Producer Theory Flashcards
(102 cards)
1
Q
2 input production function
A
Q = F(K, L)
2
Q
What is the short run?
A
When at least 1 FOP is fixed - usually K is fixed and L is variable.
3
Q
MPL =
A
Addition to output from hiring one extra worker.
dQ/dL = F’L partial derivative
4
Q
What shows diminishing marginal returns to labour?
A
dMPL/dL < 0 I.e. Q”L < 0
Concave production function.
5
Q
The slope of the production function in the SR =
A
MPL
6
Q
Diminishing returns is a … concept?
A
SHORT RUN - adding more L to fixed K causes diminishing marginal returns to labour
7
Q
Average product of labour =
A
Output per worker
APL = Q/L
8
Q
Relationship between MPL and APL
A
MPL cuts through max of APL
When MPL > APL, APL is increasing
When MPL < APL, APL is decreasing
9
Q
TP is max where…
A
MPL = 0
10
Q
MPL is max where…
A
AT the inflection point of TP where dMPL/dL = 0
11
Q
APL is max where…
A
MPL = APL
Or APL’ = 0
12
Q
How can we use the TP function to find the APL?
A
Draw a ray from the origin to the TP curve. The slope of this ray gives the APL at that point.
13
Q
Isoquants show
A
All combinations of K and L that produce a fixed level of output c.
14
Q
MRTS =
A
Marginal rate of technical substitution.
MRTS = (-) MPL / MPK
The slope of the isoquant.
15
Q
As we move down the isoquant, what happens to MRTS?
A
It decreases in absolute value
16
Q
What is the total differential of the production function? Show how this gives us the formula for the MRTS.
A
dQ = F'L dL + F'K dK dQ = MPL dL + MPK dK dQ = 0 along an isoquant dK/dL = - MPL / MPK So MRTS = the slope
17
Q
MRTS shows…
A
How many units of K can be substituted for 1 extra unit of L keeping output constant.
18
Q
3 differences between isoquants and indifferent curves
A
- Utility is ordinal, output is cardinal
- Doubling values on an indifference map doesn’t change preference ordering; doubling all inputs associated with each isoquant changes technologically feasible set
- ICs arise from utility functions = objective; isoquants from production function = constraint.
19
Q
For a production function to be concave, we require… (2)
A
- Horizontal convexity - isoquants convex to origin
2. Vertical convexity - a chord between optimal points on first and last isoquant lies within the feasible set.
20
Q
Horizontal convexity refers to…
A
Diminishing MRTS = SR concept
21
Q
Vertical convexity refers to…
A
Returns to scale = LR concept (change both inputs)
22
Q
A quasi-concave production function =
A
- Horizontal convexity holds
- But no vertical convexity - chord lies outside choice set.
23
Q
Returns to scale measures…
A
% change in output / % change in inputs.
- how much output rises if we increase all inputs by the same proportion
24
Q
Why is returns to scale a LR concept?
A
Because we vary all inputs = np fixed inputs
25
How do we work out the degree of homogeneity?
F(tK, tL) = t^k F(K, L)
26
Degree of homogeneity tells us...
Returns to scale.
K = 1 means CRS
K > 1 means IRS
K < 1 means DRS
27
How do we work out returns to scale from a Cobb-Douglas function ?
K = sum of exponents
a + b = 1 means CRS
a + b > 1 means IRS
a + b < 1 means DRS
28
If we have CRS, what is the shape of the production function?
LINEAR
29
If the production function is homogenous of degree K, what Is the degree of homogeneity of the MRTS?
K - 1
30
When we have IRS, what shape is the production function?
Increasing slope = CONVEX
31
When we have DRS, what shape is the production function?
Decreasing slope = CONCAVE
| DRS = vertical convexity
32
Returns to scale vs diminishing returns
Returns to scale = LR concept, change K & L
| Diminishing returns = SR concept, change L while K fixed
33
Can we have IRS and diminishing marginal returns?
YES e.g. If exponents of cobb Douglas are both 2/3
2/3 + 2/3 = 4/3 > 1 therefore IRS
But 0 < 2/3 < 1 therefore diminishing returns once we find second partial derivatives.
