1 Chapter 7: Costs and Cost Minimization Consumers purchase GOODS to maximize their utility. This consumption depends upon a consumer’s INCOME and the PRICE of the goods Firms purchase INPUTS to produce OUTPUT This output depends upon the firm’s FUNDS and the PRICE of the inputs

2 Chapter 7: Costs and Cost Minimization In this chapter we will cover: 7.2 Isocost Lines 7.3 Cost Minimization 7.4 Short-Run Cost Minimization

3 One of the goals of a firm is to produce output at a minimum cost. This minimization goal can be carried out in two situations: 1)The long run (where all inputs are variable) 2)The short run (where some inputs are not variable)

4 Suppose that a firm’s owners wish to minimize costs… Let the desired output be Q 0 Technology: Q = f(L,K) Owner’s problem: min TC = rK + wL K,L Subject to Q 0 = f(L,K)

5 From the firm’s cost equation: TC 0 = rK + wL One can obtain the formula for the ISOCOST LINE: K = TC 0 /r – (w/r)L The isocost line graphically depicts all combinations of inputs (labour and capital) that carry the same cost.

6 L K TC 0 /w TC 1 /w TC 2 /w TC 2 /r TC 1 /r TC 0 /r Slope = -w/r Direction of increase in total cost Example: Isocost Lines

7 Isocost curves are similar to budget lines, and the tangency condition of firms is also similar to the tangency condition of consumers: MRTS L,K = -MP L /MP K = -w/r

8 L K TC 0 /w TC 1 /w TC 2 /w TC 2 /r TC 1 /r TC 0 /r Isoquant Q = Q 0 Example: Cost Minimization Cost minimization point for Q 0 Cost inefficient point for Q 0

9 1)Tangency Condition - MP L /MP K = w/r -gives relationship between L and K 2) Substitute into Production Function -solves for L and K 3) Calculate Total Cost 4) Conclude

10 Q = 50L 1/2 K 1/2 w = $5 r = $20 MP L = 25K 1/2 /L 1/2 Q 0 = 1000 MP K = 25L 1/2 /K 1/2 1) Tangency: MP L /MP K = w/r (25K 1/2 /L 1/2 )/(25L 1/2 /K 1/2 )=w/r K/L = 5/20 L=4K

11 2) Substitution: 1000 = 50L 1/2 K 1/ = 50(4K) 1/2 K 1/2 1000=100K K = 10 L = 4K L = 4(10) L = 40

12 3) Total Cost: TC 0 = rK + wL TC 0 = 20(10) + 5(40) TC 0 = 400 4) Conclude Cost is minimized at $400 when labour is 40 and capital is 10.

13 L K 400/w 400/r Isoquant Q = 1000 Example: Interior Solution Cost minimization point 10 40

14 Q = 10L + 2K MP L = 10 MP K = 2 w = $5 r = $2 Q 0 = 200 1) Tangency Condition: MP L /MP K = w/r 10/2=5/2 10=5???? Tangency condition fails!

15 We can rewrite the tangency condition: MP L /w = MP K /r -the productivity per dollar for labour is equal to the productivity per dollar for capital -but here: MP L /w = 10/5 > MP K /r = 2/2 …the “bang for the buck” in labour is ALWAYS larger than the “bang for the buck” in capital… So you would only use labor:

16 2) Substitution (K=0) Q = 10L + 2K 200 = 10L + 2(0) 20= L 3) Total Cost TC 0 = rK + wL TC 0 = 2(0) + 5(10) TC 0 = 50 4) Conclude Cost is minimized at $50 when labour is 5 and capital is zero.

17 Example: Cost Minimization: Corner Solution L K Isoquant Q = Q 0 Cost-minimizing input combination

18 Comparative Statistics The isocost line depends upon input prices and desired output Any change in input prices or output will shift the isocost line This shift will cause changes in the optimal choice of inputs

19 A change in the relative price of inputs changes the slope of the isocost line. If MRTS L,K is decreasing, An increase in wage: -decreases the cost minimizing quantity of labour -increases the cost minimizing quantity of capital An increase in rent -decreases the cost minimizing quantity of capital -increases the cost minimizing quantity of labour.

20 Example: Change in Relative Prices of Inputs L K Isoquant Q = Q 0 Cost minimizing input combination, w=1 r=1 Cost minimizing input combination w=2, r=1 0

21 Originally, MicroCorp faced input prices of $10 for both labor and capital. MicroCorp has a contract with its parent company, Econosoft, to produce 100 units a day through the production function: Q=2(LK) 1/2 MP L =(K/L) 1/2 MP K =(L/K) 1/2 If the price of labour increased to $40, calculate the effect on capital and labour.

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24 If the price of labour quadruples from $10 to $40…  Labour will be cut in half, from 50 to 25  Capital will double, from 50 to 100

25 An increase in Q 0 moves the isoquant Northeast.  The cost minimizing input combinations, as Q 0 varies, trace out the expansion path  If the cost minimizing quantities of labor and capital rise as output rises, labor and capital are normal inputs  If the cost minimizing quantity of an input decreases as the firm produces more output, the input is called an inferior input

26 Example: An Expansion Path L K TC 0 /w TC 1 /w TC 2 /w TC 2 /r TC 1 /r TC 0 /r Isoquant Q = Q 0 Expansion path, normal inputs

27 Example: An Expansion Path L K TC 1 /w TC 2 /w TC 2 /r TC 1 /r Expansion path, labour is inferior

28 Originally, MicroCorp faced input prices of $10 for both labor and capital. MicroCorp has a contract with its parent company, Econosoft, to produce 100 units a day through the production function: Q=2(LK) 1/2 MP L =(K/L) 1/2 MP K =(L/K) 1/2 If Econosoft demanded 200 units, how would labour and capital change?

