) is differentiable, necessary condition for profit maximisation are p = λ F (y ) λ 0 FOC-PMP y is chosen so that p and F (y ) are proportional If Y is convex, FOC-PMP is not only necessary but also sufficient for profit maximisation. More than simply a technical point!! For instance, Assume L = 2 and good 1 being a net output and 2 a net input. If Y shows CRS or IRS, then y 1 = when p 1 sufficiently large relatively to p 2, and y 1 = 0 otherwise.

27 Profit maximisation 3 FOC-PMP can be rewritten as follows, for any k, l = 1,.., L and k l: F (y p ) l y = k = MRT p F (y ) kl FOC-PMP2 k y l

28 Profit function and supply correspondence Two fundamental functions/correspondences ONLY deriving from the profit maximising behaviour hypothesis are: the profit function π(p) = p y which associates to every p the maximum value of p y; the supply correspondence y(p) = {y Y : p y = π(p)} which associates to every p the profit maximising production plan y.

29 Properties of the profit and supply functions/correspondences If Y is convex, y(p) is a convex set for all p. If Y is strictly convex, y(p) is single-valued. If Y is convex, then Y = {y R L : p y π(p) for all p 0}. The profit function is a complete description of the technology.

30 Properties of the profit and supply functions/correspondences 2 π(.) is convex in prices. Let p = tp + (1 t)p for all 0 t 1. Then, π(p ) tπ(p) + (1 t)π(p ).

31 Properties of the profit and supply functions/correspondences 3 When π(.) is differentiable, can obtain the supply correspondence from the profit function, using the Hotelling s lemma or, equivalently, π(p) = y(p) π(p) p i = y i (p) for i = 1,.., L. (when i is an input, y i (p) is usually referred to as factor demand function). Hotelling s lemma is simply an application of the envelope theorem (see previous picture).

32 Properties of the profit and supply functions/correspondences 4 Dy(p) is positive semidefinite. Because of Hotelling s lemma, Dy(p) = D 2 π(p). Since π(.) is convex, its Hessian matrix must be positive semidefinite, so that also Dy(p) must be positive semidefinite. Positive semidefiniteness of Dy(p) implies...

33 Properties of the profit and supply functions/correspondences 5 1. Dy(p) is symmetric: cross-substitution effects are symmetric 2 π(.) = y l(.) = y k(.) = 2 π(.). p l p k p k p l p k p l for l, k = 1.., L very little intuition law of supply: own-price effects are nonnegative y l (.) p l 0 for l = 1.., L. optimal amount of output increases with its price and optimal amount of input decreases with its price 3. the principal-minor determinants have alternate sign, starting from positive. technical requirement for convexity of the supply function

34 Properties of the profit and supply functions/correspondences 6 π(.) is homogenous of degree one y(p) is homogenous of degree zero For all t > 0, π(tp) = t π(p) and y(tp) = y(p). A proportional change of all prices change (optimal) profits by the same proportion but does not change the (optimal) production plan. The relationship between these two results follows from Hotelling s lemma, being the factor demands the derivative of the profit function.

35 Cost minimisation A choice of inputs that minimises the cost of producing a given output is a necessary (but not sufficient) condition for profit maximisation. Result on costs of interest because often more useful than results on technology, esp. in applied work require only price-taking assumption in input markets better accommodate constant or nondecreasing returns to scale Focus on single-output technology (restrictive assumption).

36 Cost minimisation problem To minimise costs, a firm solves the problem min z w z CMP s.t q f (z) Necessary condition for z(q, w) to be the solution to CMP are, for some λ 0 and for l = 1,.., L 1, w l λ f (z ) z l or, equivalently, (with = when z l > 0) FOC-CMP w λ f (z ) and [w λ f (z )] z = 0 If f (.) is concave, these conditions are also sufficient for cost minimization.

