#### Transcript Firm: basics

THE FIRM: BASICS MICROECONOMICS Principles and Analysis Frank Cowell March 2012 Frank Cowell: Firm Basics 1 Overview The Firm: Basics The setting The environment for the basic model of the firm Input requirement sets Isoquants Returns to scale Marginal products March 2012 Frank Cowell: Firm Basics 2 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 2012 Frank Cowell: Firm Basics 3 Notation Quantities zi z = (z1, z2 , , zm ) •amount of input i q •amount of output •input vector For next presentation Prices March 2012 wi w = (w1, w2 , , wm ) •price of input i p •price of output •Input-price vector Frank Cowell: Firm Basics 4 Feasible production The basic relationship between output and inputs: •single-output, multiple-input production relation q f(z1, z2, , zm ) The production function This can be written more compactly as: Vector of inputs •Note that we use “” and not “=” in the relation. Why? •Consider the meaning of f q f(z) f gives the maximum amount of output that can be produced from a given list of inputs March 2012 distinguish two important cases... Frank Cowell: Firm Basics 5 Technical efficiency Case 1: q = f(z) Case 2: q <f(z) •The case where production is technically efficient •The case where production is (technically) inefficient Intuition: if the combination (z,q) is inefficient, you can throw away some inputs and still produce the same output March 2012 Frank Cowell: Firm Basics 6 The function f q q > f(z) q = f (z) 0 q < f(z) The production function Interior points are feasible but inefficient Boundary points are feasible and efficient Infeasible points z2 We need to examine its structure in detail March 2012 Frank Cowell: Firm Basics 7 Overview The Firm: Basics The setting The structure of the production function Input requirement sets Isoquants Returns to scale Marginal products March 2012 Frank Cowell: Firm Basics 8 The input requirement set Pick a particular output level q Find a feasible input vector z remember, we must have q f(z) Repeat to find all such vectors Yields the input-requirement set • Z(q) := {z: f(z) q} The shape of Z depends on the assumptions made about production The set of input vectors that meet the technical feasibility condition for output q We will look at four cases March 2012 First, the “standard” case Frank Cowell: Firm Basics 9 The input requirement set Feasible but inefficient z2 Feasible and technically efficient Infeasible points Z(q) q < f(z) q = f(z) q > f(z) z1 March 2012 Frank Cowell: Firm Basics 10 Case 1: Z smooth, strictly convex Pick two boundary points Draw the line between them z2 Intermediate points lie in the interior of Z Z(q) q = f(z') z q< f (z) Note important role of convexity A combination of two techniques may produce more output z q = f(z") What if we changed some of the assumptions? z1 March 2012 Frank Cowell: Firm Basics 11 Case 2: Z Convex (but not strictly) Pick two boundary points Draw the line between them z2 Intermediate points lie in Z (perhaps on the boundary) Z(q) z z A combination of feasible techniques is also feasible z1 March 2012 Frank Cowell: Firm Basics 12 Case 3: Z smooth but not convex Join two points across the “dent” z2 Take an intermediate point Highlight zone where this can occur Z(q) This point is infeasible in this region there is an indivisibility z1 March 2012 Frank Cowell: Firm Basics 13 Case 4: Z convex but not smooth z2 q = f(z) Slope of the boundary is undefined at this point z1 March 2012 Frank Cowell: Firm Basics 14 Summary: 4 possibilities for Z Standard case, but strong assumptions about divisibility and smoothness z2 Almost conventional: mixtures may be just as good as single techniques z2 z1 z1 z2 z2 Only one efficient point and not smooth Problems: the dent represents an indivisibility z1 March 2012 z1 Frank Cowell: Firm Basics 15 Overview The Firm: Basics The setting Contours of the production function Input requirement sets Isoquants Returns to scale Marginal products March 2012 Frank Cowell: Firm Basics 16 Isoquants Pick a particular output level q Think of the isoquant as an integral part of the set Z(q) Find the input requirement set Z(q) The isoquant is the boundary of Z:{ z : f (z) = q } If the function f is differentiable at z Where appropriate, use then the marginal