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 (the demand curve faced by the firm, 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)
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.
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.

5. Fig. 18.1

7. 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.
Isoquant looks very much like indifference curve, but you cannot label it arbitrarily, neither can you do any monotonic transformation of the label.

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

10. Fixed proportion (perfect complement): f(x1,x2)=min{x1,x2}

11. Fig. 18.2

12. Perfect substitutes: f(x1,x2)=x1+x2

13. Fig. 18.3

14. Cobb-Douglas f(x1,x2)=A(x1)a(x2)b, cannot normalize to a+b=1 arbitrarily

15. Some often-used assumptions 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, can free dispose of any extra inputs

16. 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]

17. Fig. 18.4

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

19. 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)

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

21. 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)

22. 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)

23. Short run: at least one factor of production is fixed
Long run: all factors of production can be varied
Can also plot the short run production function y=f(x1, k)

24. Fig. 18.5

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

26. 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)

27. MP, MRTS, returns to scale
labor, land
CRS, MP increasing of labor

28. 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