Chapter 19 Technology Key Concept: Production function

1. Chapter 19 Technology • Key Concept: Production function • On production functions, we could define some concepts which has close parallels in consumer theory. • MP MU • MRTS MRS • RTS

2. Chapter 19 Technology • First understand the technology constraint of a firm. • Later we will talk about constraints imposed by consumers and firm’s competitors, i.e., the demand curve faced by the firm or the market structure.

3. Inputs: labor and capital • Inputs and outputs measured in flow units, i.e., how many units of labor per week, how many units of output per week, etc.

4. Consider the case of one input (x) and one output (y). • To describe the tech constraint of a firm, list all the technologically feasible ways to produce a given amount of outputs.

5. The set of all combinations of inputs and outputs that comprise a technologically feasible way to produce is called a production set. • A production function measures the maximum possible output that you can get from a given amount of input.

6. Fig. 18.1

8. Isoquant is another way to express the production function. • It is a set of all possible combinations of inputs that are just sufficient to produce a given amount of output.

10. Isoquant looks very much like indifference curve, but you cannot label it arbitrarily. • Neither can you do any monotonic transformation of the label.

11. Some useful examples of production function. Two inputs, x1 and x2.

12. Fixed proportion (perfect complement) • Suppose we are producing holes and the only way to get a hole is to use one man and one shovel. Extra shovels aren’t worth anything and neither are extra men. • f(x1,x2)=min{x1,x2}

13. Fig. 18.2

14. Perfect substitutes • Suppose we are producing homework and the inputs are red pencils and blue pencils. The amount of homework produced depends only on the total number of pencils. • f(x1,x2)=x1+x2

15. Fig. 18.3

16. Cobb-Douglas f(x1,x2)=A(x1)a(x2)b, cannot normalize to a+b=1 arbitrarily • A measures the scale of production: how much output we would get if we used one unit of each input. • a and b measure how the amount of output responds to changes in the inputs.

17. Some often-assumed properties on the production function • Monotonicity: if you increase the amount of at least one input, you produce at least as much output as before • Monotonicity holds because of free disposal, that is, the firm can free dispose of any extra inputs

18. Convexity: if y=f(x1,x2)=f(z1,z2), then f(tx1+(1-t)z1,tx2+(1-t)z2)y for any t[0,1] • If you have two ways, (x1,x2) and (z1,z2), to produce y units of output, the weighted average will produce at least y units of output.

19. Suppose you could produce 1 unit of output using a1 units of factor 1 and a2 units of factor 2. • You have another way to produce 1 unit,using b1 units of factor 1 and b2 units of factor 2.

20. If you could scale up the output so (100a1, 100a2) and (100b1, 100b2) will both produce 100 units of output. • If you have (25a1+75b1, 25a2,+75b2), then you might produce 100 units by using 25 units of a technology and 75 units of b technology

21. Fig. 18.4

22. In this kind of technology, where you can scale the production process up and down easily and where separate production processes don’t interfere with each other, convexity is a natural assumption.

23. Some terms often used to describe the production function. • Marginal product: operate at (x1,x2), increase a bit of x1and hold x2, how much more y can we get per additional unit of x1?

24. Marginal product of factor 1: MP1(x1,x2)=∆y/∆x1=(f(x1+ ∆x1,x2)-f(x1,x2))/∆x1 (it is a rate, just like MU)

25. Marginal rate of technical substitution factor 1 for factor 2: operate at (x1,x2), increase a bit of x1and hold y, how much less x2 can you use? • Measures the trade-off between two inputs in production • MRTS1,2(x1,x2)=∆x2/∆x1=?

26. MRTS1,2(x1,x2)=∆x2/∆x1=? • y=f(x1,x2) • ∆y=MP1(x1,x2)∆x1+MP2(x1,x2)∆x2=0 • MRTS1,2(x1,x2)=∆x2/∆x1=-MP1(x1,x2)/MP2(x1,x2) (it is a slope, just like MRS)

27. Law of diminishing marginal product: holding all other inputs fixed, if we increase one input, the marginal product of that input becomes smaller and smaller (diminishing MU) • Diminishing MRTS: the slope of an isoquant decreases in absolute value as we increase x1 (diminishing MRS)

28. Short run: at least one factor of production is fixed • Ex: A farmer in the SR is stuck with the amount of land he has. • Long run: all factors of production can be varied • Ex: The farmer can adjust the level of the land input so as to max his profit.

30. Fig. 18.5

31. Returns to scale: if we use twice as much of each input, how much output will we get?

32. constant returns to scale (CRS): for all t>0, f(tx1,tx2)=tf(x1,x2) • Idea is if we double the inputs, we can just set two plants and so we can double the outputs • Increasing returns to scale (IRS): for all t>1, f(tx1,tx2)>tf(x1,x2) • Decreasing returns to scale (DRS): for all t>1, f(tx1,tx2)<tf(x1,x2)

34. MP, MRTS, returns to scale • labor, land • CRS, increasing MP labor • A farmer can grow the crop for the whole word in a small pot?

35. Let us think using Cobb-Douglas. • f(x1,x2)=A(labor)a(land)b • a+b=? • a>? • b<?

36. Chapter 19 Technology • Key Concept: Production function • On production functions, we could define some concepts which has close parallels in consumer theory. • MP MU • MRTS MRS • RTS