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The Firm: Basics

The Firm: Basics. Microeconomia III (Lecture 1) Tratto da Cowell F. (2004), Principles of Microeoconomics. Overview. The Firm: Basics. The setting. The environment for the basic model of the firm. Input require-ment sets. Isoquants. Returns to scale. Marginal products.

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The Firm: Basics

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  1. The Firm: Basics Microeconomia III (Lecture 1) Tratto da Cowell F. (2004), Principles of Microeoconomics

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

  3. The basics of production... • We set out 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...

  4. Notation • Quantities zi • amount of input i z = (z1, z2 , ..., zm) • input vector q • amount of output For next presentation • Prices wi • price of input i w = (w1, w2 , ..., wm) • Input-price vector p • price of output

  5. Feasible production • The basic relationship between output and inputs: q £ f (z1, z2, ...., zm ) • single-output, multiple-input production relation The production function • This can be written more compactly as: q £f (z) • Note that we use “£” and not “=” in the relation. • Consider the meaning of f Vector of inputs • f gives the maximum amount of output that can be produced from a given list of inputs distinguish two important cases...

  6. Technical efficiency • Case 1: q =f (z) • The case where production is technically efficient • The case where production is (technically) inefficient • Case 2: q < f (z) Intuition: if the combination (z,q) is inefficient you can throw away some inputs and still produce the same output

  7. q >f(z) q <f(z) z2 z1 The function  q • The production function • Interior points are feasible but inefficient • Boundary points are feasible and efficient q =f(z) • Infeasible points 0 • We need to examine its structure in detail.

  8. Overview... The Firm: Basics The setting The structure of the production function. Input require-ment sets Isoquants Returns to scale Marginal products

  9. 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) ³ 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.

  10. q =f(z) The input requirement set • Feasible but inefficient z2 • Feasible and technically efficient • Infeasible points. Z(q) q <f(z) q >f(z) z1

  11. z2 q=f(z) q< f(z) q =f(z) z1 Case 1: Z smooth, strictly convex • Pick two boundary points • Draw the line between them • Intermediate points lie in the interior of Z. Z(q) • z¢ • Note important role of convexity. • A combination of two techniques may produce more output. • z² • What if we changed some of the assumptions?

  12. z2 z1 Case 2: Z Convex (but not strictly) • Pick two boundary points • Draw the line between them • Intermediate points lie in Z (perhaps on the boundary). Z(q) • z¢ • z² • A combination of feasible techniques is also feasible

  13. z2 This point is infeasible z1 Case 3: Z smooth but not convex • Join two points across the “dent” • Take an intermediate point • Highlight zone where this can occur. Z(q) • in this region there is an indivisibility

  14. z2 z1 Case 4: Z convex but not smooth q =f(z) • Slope of the boundary is undefined at this point.

  15. z2 z2 z2 z2 z1 z1 z1 z1 Summary: 4 possibilities for Z Standard case, but strong assumptions about divisibility and smoothness Almost conventional: mixtures may be just as good as single techniques Only one efficient point and not smooth. But not perverse. Problems: the "dent" represents an indivisibility

  16. Overview... The Firm: Basics The setting Contours of the production function. Input require-ment sets Isoquants Returns to scale Marginal products

  17. Isoquants • Pick a particular output level q • Find the input requirement set Z(q) • The isoquant is the boundary of Z: { z : f(z) = q } • Think of the isoquant as an integral part of the set Z(q)... • If the function f 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 fj (z) —— fi (z) ¶f(z) fi(z) := —— ¶zi . • Gives the rate at which you can trade off one input against another along the isoquant – to maintain a constant q. Let’s look at its shape

  18. z2 ° z2 z1 ° z1 Isoquant, input ratio, MRTS • The set Z(q). • A contour of the function f. • An efficient point. • The input ratio • Marginal Rate of Technical Substitution • Increase the MRTS z2 / z1= constant MRTS21=f1(z)/f2(z) • The isoquant is the boundary of Z. • z′ • Input ratio describes one particular production technique. • z° {z: f (z)=q} • Higher input ratio associated with higher MRTS..

  19. Input ratio and MRTS • MRTS21 is the implicit “price” of input 1 in terms of input 2. • The higher is this “price”, the smaller is the relative usage of input 1. • Responsiveness of input ratio to the MRTS is a key property of f. • Given by the elasticity of substitution ∂log(z1/z2)   ∂log(f1/f2) • Can be seen as the isoquant’s “curvature”

  20. Elasticity of substitution z2 • A constant elasticity of substitution isoquant • Increase the elasticity of substitution... structure of the contour map... z1

  21. Homothetic contours • The isoquants • Draw any ray through the origin… z2 • Get same MRTS as it cuts each isoquant. z1 O

  22. Contours of a homogeneous function • The isoquants z2 • Coordinates of input z° • Coordinates of “scaled up” input tz° • tz° ° tz2 f (tz) = trf (z) • z° ° z2 trq q z1 O O ° ° tz1 z1

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

  24. 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…

  25. Case 1: IRTS q • An increasing returns to scale function • Pick an arbitrary point on the surface • The expansion path… (z1 and z2 vary in the same proportion: z2/z1 constant) z2 0 • t>1 implies • f(tz) > tf(z) • Double inputs and you more than double output z1

  26. Case 2: DRTS q • A decreasing returns to scale function • Pick an arbitrary point on the surface • The expansion path… z2 0 • t>1 implies • f(tz) < tf(z) • Double inputs and output increases by less than double z1

  27. Case 3: CRTS q • A constant returns to scale function • Pick a point on the surface • The expansion path is a ray z2 0 • f(tz) = tf(z) • Double inputs and output exactly doubles z1

  28. Relationship to isoquants q • A constant returns to scale function • Pick a point on the surface • The expansion path is a ray • Take a horizontal “slice” • Project down to get the isoquant • Repeat to get isoquant map z2 0 • f(tz) = tf(z) • Double inputs and output exactly doubles z1

  29. 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 z2 0 • The isoquant map is the projection of the set of feasible points z1

  30. Overview... The Firm: Basics The setting Changing one input at time. Input require-ment sets Isoquants Returns to scale Marginal products

  31. 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.

  32. CRTS production function again q • Now take a vertical “slice” • The resulting path for z2 = constant z2 0 Let’s look at its shape z1

  33. MP for the CRTS function f1(z) • The feasible set q • Technically efficient points f(z) • Slope of tangent is the marginal product of input 1 • Increase z1… • A section of the production function • Input 1 is essential:Ifz1=0 thenq=0 • f1(z) falls withz1 (or stays constant) iffis concave z1

  34. q q q z1 z1 z1 q z1 Relationship between q and z1 • 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…

  35. Key concepts • Technical efficiency • Returns to scale • Convexity • MRTS • Marginal product Review Review Review Review Review

  36. What next? • Introduce the market • Optimisation problem of the firm • Method of solution • Solution concepts.

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