垄断竞争DS模型的详细推导(2)

@ACj = @yj
Fw 2 <0 yj
This is the source of "monopolistic competition", i.e., the fact that there will be only one producer per variety. Pro…t Maximization Problem (PMP): Each monopolist will set up the price taking into account consumers’ demand. The pro…t function for monopolist j is:
=E
1
Integrate between 0 and n: Z , Z
n n
1
Rn
0
1
pj p1 i
(
1)
1
di
0
yj
di = E
1
1
0
yj
di
=E Z
n
Z
Rn
0 n
Rn
0
pj p1 i
(
1)
diwenku.baidu.com
1
di (1 di
1 1 1
0
p1 j di E P
)(
1
)
,Y =E
0
p1 j
,Y =V = - Production side
2
Indirect Utility Straightforward manipulation of equation (6) shows that indirect utility takes the following form: E V = P Let’ s rewrite (6) as:
1
yj
E yi = y = np pi = p
where E denotes expenditure. Then: Y = = = = y ny
1
Z
1
n
1
di
1
0
E n 1 np 1 E n 1 p
(2)
Clearly, equation (2) implies that the higher the number of varieties n, the higher the level of utility U (hence, love for variety). Utility Maximization Problem (UMP): M axY yi 0 Rn s:t: 0 pi yi
pi pj
(3)
or, using the laws of logarithms, ln , ln (yi ) yi yj = ln pi pj ln (pj )] (4) (5)
ln (yj ) =
[ln (pi )
Equation (3) represents the relative demand for any two varieties as a function of relative prices and . We can now be more explicit on the parameter : (i) j j is the (constant) elasticity of substitution between varieties: see equation (4). (ii) j j is also the (constant) price elasticity of demand: see equation (5). Now we can manipulate equation (3) in order to get an expression for yj = yj (pj ; E; P ): Multiply both sides by pi , to get: p1 pi y i = i yj pj Integrate between 0 and n: Rn
Increasing Returns to Scale (IRS) and Monopolistic Competition (MC): Dixit-Stiglitz Model1
- Preferences Z
n
1 1
U =Y =
0
yi
di
>1
(1)
Notice: equation (1) implies love for variety. To see this, guess constant price and output (we will verify these guesses later on):
(9)
There are n producers of varieties. For reasons that will be clear in a minute, each of them is a monopolist in the production of a single variety. Firm j has the following cost function: T Cj = (cyj + F ) w This cost function has the following characteristics: - labor is the only factor of production. The reward to labor is equal to the wage rate w - total costs comprise variable costs (function of the quantity produced) and …xed costs - the cost function is subadditive. Recall from your Microeconomics I class that - in the case of single output technology - the cost function is subadditive i¤ average costs (AC) decline at all levels of output. This is easily checked: ACj T Cj F = cw + yj yj 3 (10)
j
=
1
wc yj
w (cyj + F )
Impose free entry (FE). Each monopolist will then make zero pure pro…ts (in other words, it will just break even):
j
=0
,
1
wc yj = w (cyj + F )
4
where j"j is the absolute value of the elasticity of substitution (notice that not all authors use the same notation!). In order to complete the characterization of the model, we have to determine equilibrium output and number of varieties (for further details, see Krugman’ s seminal article - JIE 1979): - Equilibrium output Let’ s start by rewriting the pro…t function (11) using the pricing policy (12):
We can thus pin down equilibrium output: yj = y = ( 1) F c (13)
Equation (13) makes clear that output is the same for all monopolists. - Equilibrium number of varieties In order to pin down the number of varieties, we impose labor market clearing: L = n (cy + F ) (14) Intuition for (14) is served by rewriting it as: wL = n (T C ) This expression is quite intuitive: since labor is the only factor of production, total income (wL) is equal to total cost of production (n (T C )). Using (13) into (14), we get: h i c L=n ( 1) F + F c ,n= 5 1L F (15)
E
The Lagrangian for this problem takes the form: Z n Z n 1 1 L= yi d i pi y i
0 0
E
The necessary and su¢ cient FOC’ s for this problem are: Z n 1 1 1 1 for variety i : yi d i 1 0 Z n 1 1 1 1 for variety j : yi d i 1 0
pj p1 i di
(6)
Now, de…ne the price index as a CES aggregate of prices: 1 Z n 1 1 P pi d i
0
(7)
Equation (6) then becomes: yj = yj (pj ; E; P ) = E (demand for variety j ) pj P1 (8)
j
= pj y j
T Cj
(11)
Hence, producer j ’ s PMP is: M ax pj yj
pj
T Cj
p
j s:t: yj = E P 1
, M ax
pj
pj pj E 1 P
"
pj cE 1 P
+F w
#
FOC: (1 ) E P1 , pj Equation (12) reveals that: - each monopolist charges the same price: pj + cwE 1 pj P1
1 This
1
yi
1
1
= pi = pj
yj
1
handout: Cosimo Beverelli, HEI, Geneva. E-mail: beverel1@hei.unige.ch
1
Taking the ratio of the FOC’ s we get: yi yj
1
=
pi yi , = pj yj
1
=0 (12)
= cw
pj = p for all j
- the price is a constant markup over marginal costs. The markup is lower the higher the elasticity of substitution . In the limiting case of in…nite elasticity of substitution, the markup shrinks to one, so that each monopolist charges a price equal to marginal costs. Perfect competition can thus be represented as a limiting case of Dixit-Stiglitz-type monopolistic competition. Shortcut: It is very convenient to memorize the following formula for the monopolist’ s pricing rule: 1 1 MC p= 1 j"j
0
pi y i d i = yj
Rn
0
p1 i pj
di
Using the budget constraint, we can conveniently rewrite this expression as follows: Rn 1 p di E = 0 i yj pj , yj = E R n
0