Question

Let Q(p) = (a + bp)^(θ/(1-θ)) for the demand curve. If c is the constant marginal cost and there is a monopoly, calculate dp/dc

Answer 1

Q(p) = (a + bp)^(θ/(1-θ)) is the demand curve faced by the monopolist.

or, we can write the demand function as Q(p)^((1-θ)/θ) = a + bp;

or, p = (Q(p)^((1-θ)/θ) – a)/b.

Total Revenue (TR) = p*Q(p) = (Q(p)^((1-θ)/θ) – a)/b * Q(p);

Or, TR = (Q(p)^(1/θ) – aQ(p))/b

Differentiating TR with respect to Q(p), we get

MR = d(TR)/d(Q(p)) = 1/b Q(p)^((1-θ)/θ) – a/b.

Now, we know that the profit maximizing condition for a monopolist is MR = MC.

We know that MC = c.

Thus, setting MR = MC, we get,

1/b Q(p)^((1-θ)/θ) – a/b = c.

Now, differentiating c with respect to Q(p), we get,

dc/dQ(p) = (1-θ)/bθ * Q(p)^((1-2θ)/θ).

Now, Q(p) = (a + bp)^(θ/(1-θ)).

Differentiating Q(p) with respect to p, we get,

dQ(p)/dp = bθ/(1-θ) * (a + bp)^((2θ-1)/(1-θ)).

Now, we know that (a+bp) = Q(p)^((1-θ)/θ).

Thus, dQ(p)/dp = bθ/(1-θ) * Q(p)^((2θ-1)/θ).

Now, dp/dc

= dp/dQ(p) * dQ(p)/dc

= 1/( dQ(p)/dp) * (1/ dc/dQ(p))

= 1/( bθ/(1-θ) * Q(p)^((2θ-1)/θ)) * ((1-θ)/bθ * Q(p)^((1-2θ)/θ))

= 1.