Updated: 07 May, 2007
MICRO ECONOMICS
(ECON 601)
Lecture 8
Topics to be covered:
a- Firm’s Demands for Inputs
b- Mathematical Derivation Input Demand Functions
c- Factor Demands: Substitution and Output Effects
d- Elasticities of Demand for Inputs
ECON 601: ADVANCED micro economics
Nicholson, Chapter 21 (7th Ed.)
Firm’s Demands for Inputs
Derived Demand for Factor Inputs
Profit = TR (K, L) – TC (K, L)
First order conditions for maximizing profits,
= –= 0
= – = 0
=
=
Marginal Revenue Product (MRP)
MRPL = = = MR. MPL
MR = Marginal Revenue
The marginal revenue product from hiring an extra unit of any input is the extra revenue yielded by selling the output (marginal product) the extra unit of input produces.
MRPL = MR. MPL
MRPK = MR. MPK
Marginal Expense is equal to the change in total cost if an addition unit of input is used.
= n
= w
First order conditions for profit maximizing is simply,
MRPL = w
MRPK = n
If Firm is a Price Taker:
P = MR Therefore, Px. MPK = n
Px. MPL = w
What is the direction and size of ?
If w = PxMPL
If w¯ then MPL must fall. This occurs if L.
Single Input Case
Start with w = PxMPL
dw = Px dw
Divided by dw,
1 = Px
=
As, < 0 then < 0
Example,
Suppose production function.
Qx = 100 Px = $ 50/lb
TR= Px Qx = Px (100 ) = 5000
= 2500 L-1/2
If w = $500, then we have,
500 = 2500 L-1/2
L+1/2 500 = 2500
L+1/2 = = 5
L = 25
If w = $250, then we have,
250 = 2500 L-1/2
L+1/2 250 = 2500
L+1/2 = = 10
L = 100
If Px = $60/lb
TRs = 60 (100 ) = 6000
= 3000 L-1/2
If w = $500, then we have,
500 = 3000 L-1/2
L+1/2 500 = 3000
L+1/2 = = 6
L = 36
Two-Input Case
– If w falls:
(a) There will be a new mix of capital and labour. The quantities employed of both factors will change substitution effect.
(b) Output will also change as the MC will fall hence further increasing the demand for the input.
Substitution Effect Output Effect
It is not correct to hold output constant and first consider the substitution effect alone. A change in the wage rate changes relative factor prices changing the firms expansion path. It will also shift the firms cost curves and the profit maximizing level of output will change.
If industry price is not fixed then price will fall as more output is put on the market.
Mathematical Derivation input demand functions
L = L (Px, w, n)
K = K (Px, w, n)
= (q constant) + (firm changes in q)
Substitution Effect
By partial differentiating with the total cost function with respect to w (Shepard’s Lemma) we get the demand for an input keeping q constant,
= L’ (q, w, n)
The responsiveness of L to a change in w while holding q constant will depend on the elasticity of substitution that characterizes the firm’s production function. Extremes are sKL = ¥ and sKL = 0.
In short run substitution possibilities are not readily available but over time substitution between K and L will occur.
Let us denote hLL as the constant output wage elasticity of demand – substitution effect only.
hLL = (q constant)
hKK = (q constant)
It will be shown in the next lecture that,
hLL = - (1 – SL) sKL for the two factor case where s is the elasticity of substitution of the production function and SL is the share of labor in total costs.
For the many factor case:
hi0 = = S0si0
Output Effects
(for changes in q) = . . .
+ (-) + +
Overall (-)
eL (for changes in q) = (eLq ) (eqP ) (ePMC) (eMCw)
Output Effect will depend on:
(1) How large are the increase in costs that are being caused by the change in factor prices – depends on share of input in total cost.
(2) How much will the quantity demanded be changed by the change in prices of the output – depends on the elasticity of demand.
Output Effect
eLw (from changes in q) = eLq . eqp . epMC . eMCw
ln the long run with constant returns to scale,
eLq = 1 epMC = 1 eMCw = SL
Therefore, eLw = SL eqp
Total Impact of Changes in Wage
eLw = hLL + SL eqp = - (1 – SL) sKL + SL eqp
Where, s =
Suppose, sKL = 0.6 SL = 0.7 eqp = –1.2
eLw = – (1 – 0.7) 0.6 + .7 (-1.2)
= – 0.18 + – 0.84 = –1.02
Factor Demands: Substitution and Output Effects
Example 21.2 and 21.3
Starting from example 13.3 page 379: Computation of Supply Functions
Suppose we have production function,
q = 10 K0.25 L0.25 F0.5
Where F refers to the seating capacity of a restaurant which in the short run is fixed at F = 16. The rent for the space is R.
Therefore, short run production function equals,
q = 10 K0.25 L0.25 (16)0.5
= 40 K0.25 L0.25
Profits are given by,
P = Pq – TC
= P (40K0.25 L0.25) – vK – wL – R (where R is fixed)
(1) = 10P(K–0.75 L0.25) - v = 0
(2) = 10P(K0.25 L–0.75) - w = 0
(3) \10P K–0.75 L0.25 = v
(4) \10P K0.25 L–0.75 = w
(5) From 3/4,
(6) =
(7) =
(8) L = K or,
(9) K = L
Substitute (8) into (3)
10P K–0.75 (K)0.25 = n
(10) K–0.75 (K)0.25 =
Substitute (9) K = L() for K into (4)
10P (L)0.25 L–0.75 = w
(11) (L)0.25 L–0.75=
Now, from (10) we have,
=
=
=
Squaring both sides we have,
=
(10P)2 = K
K =
Hence, we have the demand function for capital is:
K =
The following same procedure, we derive the demand function for labor. Starting with first order condition: 10P K0.25 L–0.75 = w, and the condition that for cost minimization, = or K = L. Substituting for K in the first order conditions, we have:
=
=
=
Square both sides,
=
(10P)2 = L
L = (10P)2
Hence, the demand function for Labor is:
\L =
We have production function,
q = 40 K0.25 L0.25
Let’s substitute the value of K and L,
q = 40
= 40
= 40
q =
This is the supply curve for q expressed as a function of P, v, w.
