1. Unit 7.
Analyses of LR Production and Costs as Functions of Output (Ch. 5, 6, 8)

2. LR  Max  1. Produce Q where MR = MC 2. Minimize cost of producing Q
 optimal input combination

3. Isoquant
The combinations of inputs (K, L) that yield the producer the same level of output.
The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output.

4. Typical Isoquant

5. SR Production in LR Diagram

7. MRTS and MP MRTS = marginal rate of technical substitution
= the rate at which a firm must substitute one input for another in order to keep production at a given level
= - slope of isoquant =
= the rate at which capital can be exchanged for 1 more (or less) unit of labor
MPK = the marginal product of K =
MPL = the marginal product of L =
Q = MPK K + MPL L
Q = 0 along a given isosquant
 MPK K + MPL L = 0
 = ‘inverse’ MP ratio

8. Indifference Curve & Isoquant Slopes
Indiff Curve
Isosquant
- slope = MRS
= rate at which consumer is willing to exch Y for 1X in order to hold U constant
= inverse MU ratio
= MUX/MUY
For given indiff curve, dU = 0
Derived from diff types of U fns:
Cobb Douglas  U = XY
Perfect substitutes  U=X+Y
Perfect complements  U = min [X,Y]
- slope = MRTS
= rate at which producer is able to exch K for 1L in order to hold Q constant
= inverse MP ratio
= MPL/MPK
For given isoquant, dQ = 0
Derived from diff types of production fns:
Cobb Douglas  Q = LK
Perfect substitutes  Q=L+K
Perfect complements  Q = min [X,Y]

9. Cobb-Douglas Isoquants
Inputs are not perfectly substitutable
Diminishing marginal rate of technical substitution
Most production processes have isoquants of this shape

10. Linear Isoquants
Capital and labor are perfect substitutes

11. Leontief Isoquants Capital and labor are perfect complements
Capital and labor are used in fixed-proportions

12. Budget Line = maximum combinations of 2 goods
that can be bought given one’s income
= combinations of 2 goods whose cost
equals one’s income

13. Isocost Line = maximum combinations of 2 inputs
that can be purchased given a
production ‘budget’ (cost level)
= combinations of 2 inputs that are
equal in cost

14. Isocost Line Equation TC1 = rK + wL  rK = TC1 – wL  K =
Note: slope = ‘inverse’ input price ratio =
= rate at which capital can be exchanged for
1 unit of labor, while holding costs constant.

15. Increasing Isocost

16. Changing Input Prices

17. Different Ways (Costs) of Producing q1

18. Cost Minimization (graph)

19. LR Cost Min (math)  - slope of isoquant = - slope of isocost line  

20. SR vs LR Production

23. Assume a production process:
Q = 10K1/2L1/2
Q = units of output
K = units of capital
L = units of labor
R = rental rate for K = $40
W = wage rate for L = $10

24. Given q = 10K1/2L1/2 Q K L TC=40K+10L 40* 2* 8* 160* 100* 5* 20* 400*
3.2
232
100 2 50
580
* LR optimum for given q

25. Given q = 10K1/2L1/2, w=10, r=40
Minimum LR Cost Condition
 inverse MP ratio = inverse input P ratio
 (MP of L)/(MP of K) = w/r
 (5K1/2L-1/2)/(5K-1/2L1/2) = 10/40
 K/L = ¼
 L = 4K

26. Optimal K for q = 40? (Given L* = 4K*)
q = 40 = 10K1/2L1/2
 40 = 10 K1/2(4K)1/2
 40 = 20K
 K* = 2
 L* = 8
 min SR TC = 40K* + 10L*
= 40(2) + 10(8)
= = $160

27. SR TC for q = 40? (If K = 5)
q = 40 = 10K1/2L1/2
 40 = 10 (5)1/2(L)1/2
 L = 16/5 = 3.2
 SR TC = 40K + 10L
= 40(5) + 10(3.2)
= = $232

28. Optimal K for q = 100? (Given L* = 4K*)
Q = 100 = 10K1/2L1/2
 100 = 10 K1/2(4K)1/2
 100 = 20K
 K* = 5
 L* = 20
 min SR TC = 40K* + 10L*
= 40(5) + 10(20)
= = $400

29. SR TC for q = 100? (If K = 2) Q = 100 = 10K1/2L1/2
 SR TC = 40K + 10L
= 40(2) + 10(50)
= = $580

30. Two Different costs of q = 100

31. LRTC Equation Derivation [i.e. LRTC=f(q)]
 LRTC = rk* + wL*
= r(k* as fn of q) + w(L* as fn of q)
To find K* as fn q
from equal-slopes condition L*=f(k), sub f(k) for L into production fn and solve for k* as fn q
To find L* as fn q
from equal-slopes condition L*=f(k), sub k* as fn of q for f(k) deriving L* as fn q

32. LRTC Calculation Example
Assume q = 10K1/2L1/2, r = 40, w = 10
L* = 4K (equal-slopes condition)
K* as fn q
q = 10K1/2(4K)1/2
= 10K1/22K1/2
= 20K 
LR TC = rk* + wL* = 40(.05q)+10(.2q)
= 2q + 2q
= 4q
L* as fn q
L* = 4K*
= 4(.05 q)
L* = .2q

33. Graph of SRTC and LRTC

34. Expansion Path  LRTC

35. 

36. Technological Progress

37. Multiplant Production Strategy
Assume:
P = output price = qT
qT = total output (= q1+q2)
q1 = output from plant #1
q2 = output from plant #2
 MR = 70 – (q1+q2)
TC1 = (q1)2  MC1 = 3q1
TC2 = (q2)2  MC2 = q2

39. Multiplant  Max (#1) MR = MC1 (#2) MR = MC2 (#1) 70 – (q1 + q2) = 3q1
from (#1), q2 = 70 – 4q1
Sub into (#2),
 70 – (q – 4q1) = 70 – 4q1
 7q1 = 70
 q1 = 10, q2 = 30
 = TR – TC1 – TC2
= (50)(40)
- [ (10)2]
- [ (30)2]
= 2000 – 250 – 750 = $1000

40.  If q1 = q2 = 20?  = TR - TC1 - TC2 = (50)(40) - [100 + 1.5(20)2]
- [ (20)2]
= 2000 – 700 – 500 = $800

41. Multi Plant Profit Max (alternative solution procedure)
1. Solve for MCT as fn of qT
knowing cost min  MC1=MC2=MCT
 MC1=3q1  q1 = 1/3 - MC1 = 1/3 MCT
MC2 = q q2 = MC = MCT
 q1+q2 = qT = 4/3 MCT
 MCT = ¾ qT
2. Solve for profit-max qT
 MR=MCT
 70-qT = ¾ qT
 7/4 qT = 70
 q*T = 40  MC*T = ¾ (40) = 30

42. Multi Plant Profit Max (alternative solution procedure)
3. Solve for q*1 where MC1 = MC*T
 3q1 = 30
 q*1 = 10
4. Solve for q*2 where MC2 = MC*T
 q*2 = 30

43. Graph  Max, 2 Plants (linear MCs)

44.  Max (?), 2 Plants, nonlinear MCs