Answer to Question #242359 in Economics of Enterprise for Princess

Question

You are the manager of a firm that produces furniture. All your clients are in Kumasi (K) and Accra (A). Suppose the monthly inverse demand function for your furniture in in Accra and Kumasi combined is 𝑃 = 1200 − 4𝑄. Because of prohibitive cost of transporting furniture between Accra and Kumasi, you set up two plants, one in each town. The cost of producing furniture at each facility is given by 𝐶_A (𝑄_A ) = 8000 + 6𝑄_A² and 𝐶_K(𝑄_K) = 8000 + 6𝑄_K² where 𝑄 = 𝑄_A + 𝑄_K and 𝑄_A and 𝑄_K are the quantity of furniture produced at Accra and Kumasi respectively. Determine the profit maximizing amounts of electricity to produce in the two facilities, the optimal price and the profit. 

Answer 1

$P=1200-4Q\\TR=PQ=1200Q-4Q^2$

$MR=\frac{d(TR)}{dQ}=1200-8Q\\MR=1200-8Q_A-8Q_k\\C_A=8000+6Q_A^2\\MC_A=6(2)Q_A^{(2-1)}=12Q_A\\ C_k=8000+6Q_k^2\\MC_k=6(2)Q_k^{(2-1)}=12Q_k$

$MR=MC_A\\1200-8Q_A-8Q_k=12Q_k\\1200-8Q_k=20Q_k$

Dividing both sides by 20

$60-0.4Q_k=Q_A\\MR=MC_k\\1200-8Q_A-8Q_k=12Q_k\\1200-8Q_k=20Q_k$

$Q_k=60-0.4Q_A\\Q_k=60-0.4(60-0.4Q_k)\\Q_k=60-24+0.16Q_k\\0.84Q_k=36\\Q_k=42.86\\Q_A=60-0.4(42.86)=42.86$

Optimal price$=1200-4(42.86+42.86)=857.14$

Profit$=p(Q_A+Q_k)-C_A-C_k$

$=857.14(42.86+42.86)-(8000+6(44.86^2))\\=35428.57$