Bertrand Market Condition - Assignment Example

The paper "Bertrand Market Condition" is a great example of an assignment on macro and microeconomics. 
Where v is the power –security concern parameter of employees

C is the employer's cost function of employees

Sampling distribution of Y=Y/S-N {M, n}

For making Bayesian inference as well as the decision, The employees use of prior distribution of S as well as combines the prior and likelihood function to form posterior distribution of S

S/Y-N {m^} ^, l}

In which case,

1 is the lambda

S= {l/ (l +n) y} + {n/ l+n} m} =  Y+ {1- a} S

With

nn {+n}

In ascertaining its capability, employees solve

MaxE=Function of {B(S+V/S2)-C in S2} function of(S, Y) dS

Where,

Function of {in S2} f(S/Y) dS= {B(S^+v/S2)-C in S2}

Therefore,

The first order will be:

E/S= {B (S1^+V/S2) C/S2=0}

The optimal solution, therefore, will be

S2= {B (M^+v/c)}

The employee’s reaction function will thus be

S2/S1-N {B (S1+ [1+] S+v)/c, (Bc) n}

Question two

Q is the output of firm 1,

Q2 is the output of firm 2

Prior distribution (q) =q-N (q, p)

Noise term observed by firm 2

R=q+E

Where E is the noise term with a normal distribution

E-N (O, k)

Profit function of firm 2

P= (D-q-q2-C2) q2

The statistical decision problem of the employer, therefore, would be

Q/Y-N {M^,^]

Maximizing (P)=f{Beta(q+v)/q2-C in q2}f(Q,Y)dq={Beta(q+v)/q2-Cin q2}

F is the function.

Therefore, the first-order condition will be

p/q2={beta(q^+v/q2)-c/q2}=0

The optimal solution for Q2 will therefore be

Q2=Beta{Q^+V}/c

Question three

R is the rate of inflation,

Prior distribution (r)=r-N(r,o) and is common knowledge

Noise term=r-E

Normal distribution of =E-(N, I)

Probability of success for the representative economic agent

P=Br/Br+r2

Odd ratio of success for the government will be

{1-p/p}==(r2/Br)

Therefore the pay off function of representative economic agent will be

P=B(R+v)/r2}-C in r2

V is the power security concern for the representative economic agent

The sampling distribution Y hence will be,

For making the Bayesian inference as well as the verdict, the representative economic agent will thus appreciate prior distribution as well the combined prior and likelihood function (f) to come up with the posterior distribution of r

R/y-N(m^N^)

But

R={Y+(1+}r

n}

n/+n}

Therefore in ascertaining the ability of the representative economic agent we,

Min loss (L) =f {B(r+v)/r2-c in r2} f[r, y] dr}

The first-order condition in order to optimize the solution will be,

E/r2=B(r+v)/r2-c/r2} =0

Hence an optimal solution will be

R2= {B(R+V}/c

Question four: Cournot market collusion

Firm 1 profit function =1-c-q1-q2)

-q1+(1-c-q1-q2))=0

Q1=R1(q2)= (1-c-q1-q2)/2}

R1(q2) is a firm 1 best response function

For firm 2,

A bisymmetry is depicted

R1(q2)= (1-c-q1-q2)/2}

Solving for equilibrium yield={Q1c,q2c)]={1-c/2+y,1-c/2+y)}

Question four (b)

The pair of (q+q2) is Cournot equilibrium where neither firm can augment its profit by choosing some quantity

 Firm 1= C (q1) =20q

Firm2=C (q2) =20q2

Inverse market demand function (p) =220-(q1-q2)

Price=P (pq1-q2)

This, therefore, implies that firm 1 profit is given by

Profit=q1 (p (q1+q2)-c

In finding Cournot Nash equilibrium in order to trigger the collision, the optimal solution is to find the reaction function

Let the price function for the industry be= p (q+q2)

And firm 1 cost structure be C1 (q1)

Profit function=P (q+q2)*q-C (q)

The Nash equilibrium will, therefore,

P(q1+q2)-c{q1}+q1p1(q1+q2)=0

P(q1+q2)-c{q2}+q2p1(q1+q2)=0

Question five: Bertrand market condition (differentiated product)

Firm1 profit p=(p1-c)1/(1-y^2)-p1+yp2}

1/1+y^2{1-y}-p1yp2

-1/(1-y^2){[1-c]

P1=R1(p2)={(1-y)+c+yp2/2}

There is symmetry in firm 2=(1-y)+c+yp1/2}

Hence,p1,p2 are a strategic complement for y>0 as well as a strategic substitute for y=<0 which is the precise opposite of the Cournot model.

Solving equilibrium yield

(p1,p2)={(1-y)+c/2-y.(1-y)+c/2-y}

Question five B: Bertrand market condition (homogeneous product)

The game can be solved by finding the intersection of a firm’s reaction function

Aggregate demand p=220-(q1+q2)

P=220-40=180

The profit function ® for firm 1, therefore, will be

R=P (q1+q2) q1-(q1)

2/ (q1+q2) q1}-180q1

To get a firm 1 reaction function, we take the first-order condition

Max q1 {2/ (q1+q2) q1-180q1}

O=2/ (q1+q2)-(2q1/ (q1+q2))-180

90(q1+q2) ^2=q1+q2) ^2=q1+q2-q1

2(q1+q2) =q2^0.5

Q1=0.5q1-q1

Since firm 2 is identical to firm 1, its reaction function will be the same

Q2=0.5q1^0.5-q1