Question

2.-Consider a market with 190 consumers. Of these, 90 of them have individual (nverse) demands given by: PutQ)-10-Q, while each of the other 100 has an individual (inverse) demand of Ps(Q)-10-10Q. The cost function of the monopolist serving this market is 60 400 (a) Find the aggregate demand. Analyze the cost function and find what kind of returns to scale it exhibits. Compute the efficient total output (ignoring break-even constraints). (b) Compute the optimal linear price (and quantity) for this monopolist, and the deadweight loss

Answer 1

Aggregate demand is the sum of the individual demand of the market case I - 90 individual p_w (Q) = 10 - Q implies -p + 10 = Q Aggregate demand for 90 degree individual Q_1^T = Q_1 + Q_2 + Q_3 + .... Q_90 implies Q_1^1 = 10 - p + 10 - p + ... + 10 - p implies Q_T^' = 90 (10 - p) implies Q_T^' == 900 - 90 p__(1) case II 100 individual p = 10 - 10 Q implies - p + 10/10 = Q Aggregate demand for 100 individual Q_T^2 = 100 (10 - p/10) implies Q_T^2 = 100 - 10p__(ii) Aggregate demand Q = Q_T^' + Q_T^2 implies Q = 900 - 90 p + 100 - 10 p implies Q = 1000 - 100 p c(Q) = 6 Q - Q^2/400 increase the quantity 't' times we get c(t Q) = 6 (t) Q - (t Q)^2/400 = 6 t Q = -t^2 Q^2/400 c(t Q) = t(6 Q - t Q^2/400) i.e., mean it

Aggregate demand is the sum of the individual demand of the market case I - 90 individual p_w (Q) = 10 - Q implies -p + 10 = Q Aggregate demand for 90 degree individual Q_1^T = Q_1 + Q_2 + Q_3 + .... Q_90 implies Q_1^1 = 10 - p + 10 - p + ... + 10 - p implies Q_T^' = 90 (10 - p) implies Q_T^' == 900 - 90 p__(1) case II 100 individual p = 10 - 10 Q implies - p + 10/10 = Q Aggregate demand for 100 individual Q_T^2 = 100 (10 - p/10) implies Q_T^2 = 100 - 10p__(ii) Aggregate demand Q = Q_T^' + Q_T^2 implies Q = 900 - 90 p + 100 - 10 p implies Q = 1000 - 100 p c(Q) = 6 Q - Q^2/400 increase the quantity 't' times we get c(t Q) = 6 (t) Q - (t Q)^2/400 = 6 t Q = -t^2 Q^2/400 c(t Q) = t(6 Q - t Q^2/400) i.e., mean it