Complete Question
The complete question is shown on the first uploaded image
Answer:
a
The effect of a change in the price of a new pair of headphones on the equilibrium price of replacement tips ( dp/dpN) is
[tex]\frac{\delta p}{\delta p_N} =1[/tex]
b
The value of Q and p at equilibruim is
[tex]Q_e = 250[/tex] and [tex]p_d =[/tex] 5
The consumer surplus is [tex]C= 3125[/tex]
The producer surplus is [tex]P = 375[/tex]
Explanation:
From the question we are told that
The inverse market demand is [tex]p_d = p_N -0.1Q[/tex]
The inverse supply function is [tex]p_s = 2+ 0.012Q[/tex]
a
The effect of change in the price is mathematically given as
[tex]\frac{\delta p}{\delta p_N}[/tex]
Now differntiating the inverse market demand function with respect to [tex]p_N[/tex]
We get that
[tex]\frac{\delta p}{\delta p_N} =1[/tex]
b
We are told that [tex]p_N =[/tex]$30
Therefore the inverse market demand becomes
[tex]p_d = 30 -0.1Q[/tex]
At equilibrium
[tex]p_d = p_s[/tex]
So we have
[tex]30 -0.1Q_e = 2+ 0.012Q_e[/tex]
Where [tex]Q_e[/tex] is the quantity at equilibrium
[tex]28 = 0.112Q_e[/tex]
[tex]Q_e = \frac{28}{0.112}[/tex]
[tex]Q_e = 250[/tex]
Substituting the value of Q into the equation for the inverse market demand function
[tex]p_d = 30 - 0.1 (250 )[/tex]
[tex]p_d =[/tex] 5
Looking at the equation for [tex]p_d \ and \ p_s[/tex] we see that
For Q = 0
[tex]p_d = 30[/tex]
[tex]p_s =2[/tex]
And for Q = 250
[tex]p_d = 5[/tex]
[tex]p_s = 5[/tex]
Hence the consumer surplus is mathematically evaluated as
[tex]C = \frac{1}{2} * Q_e * (30 -5)[/tex]
Substituting value
[tex]C = \frac{1}{2} * 250 (30-5)[/tex]
[tex]C= 3125[/tex]
And
The producer surplus is mathematically evaluated as
[tex]P = \frac{1}{2} *250 * (5-2)[/tex]
[tex]P = 375[/tex]