34
Formula for perfect substitutes isoquant
Q = aL + bK
35
Slope of perfect substitutes isoquants
MRTS = a/b - constant = isoquants are straight lines
36
With perfect substitutes isoquants, returns to scale =
CONSTANT returns to scale.
37
Elasticity of substitution formula.
% change in K/L / % change in MRTS
38
Returns to scale for cobb Douglas are...
Can have CRS / IRS / DRS - depends on sum of exponents
39
Cobb Douglas is linear in...
Logs - MRTS is just alpha / beta
40
Formula for perfect compliments
Q(L, K) = MIN {aL, bK}
41
MRTS and shape of isoquants for perfect compliments
MRTS = infinity on vertical, zero on horizontal, undefined at kink.
L shaped isoquants.
42
Returns to scale for perfect compliments
CRS
43
Technological progress causes... (2)
1. The isoquant will shift downwards - can produce same output with fewer inputs.
2. The shape of the isoquant may also change.
44
Name 3 types of tech progress
1. Neutral
2. Labour saving
3. Capital saving
45
How does neutral tech progress affect isoquants?
Causes PARALLEL shift since shape unchanged - MPL & MPK affected equally.
46
How does labour saving tech progress affect isoquants?
Isoquants shift inwards
| MPK increases relative to MPL = MRTS decreases in absolute value = isoquants flatter.
47
How does capital saving tech progress affect isoquant?
Isoquants shift inwards
| MPL increases relative to MPK = MRTS increases in absolute value = more steep
48
Economic costs =
Explicit + implicit costs
49
Sunk costs =
Costs already incurred so cannot be avoided. They should not affect current decision making as already incurred.
50
Isocost curves show
All combinations of K & L that a firm can afford to purchase given fixed input prices w & r and a total allowable cost level. TC constant along a given isocost curve.
51
Slope of isocost line =
- w/r - ratio of input prices.
52
If w/r changes, what happens to isocost lines?
Pivots around the isoquant = change in slope + both intercept.
53
Firms 2 step problem
1. Minimise costs for a given level of output = L* and K*
| 2. Then choose output to maximise profit.
54
What is the objective & constraint for cost minimisation?
```
Objective = minimise TC
Constraint = output fixed at Q0
```
55
Diagrammatically, what does cost minimisation do?
We shift in the isocost curve until it is tangential to the isoquant. This minimises cost given that we must produce Q0 output.
56
All points on the isoquant are... but not necessarily...
All points are technologically efficient.
| But not all are economically efficient I..e cost minimising
57
Tangency condition for cost minimisation
Slope of isoquant = slope of isocost
| MRTS = MPL / MPK = w/r
58
Cost minimisation yields...
Conditional input demands - L* (r, w, Q0) and K* (r, w, Q0)
- depend on output
59
Min TC =
wL* + rK*
60
What does the Lagrange multiplier for cost minimisation represent?
Lambda = how much the optimal value changes if we relax the constraint marginally = how much TC increases if we increase output marginally = Marginal cost
61
Expansion path shows
The set of optimal combinations of L* and K* as we increase output and the isoquants shift out.
62
The expansion path is a straight line if
MRTS unchanged along optimal points = parallel shifts = homothetic.
63
How do we find corner solutions to cost minimisation?
Compare MPL/w and MPK/r.
| Whichever is higher use only that input and zero of the other.
64
What happens to the isocost/isoquant diagram if we change input prices?
Isoquant unchanged.
Isocost pivots around the isoquant = slope + both intercepts change
New optimal point for cost minimisation.
65
In order for change in input prices analysis to hold, what two assumptions do we need?
1) K, L > 0 = interior solution
| 2) Isoquants are convex
66
What causes a shift of the input demand curve?
If we increase output and shift isoquant out, the only way to reach this new isoquant is to all TC to rise = demand for labour rises at any given wage rate.
67
Substitution between K and L if perfect compliments. Implication for input demand curves.
Zero substitution. If input price changes, no change in optimal = perfectly inelastic input demand curves.
