To better quality, It can be down load @ PDF
This post deals with three issues, the first is about how to solve in a numeric way a monopolistic competition model, the second is to highlight an errata on the book called "The Monopolistic Competition Revolution in Retrospect" edited by Brakman and Heijdra, 2004, this book is excellent but its figure 1.1 does not represent equations through introduction as the authors mention such as pg 17 after eq 1.17, and third I propose a special case model (as exercise) where salary w is endogenous and technologies are not lineal as model worked in detail in this post. It is important to highlight this post main point is to understand how monopolistic competition works in order to apply to international economics models mainly or geographical economics models also. My thought is throughout numerical exercises one gets on well with these models.
1. The model
1.1.
The consumer side (representative agent)
The model takes into account two goods, one is homogeneous "Z" and the other is a compound good "Y", therefore the representative agent wants to solve this problem:
$\underset{\{Z,Y\}}{Max}$
$U(Z,Y(X_{i}))=Z^{\delta}Y^{1-\delta}\ \ with \ \delta\in(0,1)$
$s.a$
$P_{Z}Z+P_{Y}Y=M$,
this problem is worked in textbooks on microeconomics, it does not require time to solve it in detail, therefore the solution takes the form of:
$Z(P_{Z},M)=\frac{\delta M}{P_{Z}}$
$Y(P_{Y},M)=\frac{(1-\delta)M}{P_{Y}}$.
The compound commodity "Y" is determined by different commodities $X_i$:
$Y=[\sum_{i=1}^{N}X_{i}^{\frac{(\sigma-1)}{\sigma}}]^{\frac{\sigma}{(\sigma-1)}}$,
when one takes into account the compound good (commodity) Y, the problem that has to be sorted out is:
$\underset{\{X_{i}\}_{i=1}^{N}}{Max}$
$Y=[\sum_{i=1}^{N}X_{i}^{\frac{(\sigma-1)}{\sigma}}]^{\frac{\sigma}{(\sigma-1)}}$
$s.a$
$P_{Y}Y=(1-\delta)M$,
or in explicit form
$\sum_{i=1}^{N}P_{i}X_{i}=(1-\delta)M$,
the solution to this problem is:
$X_{i}=\left(\frac{P_{i}}{P_{Y}}\right)^{-\sigma}\frac{(1-\delta)M}{P_{Y}}$ or the inverse demand $P_{i}=P_{Y}^{\frac{(\sigma-1)}{\sigma}}\left(\frac{X_{i}}{(1-\delta)M}\right)^{\left(\frac{-1}{\sigma}\right)}$,
the price index solution derived from this solution is:
$P_{Y}=\left(\sum_{i=1}^{N}P_{i}^{(1-\sigma)}\right)^{\frac{1}{1-\sigma}}$.
1.2.
The firms side, monopolistic competition market
Now firms want to maximize their profits, each firm "i" must solve this problem:
$\underset{X_i}{Max}$
$\pi=P_{i}X_{i}-wL_{i}$,
subject to its technology:
\[ X_{i}= \left\{
\begin{array}{ll}
\hspace{0.8cm} 0 \hspace{1.5cm} if \quad L_{i} \leq F\\
\frac{1}{k_{Y}}(L_{i}-F) \quad if \quad L_{i} \geq F
\end{array} \right.\]
The solution to each firm takes into account a monopolistic solution
Marginal Income=Marginal Cost, therefore solution is according to:
$X_{i}=\left(\frac{wk_{Y}\sigma}{\sigma-1}\right)^{-\sigma}((1-\delta)M)P_{Y}^{(\sigma-1)}$
$P_{i}=\left(\frac{\sigma}{\sigma-1}\right)wk_{Y}$
$P_{Y}=\left(\sum_{i=1}^{N}P_{i}^{(1-\sigma)}\right)^{\frac{1}{1-\sigma}}$,
if all firms charge same price, then,
$P_{Y}=Nwk_{Y} \left(\frac{\sigma}{\sigma-1}\right)$.