rate of technical subscript to denote partial fj (z) derivatives. So substitution is the slope at z: —— f(z) fi (z) fi(z) := —— zi . Gives rate at which you trade off Let’s look at one input against another along the its shape isoquant, maintaining constant q 10 Oct 2012 Frank Cowell: Firm Basics 17 Isoquant, input ratio, MRTS The set Z(q) A contour of the function An efficient point z2 The input ratio Marginal Rate of Technical Substitution z2 / z1= constant MRTS21=f1(z)/f2(z) z2° Increase the MRTS The isoquant is the boundary of Z z′ z° {z: f (z)=q} z1° March 2012 z1 Input ratio describes one production technique MRTS21: implicit “price” of input 1 in terms of 2 Higher “price”: smaller relative use of input 1 Frank Cowell: Firm Basics 18 MRTS and elasticity of substitution Responsiveness of inputs to MRTS is elasticity of substitution prop change input ratio - = prop change in MRTS input-ratio MRTS input-ratio MRTS z2 ∂log(z1/z2) = ∂log(f1/f2) z2 s=½ s=2 z1 March 2012 z1 Frank Cowell: Firm Basics 19 Elasticity of substitution z2 A constant elasticity of substitution isoquant Increase the elasticity of substitution... structure of the contour map... z1 March 2012 Frank Cowell: Firm Basics 20 Homothetic contours z2 O March 2012 The isoquants Draw any ray through the origin… Get same MRTS as it cuts each isoquant z1 Frank Cowell: Firm Basics 21 Contours of a homogeneous function The isoquants z2 Coordinates of input z° Coordinates of “scaled up” input tz° tz° tz2° z2° f(tz) = trf(z) z° trq q O March 2012 z1° tz1° z1 Frank Cowell: Firm Basics 22 Overview... The Firm: Basics The setting Changing all inputs together Input requirement sets Isoquants Returns to scale Marginal products March 2012 Frank Cowell: Firm Basics 23 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 2012 Frank Cowell: Firm Basics 24 Case 1: IRTS An increasing returns to scale function Pick an arbitrary point on the surface q The expansion path… 0 t>1 implies f(tz) > tf(z) z2 Double inputs and you more than double output March 2012 Frank Cowell: Firm Basics 25 Case 2: DRTS A decreasing returns to scale function q Pick an arbitrary point on the surface The expansion path… 0 t>1 implies f(tz) < tf(z) z2 Double inputs and output increases by less than double March 2012 Frank Cowell: Firm Basics 26 Case 3: CRTS A constant returns to scale function Pick a point on the surface q The expansion path is a ray 0 f(tz) = tf(z) z2 Double inputs and output exactly doubles March 2012 Frank Cowell: Firm Basics 27 Relationship to isoquants q Take any one of the three cases (here it is CRTS) Take a horizontal “slice” Project down to get the isoquant Repeat to get isoquant map 0 z2 The isoquant map is the projection of the set of feasible points March 2012 Frank Cowell: Firm Basics 28 Overview The Firm: Basics The setting Changing one input at time Input requirement sets Isoquants Returns to scale Marginal products March 2012 Frank Cowell: Firm Basics 29 Marginal products • Pick a technically efficient input vector Remember, this means a z such that q= f(z) Keep all but one input constant Measure the marginal change in output w.r.t. this input • The marginal product f(z) MPi = fi(z) = —— zi . March 2012 Frank Cowell: Firm Basics 30 CRTS production function again q Now take a vertical “slice” The resulting path for z2 = constant z2 0 Let’s look at its shape March 2012 Frank Cowell: Firm Basics 31 MP for the CRTS function q f1(z) f(z) The feasible set Technically efficient points Slope of tangent is the marginal product of input 1 Increase z1… A section of the production function Input 1 is essential: If z1= 0 then q = 0 z1 March 2012 f1(z) falls with z1 (or stays constant) if f is concave Frank Cowell: Firm Basics 32 Relationship between q and z1 q q We’ve just taken the conventional case z1 But in general this curve depends on the shape of f Some other possibilities for the relation between output and one input… q z1 q z1 z1 March 2012 Frank Cowell: Firm Basics 33 Key concepts Review Technical efficiency Review Returns to scale Review Convexity Review MRTS Review Marginal product March 2012 Frank Cowell: Firm Basics 34 What next? Introduce the market Optimisation problem of the firm Method of solution Solution concepts March 2012 Frank Cowell: Firm Basics 35