Analysis of Bertrand Duopoly Model with Mathematical Equations

The assignment explores Bertrand's Duopoly model by deriving demand functions, establishing best response equations, and calculating equilibrium prices and profits for two firms. It begins with defining demand curves for each firm based on given parameters. The next step involves determining the Marginal Revenue (MR) from Total Revenue (TR), followed by setting MR equal to Marginal Cost (MC) to obtain best response functions for both firms. Solving these equations simultaneously yields equilibrium prices, which are then used to determine quantities and profits. Ultimately, it compares market power and profitability between the two firms based on their output levels and profit margins.

q1= 100-p1 +p2/ 2
q2 = 200-4p2 +2p1
MC1=10
MC2= 20
Q1
Good 1 is a substitute for 2 when a price rise of good 2 leads to an increase in demand for good 1. In
this case we consider demand curve for good 1. If the price of good 2 rise (p2 rises) then it leads to
an increase in q1. This is because the sign for p2 is positive. So, good 1 and 2 are substitutes.
Also dq1/dp2 = +1/2, which shows the above logic in mathematical terms also.
Q2
Next we look at demand curve for good 2. The law of demand states that when price of a good rises
its demand must fall, assuming other things remain constant (ceteris paribus). In this equation as p2
rises q2 will fall as dq2/dp2 = -4. The negative sign before the coefficient of p2 shows that 2 follows
the demand law.
Q3
To get the best response function we use the condition MC= MR
MC1= 10
To get MR1 we need TR1
TR1= p1*q1 = 100p1 -p12 + (p1*p2)/2
MR1 = dTR1/dp1 = 100 -2p1 +p2
So the best response function is :
100-2p1 +p2 = 10
2p1 –p2 = 90 eq 1
Q4
MC2= 20
To get MR2 we need TR2
TR2= p2*q2 =200p2 - 4p22 + 2*(p1*p2)
MR2 = dTR2/dp2 = 200 -8p2 +2p1
So the best response function is :
200-8p2 +2p1 = 20
2p1 – 8p2 = -180 eq2

Q5
To get equilibrium prices we solve eq 1 and eq2 simultaneously.
2p1 – 8p2 = -180
2p1 –p2 = 90
p2 = 270/7 = 38.5
p1 = 450/7 = 64.3
Q6
Profits = Tr- Tc for each firm
Using demand curves we get
q1 = 100 – 64.3 +.5*38.5 = 55
q2 = 200-4*38.5 + 2*64.3 =174.2857
profits 1= 55* ( 64.3-10) = 2985.714
profits2 = 174.2857*(38.5 -20) = 3236.735
Q7
As firm 2 sells much more than firm 1 ( q1 < q2) it enjoys greater market power.
Its profits are also higher than those of firm 1.

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