1.3.
Simulations
Until now the model is not closed yet but one can simulate, therefore:
Marginal Cost:
$MC=wk_{Y}$
Average Cost:
$AC=wk_{Y}+\frac{wF}{X_{i}}$
Inverse Demand:
$P_{i}=P_{Y}^{\frac{(\sigma-1)}{\sigma}}\left(\frac{X_{i}}{(1-\delta)M}\right)^\frac{-1}{\sigma}$
Marginal Income:
$MI=P_{Y}^{\frac{(\sigma-1)}{\sigma}}\left(\frac{1}{(1-\delta)M}\right)\left(\frac{\sigma-1}{\sigma}\right)X_{i}^{\frac{-1}{\sigma}$.
1.3.1. The case of positive profits
Consumer preferences:
$\delta=0.5$,
demand elasticity:
$\sigma=2$,
wage:
$w=90$,
technology (productivity):
$k_{y}=0.3$,
fist costs:
$F=0.1$,
market size (number of firms):
$N=50$,
consumer income:
$M=9.000$.
The figure 1.1 shows the positive profits of 36 for firm "i", the total number of firms are 50 and each one show the same economic structure, moreover the average cost for each firm is convex as marginal income and the demand do also, this figure is an example of the correct one that should be in the textbook I noted before.
The optimal volume of each firm produces:
$X_{i}^{*}= \frac{5}{3} \approx 1.66667,$
the optimal price charged is:
$P_{i}^{*}=54$,
the optimal average cost:
$AC_{i}^{*}=32.4$,
the optimal marginal cost:
$MC_{i}^{*}=27$,
total optimal cost:
$TC_{i}^{*}=54$,
the optimal index price is:
$P_{Y}^{*}=1.08$.
It is right to point out that each firm faces the same demand and therefore they produce same volume of X.
1.3.2. The case of negative profits
Now, there is a scenario where profits are negative, to get it, the main point is an increase of number of firms and realize that marginal cost and average cost graphics do not change but marginal income and the demand move to inside of Cartesian plane due to more similar products in the market. According to this:
$N=300$
The variable that makes to go inside this two curves is "Py" due to there are more product this index price goes down, but it is interesting to highlight that price that charges each firm is still the same, this price does not depend on number of firms but the volume does.
Under these 300 firms the optimal solutions are:
The optimal volume of each firm produces:
$X_{i}^{*}= \frac{5}{18} \approx 0.277778$,
the optimal price charged is:
$P_{i}^{*}=54$,
the optimal average cost:
$AC_{i}^{*}=59.4$,
the optimal marginal cost:
$MC_{i}^{*}=27$,
total optimal cost:
$TC_{i}^{*}=16.5$,
the optimal index price is:
$P_{Y}^{*}=0.18$.
It is right to point out that each firm faces the same demand and therefore they produce same volume of X and get same profits -1.5.
1.4. Chamberlain thought: the case of zero profits, closing the model
1.4.1.
Market for Z
Supply side
To close this model it is necessary work on homogeneous good (commodity) "Z" market. The supply side shows the following technology:
$Z=\frac{1}{k_{Z}}L_{Z}$,
therefore, profits are:
$\pi=P_{Z}Z-wL_{Z}$
or
$\pi=P_{Z}Z-wk_{Z}Z$,
in this case average and marginal cost are equal to $wk_{Z}$.
This market works under perfect competition and zero profits, therefore price charged is:
$\pi=P_{Z}Z-wk_{Z}Z=0\rightarrow0=Z(P_{Z}-wk_{Z})$
or
$P_{Z}=wk_{Z}$.
Demand side and equilibrium
As soon one gets price level, the demand gives information on optimal volume of Z:
$Z(P_{z},M)=\frac{\delta M}{P_{Z}}$,
$Z^{*}=\frac{\delta M}{wk_{Z}}$.