68
Substitution between K and L if perfect substitutes. Implication for input demand curves.
If W rises, move from all L to all K = infinite substitution
Perfect elastic input demand curve.
69
Homogeneity of TC function in input prices
K = 1 - double input prices = double TC
70
If we have CRS production function, our TC curve is...
| Economies of scale?
LINEAR - MC = AC = a constant
No Economies / diseconomies of scale
71
If we have DRS production function, our TC curve is...
| Economies of scale?
DRS = concave production function = convex cost function
Increasing gradient of TC curve
MC is increasing and AC increasing = diseconomies of scale.
72
MES where
MC = AC
73
If we have IRS production function, our TC curve is...
| Economies of scale?
IRS production function = convex = concave TC function
MC and AC are decreasing
Therefore Economies of scale
74
If the production function is concave, the TC function is
Convex
75
If input prices change, what happens to TC?
Change in input prices = pivot of isocost around isoquant = new optimal point substituting away from more expensive input. But TC may still rise .
76
MIN TC Cost minimisation in the SR
MIN TC = wL* + r(K bar)
- K is fixed
- we find L* given that K must be K bar
77
SR vs LR costs
SR costs tend to be higher than LR costs as K is fixed = we cannot reach cost minimising combination.
78
Expansion path in SR
Horizontal line at K bar
79
SR average total cost =
average fixed cost (rK bar) + average variable cost (wL*)
80
The long run cost curve is known as an...
Envelope curve - it envelopes the SR cost curves as we change K bar
81
Method for stage 2 of firms problem i.e. Profit max
Find Q* that maximises Pi*
Unconstrained maximisation - do not need Lagrangian
FOC: dPi/dQ = P - MC = 0
P = MC
82
Method for firms 1 step problem
```
Work out L* K* and Q* all at once. Plug in production function.
Pi = pF(K, L) - wL- rK
FOCs: (1) dPi/dL = 0 and (2) dPi/dK = 0
Use (1) to find expansion path for K
Then plug into (2)
```
83
The firms one step problem yields...
Unconditional factor demands L* (p, r, w) and K* (p, r, w)
| Do NOT depend on Q.
84
The supply and profit functions from one step problem
```
Q* = F(K*, L*)
Pi* = pQ* - wL* - rK*
```
85
How do we know whether to use the 2 step or 1 step problem?
2 step if Q asks for conditional factor demands
| 1 step if Q asks for unconditional factor demands / supply function
86
Profit is zero when... (2)
```
TR = TC
AR = AC
```
87
Profit is increasing in...
Price
88
Profit is decreasing in...
r and w (input prices)
89
Pi(r, w, p) is homogeneous of degree...
1 in p, w, r.
90
Isoprofit curves show
All combinations of output and inputs (with p, r, w fixed) that generate the same level of profit.
91
To find isoprofit curves, do we use 1 or 2 step problem?
use 1 step as we get unconditional input demands
92
What is on axes for isoprofit curves?
Y axis = output (Q)
| X axis = L=K or just L & assume K fixed
93
Formula for isoprofit lines
Pi = pQ - wL - r(K Bar)
rearrange to give
Q = Pi/p + (w/p) L + (r/p) K bar
94
Slope of isoprofit lines if we have L on X axis and assume K fixed
Slope = w/p
95
Tangency condition based on isoprofit diagram
Slope of isoprofit = slope of production function with just labour
W/p = MPL
W = p MPL
- Extra revenue from 1 more worker = cost of this worker
96
P MPL =
The marginal revenue product of labour
97
If we change L, by how much should we change Q so that we remain on the same isoprofit line?
Change in Q = MPL change in L.
98
A firm's LR supply curve =
LRMC curve above LRAC curve
99
A firm's supply curve in the SR =
MC curve above AVC
100
Shutdown condition in SR
Shutdown if AR (P) < AVC
101
Shape of supply curve depends on (2)
Shape of MC - SR or LR?
| Whether there are fixed costs.
102
How does the LR supply curve compare to the SR supply curve?
LR supply curve is flatter than the SR supply curve (due to K bar in SR)
