#### Transcript Firm: Basics

Frank Cowell: Microeconomics October 2011 The Firm: Basics MICROECONOMICS Principles and Analysis Frank Cowell Overview... The Firm: Basics Frank Cowell: Microeconomics The setting The environment for the basic model of the firm. Input requirement sets Isoquants Returns to scale Marginal products The basics of production... Frank Cowell: Microeconomics Some of the 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... Notation Frank Cowell: Microeconomics Quantities zi z = (z1, z2 , ..., zm ) •amount of input i q •amount of output •input vector For next presentation Prices wi w = (w1, w2 , ..., wm ) •price of input i p •price of output •Input-price vector Feasible production Frank Cowell: Microeconomics The basic relationship between The production output and function inputs: •single-output, multiple-input production relation q f(z1, z2, ...., zm ) This can be written more compactly •Note that we use “” and not Vector of inputs as: “=” in the relation. Why? q f(z) •Consider the meaning of f f gives the maximum amount of output that can be produced from a given list of inputs distinguish two important cases... Technical efficiency Frank Cowell: Microeconomics 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 The function f Frank Cowell: Microeconomics q q >f (z) q =f (z) 0 q <f The production function Interior points are feasible but inefficient Boundary points are feasible (z) and efficient Infeasible points z2 We need to examine its structure in detail. Overview... The Firm: Basics Frank Cowell: Microeconomics The setting The structure of the production function. Input requirement sets Isoquants Returns to scale Marginal products The input requirement set Frank Cowell: Microeconomics 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: f(z) q} The shape of Z depends on the assumptions made about production... We will look at four cases. remember, we must have q f(z) The set of input vectors that meet the technical feasibility condition for output q... First, the “standard” case. The input requirement set Frank Cowell: Microeconomics Feasible but inefficient z2 Feasible and technically efficient Infeasible points. Z(q) q < f (z) q = f (z) q > f (z) z1 Case 1: Z smooth, strictly convex Frank Cowell: Microeconomics Pick two boundary points Draw the line between them z2 Intermediate points lie in the interior of Z. Z(q) q = f (z') z Note important role of convexity. q< f (z) A combination of two techniques may produce more output. z q = f (z") z1 What if we changed some of the assumptions? Case 2: Z Convex (but not strictly) Frank Cowell: Microeconomics Pick two boundary points Draw the line between them z2 Intermediate points lie in Z (perhaps on the boundary). Z(q) z z z1 A combination of feasible techniques is also feasible Case 3: Z smooth but not convex Frank Cowell: Microeconomics Join two points across the “dent” Take an intermediate point z2 Highlight zone where this can occur. Z(q) This point is infeasible in this region there is an indivisibility z1 Case 4: Z convex but not smooth Frank Cowell: Microeconomics z2 q = f (z) Slope of the boundary is undefined at this point. z1 Summary: 4 possibilities for Z Frank Cowell: Microeconomics Standard case, but strong assumptions about divisibility and smoothness z2 z1 z2 z1 Problems: the "dent" represents an indivisibility z1 Almost conventional: mixtures may be just as good as single techniques z2 Only one efficient point and not smooth. But not perverse. z2 z1 Overview... The Firm: Basics Frank Cowell: Microeconomics The setting Contours of the production function. Input requirement sets Isoquants Returns to scale Marginal products Isoquants Frank Cowell: Microeconomics Pick a particular output level q Find the input requirement set Z(q) The isoquant is the boundary of Z: Think of the isoquant as an integral part of the set Z(q)... { z : f (z) = q } If the function f is differentiable at z Where appropriate, use subscript to denote partial then the marginal rate of technical derivatives. So substitution is the slope at z: fj (z) f(z) —— f (z) := —— i fi (z) zi . Gives the rate at which you can trade off one input against another along the isoquant, to maintain constant output q Let’s look at its shape Isoquant, input ratio, MRTS Frank Cowell: Microeconomics The set Z(q). A contour of the function f. 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° 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 MRTS and elasticity of substitution Frank Cowell: Microeconomics z2 Responsiveness propsubstitution change input ratio prop change in MRTS of inputs to MRTS is elasticity of D input-ratio D MRTS = input-ratio MRTS z2 s=½ ∂log(z1/z2) = ∂log(f1/f2) s=2 z1 z1 Frank Cowell: Microeconomics Elasticity of substitution z 2 A constant elasticity of substitution isoquant Increase the elasticity of substitution... structure of the contour map... z1 Homothetic contours Frank Cowell: Microeconomics The isoquants Draw any ray through the origin… z2 Get same MRTS as it cuts each isoquant. O z1 Contours of a homogeneous function Frank Cowell: Microeconomics The isoquants z2 Coordinates of input z° Coordinates of “scaled up” input tz° tz° tz2° z2° f(tz) = trf(z) z° trq q O z1° tz1° z1 Overview... The Firm: Basics Frank Cowell: Microeconomics The setting Changing all inputs together. Input requirement sets Isoquants Returns to scale Marginal products Let's rebuild from the isoquants Frank Cowell: Microeconomics 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: Take three standard cases: Focus on the expansion path. q plotted against proportionate increases in z. Increasing Returns to Scale Decreasing Returns to Scale Constant Returns to Scale Let's do this for 2 inputs, one output… Case 1: IRTS Frank Cowell: Microeconomics q An increasing returns to scale function Pick an arbitrary point on the surface The expansion path… 0 z2 t>1 implies f(tz) > tf(z) Double inputs and you more than double output Case 2: DRTS Frank Cowell: Microeconomics q A decreasing returns to scale function Pick an arbitrary point on the surface The expansion path… 0 z2 t>1 implies f(tz) < tf(z) Double inputs and output increases by less than double Case 3: CRTS Frank Cowell: Microeconomics q A constant returns to scale function Pick a point on the surface The expansion path is a ray 0 z2 f(tz) = tf(z) Double inputs and output exactly doubles Relationship to isoquants Frank Cowell: Microeconomics 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 Overview... The Firm: Basics Frank Cowell: Microeconomics The setting Changing one input at time. Input requirement sets Isoquants Returns to scale Marginal products Marginal products Frank Cowell: Microeconomics Remember, this means a z such that q= f(z) Pick a technically efficient input vector Keep all but one input constant Measure the marginal change in output w.r.t. this input f(z) MPi = fi(z) = —— zi . The marginal product CRTS production function again Frank Cowell: Microeconomics q Now take a vertical “slice” The resulting path for z2 = constant 0 z2 Let’s look at its shape MP for the CRTS function Frank Cowell: Microeconomics 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 f1(z) falls with z1 (or stays constant) if f is concave Relationship between q and z1 Frank Cowell: Microeconomics 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 z1 q z1 Key concepts Frank Cowell: Microeconomics Review Review Review Review Review Technical efficiency Returns to scale Convexity MRTS Marginal product What next? Frank Cowell: Microeconomics Introduce the market Optimisation problem of the firm Method of solution Solution concepts.