Labor market
Now the optimal level of $L_{Z}$:
$\frac{\delta M}{wk_{Z}}=\frac{1}{k_{Z}}L_{Z}^{*}$
$L_{Z}^{*}= \frac{\delta M}{w}$
1.4.2
Monopolistic competition, market for X
Now we are going to work on monopolistic competition market, in this case due to positive profits and negative profits what are incentives to increase and decrease the number of firms respectively, therefore in the long run profits for each firm is zero.
Due to any moment firms maximize their benefits without pay attention to the number of firms and each firm face same demand and equal technology, therefore they take this price (it is suppressed "i" due to all of them face same economic structure):
$P_{i}^{*}=P=\left(\frac{\sigma}{\sigma-1}\right)wk_{Y}$,
as a special case (long run) we have profit zero:
$\pi^{*}_{i}=\pi^{*}=w\left(\left(\frac{1}{\sigma-1}\right)k_{Y}X_{i}^{*}-F\right)=0$
therefore,
$X_{i}^{*}=X^{*}=\frac{(\sigma-1)F}{k_{Y}}.$
The labor demand that makes this happens is sorted out through production function, therefore:
$\frac{(\sigma-1)F}{k_{Y}}=\frac{1}{k_{Y}}(L^{*}-F)$
$L^{*}=\sigma F$
1.4.3.
Labor market H and optimal number of firms
It is important to highlight that each firm hires same amount of labor and produce same volume of X, therefore the subscript "i" is suppressed to determine how total labor force (H) is shared between production of N types of X and production of Z.
Total labor H is given by:
$H=NL+L_{Z}$,
total income:
$M=w(NL+L_{Z})+N \pi$,
due to zero profits in long run, $\pi=0$,
$M^{*}=w(NL+L_{Z})$
or
$M=wH$.
Now we can get the optimal shares of labor H for each market Z and X:
$L_{Z}^{*}= \frac{\delta M^{*}}{w}$,
therefore:
$L_{Z}^{*}=\delta H$.
Now for labor market in X:
the supply side is given by
$NL^{*}=(1-\delta)H$,
and the demand side is given by
$L^{*}=\sigma F$,
therefore the number of firms make possible fits this labor market is given by
$N^{*}=\frac{(1-\delta)H}{L^{*}}$,
the optimal number of firms is
$N^{*}=\frac{(1-\delta)H}{\sigma F}$.
1.4.4. The case of zero profits, a simulation
We follow with the parameters already given, but there are some that are endogenous now, they are N and M. The figure 1.3 shows results of this simulation, as one can realize, total income is equal to total cost. As it was highlighted before, the curves that showed a move are IM and the demand due to costumers pay attention to new variety of products.
The optimal volume of each firm produces:
$X^{*}= \frac{1}{3} \approx 0.3333$,
the optimal price charged is:
$P^{*}=54$,
the optimal average cost:
$AC_{i}^{*}=54$,
the optimal marginal cost:
$MC_{i}^{*}=27$,
total optimal cost:
$TC_{i}^{*}=18$,
the optimal index price is:
$P_{Y}^{*}=0.216$.
the optimal number of firms:
$N^{*}=250$
the optimal income:
$M^{*}=9.000$
It is right to point out that each firm faces the same demand and therefore they produce same volume of X and get same profits 0.
1.5. Equation summarize
Optimal values,
Z Production:
$Z^{*}=\frac{\delta H}{k_{Z}}$,
Z Price:
$P_{Z}^{*}=wk_{Z}$,
Z labor:
$L_{Z}^{*}=\delta H$,
X Production:
$X^{*}=\frac{(\sigma-1)F}{k_{Y}}$,
X price:
$P^{*}=\left(\frac{\sigma}{\sigma-1}\right)wk_{Y}$,
X labor:
$L^{*}=\sigma F$,
number of firms:
$N^{*}=\frac{(1-\delta)H}{\sigma F}$,
total labor in X market:
$NL^{*}=(1-\delta)H$.
