Answer to Question #243542 in Microeconomics for Kuds

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

Solution:

Profit maximizing level of output: MR = MC

Derive MR:

TR = P x Q = (1,200 – 4Q) Q = 1,200Q – 4Q²

TR = 1,200Q – 4Q²

MR = $\frac{\partial TR} {\partial Q}$ = 1,200 – 8Q

MR = 1,200 – 8Q_A – 8Q_K

 

MR = MC₁ and MR = MC₂

TC₁ (Q_A) = 8000 + 6Q_A²

MC₁ (Q_A) = $\frac{\partial TC_{1} } {\partial Q_{A} }$ = 12Q_A

TC₂ (Q_K) = 8000 + 6Q_K²

MC₂ (Q_K) $\frac{\partial TC_{2} } {\partial Q_{K} }$ = 12Q_K

Q = Q_A + Q_K

Set the two equations:

MC₁ (Q_A) = MR

MC₂ (Q_K) = MR

 

1). 1,200 – 8Q_A – 8Q_K = 12Q_A

1200 – 8Q_K = 12Q_A + 8Q_A

1200 – 8Q_K = 20Q_A

Divide all sides by 20:

60 – 0.4Q_K = Q_A

Q_A = 60 – 0.4Q_K

Substitute for Q_K in the second function:

2.). 1200 – 8Q_A – 8Q_K = 12Q_K

1200 – 8(60 – 0.4Q_K) – 8Q_K = 12Q_K

1200 – 480 + 3.2Q_K – 8Q_K = 12Q_K

720 = 12Q_K + 8Q_K – 3.2Q_K

720 = 16.8Q_K

Q_K = 43

Q_A = 60 – 0.4Q_K = 60 – 0.4(43) = 60 – 17 = 43

Q_A = 43

Q = 43 + 43 = 86

The profit-maximizing amounts of electricity to produce in the two facilities = 86 Units

The optimal price:

P = 1200 – 4Q = 1200 – 4(86) = 1200 – 344 = 856

The optimal price = 856

 

Profit = TR – TC

TR = P $\times$ Q = 856 $\times$ 86 = 73,616

TC = TC₁ + TC₂

TC₁ = 8000 + 6𝑄_A² = 8000 + 6(43²) = 8000 + 11094 = 19,094

TC₂ = = 8000 + 6𝑄_K² = 8000 + 6(43²) = 8000 + 11094 = 19,094

TC = 19,094 + 19,094 = 38,188

Profit = 73,616 – 38,188 = 35,428

Profit = 35,428