37 Cost minimisation problem 2 In case of interior solutions, FOC-CMP can be re-written as follows, for l, k = 1,.., L and l k, w l w k = f (z l,z k ) z k f (z l,z k ) z l = MTRS lk (FOC-CMP2) which is clearly a special case of the condition FOC-PMP2 for profit maximisation and which has a nice graphical interpretation.

38 FOC-CMP

39 From CMP Two fundamental functions/correspondences deriving from cost minimisation problem: the conditional factor demand correspondence z(q, w) which associates to every q and w the cost minimising input demand the cost function c(q, w) = w z(q, w) which associates to every q and w the minimum production cost

40 Properties of cost fct and conditional demand factor fct 1 c(.) is concave in w.

41 Properties of cost fct and conditional demand factor fct 2 If the sets {z > 0 : f (z) q} are convex for every q, then Y = {( z, q) : w z c(w, q) for all w > 0}. The cost function is a complete description of the technology.

42 Properties of cost fct and conditional demand factor fct 3 When c(.) is differentiable, can obtain the conditional factor demand correspondence from the cost function, using w c(w, q) = z(w, q) Shepard s lemma or, equivalently, c(w, q) w i = z i (w, q) for i = 1,.., L. Similarly to Hotelling s lemma, Shepard s lemma is simply an application of the envelope theorem (see previous picture).

43 Properties of cost fct and conditional demand factor fct 4 D w z(w, q) is symmetric negative semidefinite. Because of Shepard s lemma,d w z(w, q) = D 2 c(w, q). Since c(.) is concave in w, its Hessiam matrix must negative semidefinite, so that also D w z(w, q) must be negative semidefinite. Negative semidefiniteness of D w z(w, q) implies...

44 Properties of cost fct and conditional demand factor fct 4 1. D w z(w, q) is symmetric: 2 c(.) = z l(.) = z k(.) = 2 c(.). w l w k w k w l w k w l very little intuition the conditional factor demand are (weakly) downward sloping: z i (.) w i = 2 c(.) w i 0 for i = 1.., L. law of demand for inputs the principal-minor determinants have alternate sign, starting from negative. technical requirement for concavity of the cost function

45 Properties of cost fct and conditional demand factor fct 5 c(.) is homogeneous of degree one in w: c(q, αw) = α c(q, w); z(q, w) is homogeneous of degree zero in w: z(q, αw) = z(q, w) An equally proportional change of all input prices causes an equal change in total cost but not a change in factor demands. These two results depend on the Shepard s lemma, being the conditional factor demands the derivative of the cost function.

46 Properties of cost fct and conditional demand factor fct 6 c(.) is nondecreasing in w: if w > w, then c(q, w ) > c(q, w). The total cost of producing q can only increase when at least one of the input prices increases. This again depends from the Shepard s lemma, since c(w,q) w i = zi (w, q) 0.

47 Using the cost function Using the cost function, we can rewrite the profit maximisation problem as follows max q 0 p q c(w, q) Since input are optimally chosen, focus is now on the choice of output only!!! When the technology is single-output, condition necessary for q to be optimal is p c(w, q ) q 0 with strict equality if q > 0

48 Competitive firms 1 The following figures describe the optimal behaviour of a competitive firm under different technological conditions. Let 1 output p > 0 and w 0 C(q) = c(q, w); AC(q) = C(q)/q; C (q) = dc(q)/dq

49 Competitive firms and strictly decreasing returns to scale (convex)

50 Competitive firms and constant returns to scale (convex)

51 Competitive firms and non convex technology

52 Competitive firms and strictly convex variable costs with nonsunk setup costs

53 Competitive firms and constant returns variable costs with nonsunk setup costs

54 Competitive firms and strictly convex variable costs with sunk setup costs

Microeconomics I. September, c Leopold Sögner

Microeconomics I. September, c Leopold Sögner Microeconomics I c Leopold Sögner Department of Economics and Finance Institute for Advanced Studies Stumpergasse 56 1060 Wien Tel: +43-1-59991 182 soegner@ihs.ac.at http://www.ihs.ac.at/ soegner September,