The Firm’s Decision in Space

1. The Firm’s Decision in Space

2. Production theory • A firm is characterized by it’s technology represented by the production function • Y=f(x1, x2) • It is a price taker in the products and inputs markets and it faces prices py w1 and w2. • It chooses how much to produce given these prices in order to maximize profits • Two step process • minimize the cost of producing any given output • Choses output optimally given prices and it’s marginal cost function

3. Cost Minimization • A firm first computes how to produce any given output level y ³ 0 at smallest possible total cost. • c(y) denotes the firm’s smallest possible total cost for producing y units of output. • c(y) is the firm’s total cost function. • When the firm faces given input prices w = (w1,w2,…,wn) the total cost function will be written as c(w1,…,wn,y).

4. The Cost-Minimization Problem • Consider a firm using two inputs to make one output. • The production function isy = f(x1,x2). • Take the output level y ³ 0 as given. • Given the input prices w1 and w2, the cost of an input bundle (x1,x2) is w1x1 + w2x2.

5. The Cost-Minimization Problem • For given w1, w2 and y, the firm’s cost-minimization problem is to solve subject to

6. The Cost-Minimization Problem • The levels x1*(w1,w2,y) and x2*(w1,w2,y) in the least-costly input bundle are the firm’s conditional demands for inputs 1 and 2. • The (smallest possible) total cost for producing y output units is therefore

7. Iso-cost Lines x2 c” º w1x1+w2x2 c’ º w1x1+w2x2 c’ < c” x1

8. Iso-cost Lines x2 Slopes = -w1/w2. c” º w1x1+w2x2 c’ º w1x1+w2x2 c’ < c” x1

9. The y’-Output Unit Isoquant x2 All input bundles yielding y’ unitsof output. Which is the cheapest? f(x1,x2) º y’ x1

10. The Cost-Minimization Problem x2 All input bundles yielding y’ unitsof output. Which is the cheapest? f(x1,x2) º y’ x1

11. The Cost-Minimization Problem x2 All input bundles yielding y’ unitsof output. Which is the cheapest? f(x1,x2) º y’ x1

12. The Cost-Minimization Problem x2 All input bundles yielding y’ unitsof output. Which is the cheapest? f(x1,x2) º y’ x1

13. The Cost-Minimization Problem x2 All input bundles yielding y’ unitsof output. Which is the cheapest? x2* f(x1,x2) º y’ x1* x1

14. The Cost-Minimization Problem At an interior cost-min input bundle:(a) x2 x2* f(x1,x2) º y’ x1* x1

15. The Cost-Minimization Problem At an interior cost-min input bundle:(a) and(b) slope of isocost = slope of isoquant x2 x2* f(x1,x2) º y’ x1* x1

16. The Cost-Minimization Problem At an interior cost-min input bundle:(a) and(b) slope of isocost = slope of isoquant; i.e. x2 x2* f(x1,x2) º y’ x1* x1

17. A Cobb-Douglas Example of Cost Minimization • A firm’s Cobb-Douglas production function is • Input prices are w1 and w2. • What are the firm’s conditional input demand functions?

18. A Cobb-Douglas Example of Cost Minimization At the input bundle (x1*,x2*) which minimizesthe cost of producing y output units: (a)(b) and

19. A Cobb-Douglas Example of Cost Minimization (a) (b)

20. A Cobb-Douglas Example of Cost Minimization (a) (b) From (b),

21. A Cobb-Douglas Example of Cost Minimization (a) (b) From (b), Now substitute into (a) to get

22. A Cobb-Douglas Example of Cost Minimization (a) (b) From (b), Now substitute into (a) to get

23. A Cobb-Douglas Example of Cost Minimization (a) (b) From (b), Now substitute into (a) to get So is the firm’s conditional demand for input 1. Is the conditional demand for input 2

24. A Cobb-Douglas Example of Cost Minimization So the cheapest input bundle yielding y output units is

25. A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is

26. Output Decision • Marginal cost equals marginal revenue • In the case of price takers the marginal revenue is the price of output • Marginal cost is the derivative of C(y) with respect to y • Marginal cost curve is also known as the supply curve

27. Taking Space into Account • The firm is now characterized by the technology and it is still a price taker in all markets • It can buy inputs at constant location-specific prices. • It sells at a fixed output price • Locations are spatially separated and the firm incurs linear transportation costs

28. Two Inputs and One Market • consider the decision of a locational unit with two transferable inputs (x1 located at S1 and x2 located at S2) and one transferable output with a market located at M. • Limit consideration to locations I and J, which are equidistant from the market • The arc IJ includes additional locations at that same distance from the market

29. Visually:

30. Incorporating Distance Into Prices • Their delivered prices are respectively p’1=p1 + r1d1       and p’2=p2 + r2d2 • where p1 and p2 are the prices of each input at is source, • r1 and r2 represent transfer rates per unit distance for these inputs. • The distance from each source to a particular location such as I or J is given by d1and d2. • Location I is closer than J to the source of x1, but farther away from the source of x2. • So x1 is relatively cheaper at I and x2 is relatively cheaper at J.

31. Iso-outlay lines • The total outlay (TO) of the locational unit on transferable inputs is TO=p’1x1 +p’2x2                          (2)  This equation may be reexpressed as • x1=(TO / p’1) – (p’2 / p’1)x2                 (3) • For any given total outlay (TO), the possible combinations of the two inputs that could be bought are determined by equation (2), • These can be plotted by equation (3) as an iso-outlay line

32. Iso-Outlay lines (contd.) • The iso-outlay line is linear. It has the form x1=a + ßx2, where the slope (ß) is - (p'2/p'1), and the vertical intercept (a ) is (TO/p'1) • Locations have different sets of delivered prices, so the combinations of inputs x1and x2 that any given outlay TO can buy vary by location

33. Location Decision and Inputs

34. The choice • Consider the isoquant Q0, it indicates all possible combinations of inputs that produce that quantity. • It is clear here that the cheapest way to produce Q0 is to locate in I