Frank Cowell: Firm Basics THE FIRM: BASICS MICROECONOMICS Principles and Analysis Frank Cowell March

Frank Cowell: Firm Basics Overview  March The setting Input require- ment sets Returns to scale Marginal products The Firm: Basics The environment for the basic model of the firm Isoquants

Frank Cowell: Firm Basics The basics of production   Some elements needed for an analysis of the firm Technical efficiency Returns to scale Convexity Substitutability Marginal products  This is in the context of a single-output firm  ...and assuming a competitive environment  First we need the building blocks of a model  March

Frank Cowell: Firm Basics Notation March  Quantities zizi amount of input i z = (z 1, z 2, , z m ) input vector amount of output q  Prices price of input i w = (w 1, w 2, , w m ) Input-price vector price of output p For next presentation wiwi

Frank Cowell: Firm Basics Feasible production March  The basic relationship between output and inputs: q   (z 1, z 2, , z m )  This can be written more compactly as: q   (z) single-output, multiple-input production relation   gives the maximum amount of output that can be produced from a given list of inputs Note that we use “  ” and not “=” in the relation. Why? Consider the meaning of  Vector of inputs The production function distinguish two important cases...

Frank Cowell: Firm Basics Technical efficiency March The case where production is technically efficient The case where production is (technically) inefficient  Case 1: q   (z)  Case 2: q  (z) Intuition: if the combination (z,q) is inefficient, you can throw away some inputs and still produce the same output

Frank Cowell: Firm Basics z2z2 z1z1 q >  (z)  Boundary points are feasible and efficient The function  March q 0  The production function  Interior points are feasible but inefficient  Infeasible points q <  (z) q =  (z)  We need to examine its structure in detail

Frank Cowell: Firm Basics Overview  March The setting Input require- ment sets Returns to scale Marginal products The Firm: Basics The structure of the production function Isoquants

Frank Cowell: Firm Basics The input requirement set March remember, we must have q   (z)  Pick a particular output level q  Find a feasible input vector z  Repeat to find all such vectors  Yields the input-requirement set Z(q) := {z:  (z)  q} The set of input vectors that meet the technical feasibility condition for output q  The shape of Z depends on the assumptions made about production  We will look at four cases First, the “standard” case

Frank Cowell: Firm Basics z1z1 z2z2 The input requirement set March  Feasible but inefficient  Feasible and technically efficient  Infeasible points Z(q)Z(q) q <  (z) q =  (z) q >  (z)

Frank Cowell: Firm Basics z1z1 z2z2 Case 1: Z smooth, strictly convex March  Pick two boundary points  Draw the line between them  Intermediate points lie in the interior of Z  A combination of two techniques may produce more output  What if we changed some of the assumptions? z z  Z(q)Z(q) q<  (z) q =  (z") q =  (z')  Note important role of convexity

Frank Cowell: Firm Basics Case 2: Z Convex (but not strictly) March z1z1 z2z2  A combination of feasible techniques is also feasible Z(q)Z(q) z z   Pick two boundary points  Draw the line between them  Intermediate points lie in Z (perhaps on the boundary)

Frank Cowell: Firm Basics Case 3: Z smooth but not convex March z1z1 z2z2  in this region there is an indivisibility  Join two points across the “dent” Z(q)Z(q)  Take an intermediate point  Highlight zone where this can occur This point is infeasible

Frank Cowell: Firm Basics z1z1 z2z2 Case 4: Z convex but not smooth March  Slope of the boundary is undefined at this point q =  (z)

Frank Cowell: Firm Basics z1z1 z2z2 z1z1 z2z2 z1z1 z2z2 z1z1 z2z2 Summary: 4 possibilities for Z March Only one efficient point and not smooth Problems: the dent represents an indivisibility Standard case, but strong assumptions about divisibility and smoothness Almost conventional: mixtures may be just as good as single techniques

Frank Cowell: Firm Basics Overview  March The setting Input require- ment sets Returns to scale Marginal products The Firm: Basics Contours of the production function Isoquants

Frank Cowell: Firm Basics Isoquants 17  Pick a particular output level q  Find the input requirement set Z(q)  The isoquant is the boundary of Z: { z :   (z) = q } Think of the isoquant as an integral part of the set Z(q)  (z)  i (z) := ——  z i.  j (z) ——  i (z)  If the function  is differentiable at z then the marginal rate of technical substitution is the slope at z: Where appropriate, use subscript to denote partial derivatives. So  Gives rate at which you trade off one input against another along the isoquant, maintaining constant q Let’s look at its shape 10 Oct 2012

