Question

Amanufacturer U supplies to two downstream retailers D and D, at a whole- sale price of w. Both retailers supply a homogenous product in the down- stream market and compete in quantities, with the final market price being represented by p=1+s-91-92, where 91 and 92 represents the downstream quantities that D, and Dsupply. All other production costs are assumed to be zero for simplicity. In addition, s= e, +e, represents the combined effort that each of D, and D, may exert to expand the market demand (think of this as improving perceived quality). Each downstream retailer's manufacturing cost of purchasing q; units from U and exerting effort level e; is given by the following function. C(qi,e)= waite We will first begin by considering that no vertical relations nor vertical integration has been implemented. a. (Write out the profit maximisation problem of each of D, and D2 b. (Write out the first order conditions of each firm. c. (Find the equilibrium quantities supplied by either firm, effort levels, total downstream quantity supplied and profits of D, and D, in terms of w.

Answer 1

A. For D1:

Profit Maximisation problem = PXQ1 - C(Q1)

= q1 x ( 1 + s - q1-q2) - w(q1) - (e1)^2

Profit Maximisation for D2 = PXQ2 - C(Q2)

= q2 x (1+s -q1-q2) - w(q2) - (e2)^2

B. First Order conditions will be:

For D1: d/dq1 [q1 x(1+s-q1-q2) - wq1 - (e1)^2]

Differentiate the condition with respect to q1

= (1+s-q1-q2) - w : first order differentiation

equate the equation to 0

1+s-q1-q2 - w = 0

= w = 1+s+q1+q2

Follow the same for D2:

d/dq2 [q2x(1+s-q1-q2) - wq2 - (e2)^2]

= (1+s-q1-q2) - w : first order differentiation for D2

equate it with 0

1+s-q1-q2=w

this implies for profit to maximise the wage should be equal to 1+s+e1+e2

C. At equilibrium : MR=MC

In case of D1: MR1 = d/dq1( q1 x (1+s-q1-q2)) = 1+s-q1-q2

MC1 = d/dq1 ( wq1+(e1)^2)= w

This implies that MR1=MC1 = w= 1+s-q1-q2

Similarly in the case of D2 :

MR2=MC2=w=1+s-q1-q2

This means for equilibrium: w = 1+s-q1-q2 = 1+e1+e2-q1-q2 for booth D1 and D2.