The Firm: Basics

1. The Firm: Basics MICROECONOMICS Principles and Analysis Frank Cowell October 2011

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... • 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. Why? • 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 constant output 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 production technique • z° {z: f (z)=q} • MRTS21: implicit “price” of input 1 in terms of 2 • Higher “price”: smaller relative use of input 1

19. z2 z2 z1 z1 MRTS and elasticity of substitution ∂log(z1/z2) =   ∂log(f1/f2) • Responsiveness of inputs to MRTS is elasticity of substitution prop change input ratio   prop change in MRTS D input-ratio D MRTS =     input-ratio MRTS s = ½ s = 2

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… 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 • 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

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

30. 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.

31. 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

32. 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

33. 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…

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

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