Frank Cowell: Firm Basics z1z1 z2z2  The set Z(q ) Isoquant, input ratio, MRTS March  A contour of the function  {z:  (z)=q}  An efficient point  Input ratio describes one production technique z2z2 ° z1z1 °  z°  The input ratio z 2 / z 1 = constant  Marginal Rate of Technical Substitution  The isoquant is the boundary of Z  Increase the MRTS  z′ MRTS 21 =  1 (z)/  2 (z)  MRTS 21 : implicit “price” of input 1 in terms of 2  Higher “price”: smaller relative use of input 1

Frank Cowell: Firm Basics MRTS and elasticity of substitution  Responsiveness of inputs to MRTS is elasticity of substitution March ∂log(z 1 /z 2 ) =   ∂log(f 1 /f 2 ) prop change input ratio -  = prop change in MRTS  input-ratio  MRTS    input-ratio MRTS z1z1 z2z2  = ½ z1z1 z2z2  = 2

Frank Cowell: Firm Basics Elasticity of substitution March z1z1 z2z2 structure of the contour map...  A constant elasticity of substitution isoquant  Increase the elasticity of substitution...

Frank Cowell: Firm Basics Homothetic contours March  The isoquants O z1z1 z2z2  Draw any ray through the origin…  Get same MRTS as it cuts each isoquant

Frank Cowell: Firm Basics Contours of a homogeneous function March  The isoquants O z1z1 z2z2  Coordinates of input z°  Coordinates of “scaled up” input tz ° O tz 1 ° tz 2 ° z2z2 ° z1z1 °  tz°  z° z° q trqtrq  (tz) = t r  (z)

Frank Cowell: Firm Basics Overview... March The setting Input require- ment sets Returns to scale Marginal products The Firm: Basics Changing all inputs together Isoquants

Frank Cowell: Firm Basics Let's rebuild from the isoquants  The isoquants form a contour map  If we looked at the “parent” diagram, what would we see?  Consider returns to scale of the production function  Examine effect of varying all inputs together: Focus on the expansion path q plotted against proportionate increases in z  Take three standard cases: Increasing Returns to Scale Decreasing Returns to Scale Constant Returns to Scale  Let's do this for 2 inputs, one output  March

Frank Cowell: Firm Basics Case 1: IRTS March z2z2 q z1z1 0  An increasing returns to scale function  t>1 implies  ( tz) > t  ( z)  Pick an arbitrary point on the surface  The expansion path…  Double inputs and you more than double output

Frank Cowell: Firm Basics Case 2: DRTS March q z1z1  A decreasing returns to scale function  Pick an arbitrary point on the surface  The expansion path… z2z2 0  t>1 implies  ( tz) < t  ( z)  Double inputs and output increases by less than double

Frank Cowell: Firm Basics Case 3: CRTS March z2z2 q 0  A constant returns to scale function  Pick a point on the surface  The expansion path is a ray z1z1   (tz) = t  (z)  Double inputs and output exactly doubles

Frank Cowell: Firm Basics Relationship to isoquants March z2z2 q 0  Take any one of the three cases (here it is CRTS)  Take a horizontal “slice” z1z1  Project down to get the isoquant  Repeat to get isoquant map  The isoquant map is the projection of the set of feasible points

Frank Cowell: Firm Basics Overview  March The setting Input require- ment sets Returns to scale Marginal products The Firm: Basics Changing one input at time Isoquants

Frank Cowell: Firm Basics Marginal products March  Measure the marginal change in output w.r.t. this input  (z) MP i =  i (z) = ——  z i.  Pick a technically efficient input vector  Keep all but one input constant Remember, this means a z such that q=  (z) The marginal product

Frank Cowell: Firm Basics CRTS production function again March z2z2 q 0  Now take a vertical “slice” z1z1  The resulting path for z 2 = constant Let’s look at its shape

Frank Cowell: Firm Basics MP for the CRTS function March z1z1 q (z)(z)  The feasible set  A section of the production function  Technically efficient points  Input 1 is essential: If z 1 = 0 then q = 0  Slope of tangent is the marginal product of input 1 (z)(z) (z)(z)  Increase z 1 …   1 ( z ) falls with z 1 (or stays constant) if  is concave

Frank Cowell: Firm Basics Relationship between q and z 1 March z1z1 q z1z1 q z1z1 q z1z1 q  We’ve just taken the conventional case  But in general this curve depends on the shape of   Some other possibilities for the relation between output and one input…

Frank Cowell: Firm Basics Key concepts  Technical efficiency  Returns to scale  Convexity  MRTS  Marginal product March Review

Frank Cowell: Firm Basics What next?  Introduce the market  Optimisation problem of the firm  Method of solution  Solution concepts March