If, w = 4 and v = 4, then production function (supply curve) becomes,
q =
q =
q = 100P
If P = 1, then q = 100.
Given the demand function for labor of L =
If p = 1, w = 4, v = 4, the demand for Labor is,
L = = = = = = 6.25 workers
Let’s see, if, w = 9 and v = 4, then level of production is:
q =
q =
q =
If P = 1, then q = 66.7 only.
Now the demand for labor is:
L = = = = = 1.9 workers
Now suppose we hold q constant at 100
If, = and q = 100
Cost minimizing means that,
= =
4K = 9L
\K = L
Substitute into production function q = 100 = 10 K0.25 L0.25 F0.5 where F = 16
10 = K0.25 L0.25 F0.5
Substitute for K = (9/4)L
10 = 4 (L)0.25 L0.25
10/4= L0.25 L0.25
= L0. 5
Squaring both sides,
L = = = = 4.17
6.25 to 4.17 is the substitution effect.
Let us now derive the constant output demand function for Labor L’ using Shephard’s Lemma and the total cost function.
(1) TC = vK + wL + R (R is held constant)
(2) K =
(3) L =
q = 40 which can be expressed as
(4) 10P =
Substitute 2 and 3 into 1
(5) TC = v+ w+ R
Substitute 4 into 5,
TC= + + R
= + + R
= + + R
= 2 + R
TC = + R
By Shephard’s Lemma,
The demand for Labor L’ (substitution effect only) =
(6) = L’ = =
For, q = 100
L’ =
L’ = 6.25 v0.5 w–0.5
For, v = $4, w = $4
L’ = 6.25 (2) (1/2) = 6.25
For, v = $4, w = $9, q = 100
L’ = 6.25 (2) (1/3) = = 4.17
Note: The constant output input demand function L’ = allows us to hold output (q) constant while the demand function for labor L = implicitly allows q to change.
Elasticities of Demand for Inputs in the short-run:
At least one other factor of production is held constant. Here capital and labor are variable, F is held constant.
Substitution Effects:
Constant output demand for labor function is,
L’ =
The constant output wage elasticity of demand is,
hLL =
=
= *
= –1/2
We know from above that hLL = – (1 – SL) sKL
In this case, sKL= 1 (Cobb-Douglas) production function
Let’s use, SL to denote the share of labor costs in total variable costs,
Hence, we have,
hLL = –(1 – SL) 1 = –0.5
Hence,
–1 + SL = –0.5
SL = 0.5
Output Elasticities
How does a change in w affect the demand for L through induced output changes? We can write, (Equation 21.24)
(1)
as P = MC for project maximization, we have
Hence, phrasing it in elasticity terms yields,
eLw (from changes in q) = eLq * eqP * ePMC * eMCw
Demand elasticity is assumed to be eqP = –1 and as marginal costs are a linear function of q (see below), hence P = MC and eP,MC = 1, therefore the two middle terms, eqP and ePMC terms have a product of –1. From the total cost function we find that eMCw = 0.5.
Derivation
TC = + R
MC =
MC =
eMCw =
= .
= ½ = 0.5
Since the production function used in example exhibits diminishing returns to scale for the short run, q =2 for movements along the expansion path.
Derivation
L =
eLq = 2
In summary then,
eLw (for changes in q) = (2) (–1) (0.5) = –1
The total elasticity of demand (including substitution and output effects) is,
hTLL = – (1 – SL) sKL + eLw
= –0.5 – 1.0 = –1.5
This can be derived directly from L = == –1.5
Long-Run Elasticity of Demand for Labor
When wage changes affect all firms, equation 1 must be reinterpreted once again. In the long run capital and any other factors are also allowed to change, hence,
eLq = 1
ePMC = 1
eMCw = SL’
So the output effect can be written as
(eLq) (eqP) (ePMC) (eMCw)
eLw (from changes in q) = (1) (eqP) (1) (SL) = SLeqP
and the total wage elasticity of demand is
eLw = hLL + SLeqP = –(1 – SL)s + SLeqP
Since each firm maintains a constant share of industry output, eqP is identical to the market elasticity of demand for these firms’ outputs (eqP) = -1. Hence, above equation shows explicitly how eLw depends on the various factors listed previously.
If as in this example, the share of labor SL in long run total costs is 0.25 (see production function).
SL = 0.25 sKL = 1 eqP = –1
then, eLw = –(1 – 0.25) 1 + 0.25 (–1)
eLL = –0.75 – 0.25 = –1
For example, if labor constitutes 75 percent of the costs in an industry characterized by a Cobb-Douglas production i.e. s = 1 and the elasticity of demand for the industry’s output is –2, eLw will be [= –(1 – SL)s + (SLexPx) = –0.25 + 0.75 (–2)]= –1.75. Notice that in this case the wage elasticity is largely determined by the elasticity of demand for the good labor produces. On the other hand, for an input that constitutes a small share of total costs, the demand elasticity will be determined mainly by the elasticity of substitution of that input for other inputs.
The End
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