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30 If the output required doubled from 100 to  Labour will double, from 50 to 100  Capital will double, from 50 to 100 (Constant Returns to Scale)

31 Input Demand Functions The demand curve for INPUTS is a schedule of amount of input demanded at each given price level This demand curve is derived from each individual firm minimizing costs: Definition: The cost minimizing quantities of labor and capital for various levels of Q, w and r are the input demand functions. L = L*(Q,w,r) K = K*(Q,w,r)

32 0L K L w L*(Q 0,w,r) Q = Q 0 W 1 /rW 2 /r W 3 /r L 1 L 2 L 3 Example: Labour Demand Function When input prices (wage and rent, etc) change, the firm maximizes using different combinations of inputs. As the price of inputs goes up, the firm uses LESS of that input, as seen in the input demand curve

33 0L K L w L*(Q 0,w,r) Q = Q 0 L 1 L 2 L 3 A change in the quantity produced will shift the isoquant curve. This will result in a shift in the input demand curve. Q = Q 1 L*(Q 1,w,r)

34 1)Use the tangency condition to find the relationship between inputs: MP L /MP K = w/r K=f(L) or L=f(K) 2) Substitute above into production function and solve for other variable: Q=f(L,K), K=f(L) =>L=f(Q) Q=f(L,K), L=f(K) =>K=f(Q)

35 Q = 50L 1/2 K 1/2 MP L =25(K/L) 1/2, MP K =25(L/K) 1/2 1)Tangency Condition: MP L /MP K = w/r K/L = w/r K=(w/r)L or L=(r/w)K

36 Labor and capital are both normal inputs Labor is a decreasing function of w Labor is an increasing function of r 2) Production Function Q 0 = 50L 1/2 K 1/2 Q 0 = 50L 1/2 [(w/r)L] 1/2 L*(Q,w,r) = (Q 0 /50)(r/w) 1/2 or Q 0 = 50 [(r/w)K] 1/2 K 1/2 K*(Q,w,r) = (Q 0 /50)(w/r) 1/2

37 Price elasticity of demand can be calculated for inputs similar to outputs:

38 JonTech produces the not-so-popular J-Pod. JonTech faces the following situation: Q*=5(KL) 1/2 =100 MRTS=K/L. w=$20 and r=$20 Calculate the Elasticity of Demand for Labour if wages drop to $5.

39 Initially: MRTS=K/L=w/r K=20L/20 K=L Q=5(KL) 1/2 100=5K 20=K=L

40 After Wage Change: MRTS=K/L=w/r K=5L/20 4K=L Q=5(KL) 1/2 100=10K 10=K 40=L

41 Price Elasticity of Labour Demand:

Short Run Cost Minimization Cost minimization occurs in the short run when one input (generally capital) is fixed (K*). Total variable cost is the amount spent on the variable input(s) (ie: wL) -this cost is nonsunk (can be avoided) Total fixed cost is the amount spent on fixed inputs (ie: rK*) -if this cost cannot be avoided, it is sunk -if this cost can be avoided, it is nonsunk (ie: rent factory to another firm)

43 Short Run Cost Minimization Cost minimization in the short run is easy: Min TC=wL+rK* L s.t. the constraint Q=f(L,K*) Where K* is fixed.

44 Short Run Cost Minimization Example: Minimize the cost to build 80 units if Q=2(KL) 1/2 and K=25. Q=2(KL) 1/2 80=2(25L) 1/2 80=10(L) 1/2 8=(L) 1/2 64=L Notice that price doesn’t matter.

45 Short Run Cost Minimization L K TC 1 /w TC 2 /w TC 2 /r TC 1 /r Long-Run Cost Minimization K* Short-Run Cost Minimization

46 Short Run Expansion Path Choosing 1 input in the short run doesn’t depend on prices, but it does depend on quantity produced. The short run expansion path shows the increased demand for labour as quantity produced increases: (next slide) The demand for inputs will therefore vary according to quantity produced. (The demand curve for inputs shifts when production changes)

47 Example: Short and Long Run Expansion Paths L K TC 0 /w TC 1 /w TC 2 /w TC 2 /r TC 1 /r TC 0 /r Short Run Expansion Path Long Run Expansion Path K*

48 Short Run and Many Inputs If the Short-Run Minimization problem has 1 fixed input and 2 or more variable inputs, it is handled similarly to the long run situation:

49 Chapter 7 Key Concepts  The Isocost line gives all combinations of inputs that have the same cost  Costs are minimized when the Isocost line is tangent to the Isoquant  When input costs change, the minimization point (and minimum cost) changes  When required output changes, the minimization point (and minimum cost) changes  The creates the expansion path

50 Chapter 7 Key Concepts  Individual firm choice drives input demand  As input prices change, input demanded changes  There are price elasticities of inputs  In the short run, at least one factor is fixed  Short run expansion paths differ from long run expansion paths