1.6. Exercises
For those who wants to practice the following is a endogenous salary model, as you realized, the model decried above has a exogenous salary w and technologies are lineal, therefore I propose the following model where technology of homogeneous commodity Z is decreasing returns to scale with sunk cost and technology of compound commodity Xi is lineal (constant returns to scale) with sunk cost also. In this model you will get two levels of salary, for hiring labor in producing Z, you must pay wz and wx is paid in compound commodity market. Enjoyed and any suggestion is welcome through comments box below.
1. Suppose there is an economy which is described by following:
a.
Consumer. There is a representative consumer who faces the following problem:
$\underset{\{Z,Y, O\}}{Max}$
$U[Z,Y,O]=Z^{\frac{1}{3}}Y^{\frac{1}{3}}O^{\frac{1}{3}}$
s.a
$P_{Z}Z+P_{Y}Y=wL+120$
$ L+O=24$.
This type of utility takes into account the labor supply (L) and the trade off between labor and leisure (O), there is an initial wealth due to profits earned before but from this point you will face profits equal to zero in both markets, solve this problem for this consumer. (Hit: in this case the labor supply is supplied according to level of salary, show it).
b.
Consumer. Y is compound commodity where:
$Y=[\sum_{i=1}^{N}X_{i}^{\frac{1}{2}}]^{2}$,
solve for $P_{Y}, X_{i}$ and find the inverse demand $P_{i}$ also.
c. Production homogeneous commodity Z. In this case you have to find optimal volume of Z, Lz, Pz and wz what makes profits zero (Hit: you have to maximize benefits and get Z(Pz,wz) and L(Pz,wz), then get optimal Pz(wz) what makes zero profits through profits function $\pi_{Z}=0$, therefore get the optimal volume of Z and wz throughout equality between supply and demand).
the technology you have to use is:
$Z(L_{Z})=L_{Z}^{\frac{1}{2}}$
and its Total Cost (TC) is:
$CT(Z)=100+wZ^{2}$
where 100 can be taken as sunk cost.
d. Production Xi. In this case you have to find optimal volume of Xi=X, Pi=Px, Li=Lx, wx, Py and N what makes profits zero. (Hit: you have to maximize benefits as monopoly and get X and Px, moreover you must carry Py throughout all optimization problem and at the end take into account that Pi=Px for all firms. Thought all this you will get X(N,wx), Px(N,wx) and Lx(N,wx), use NLx+Lz=L to get optimal N and finally labor market to get wx and you have finished, remember to take into account the volume of labor taken in production of homogeneous commodity Lz). Technology for every firm is:
$X_{i}(L_{i})=10,000L_{i}- 10, \ \ if \ \ L_{i}\geq \frac{1}{1000}$
zero in other cases
e. Plot. Finally, plot AC(Z), MC(Z) and Z (demand); moreover plot AC(X), MC(X), MI(X) and X(demand). In both cases point out the solution and write main conclusions making a parallel with above model. For labor market plot supply (vertical line) and both labor demands.
1.6.1 Solution
Homogeneous market:
$P_{Z}=40(2)^{\frac{1}{2}}$
$w_{Z}=20$
$L_{Z}=5$
$Z=5^{\frac{1}{2}}$
Compound commodity market:
$w_{X}=\frac{80}{3}$
$N=4,750$
$X_{i}=X=10$
$L_{i}=L=\frac{1}{500}$
$P_{i}=P=\frac{2}{375}$
$P_{Y}=\frac{1}{890,625}$.
You should notice that labor market in production of homogeneous good pays a lower salary than labor market in compound commodity, moreover in the first market labor demand is total inelastic, therefore just an amount of workers can get this salary and the others get a higher salary in the second market (the total labor taken by these two markets is TL=14.5 hours out of 24).
