Imagine that your are selling spring water from your own spring (or any product whose cost of production is negligible). Suppose now that two of yourneighbors open competing spring water businesses. Now all of you must share the market.Use the price function p = 32 - 0.01x (for 0 sxs 3200) and analyze, comparing a monopoly with competition among three suppliers (a "triopoly").(a) For a monopoly, calculate the quantity x that maximizes your revenue. Also calculate the price p and the revenue R.x =P =R =(b) For a triopoly, find the quantities x, y, and z for the three suppliers that maximize revenue for each. Also calculate the price p and the threerevenues. [Hint: Find the three revenue functions, one for each supplier, and maximize each with respect to that supplier's variable.]y =P =revenuevenue,revenue, =

Given- Imagine that your are selling spring water from your own spring (or any product whose cost of production is negligible). Suppose now that two of your neighbors open competing spring water businesses. Now all of you must share the market. To find- Use the price function p = 32 - 0.01x for 0 ≤ x ≤ 3200 and analyze, comparing a monopoly with competition among three suppliers (a tripoly) For a monopoly, calculate the quantity x that maximizes your revenue. Also calculate the price p and the Revenue R. For a triopoly, find the quantities x, y, and z for the three suppliers that maximize revenue for each. Also calculate the price p and the three revenues.Explanation- Given that, The price function p = 32 - 0.01x for 0 ≤ x ≤ 3200 a.) Let us assume that, Total number of quantity produced = x So, Revenue is given by: Rx = px = 32 - 0.01xx = 32x - 0.01x2⇒Rx = 32x - 0.01x2 For maximize Revenue, Put dRdx = 0 Now, dRdx = ddx32x - 0.01x2 = 32 - 0.02x⇒dRdx = 32 - 0.02xSo,dRdx = 0⇒32 - 0.02x = 0⇒0.02x = 32⇒x = 1600 Therefore, p = 32 - 0.011600 = 32 - 16 = 16⇒p = 16 and R = 321600 - 0.0116002 = 51200 - 25600 = 25600⇒R = 25600 So, we get x = 1600p = 16R = 25600b.) Let us assume that, Quantity produced by the supplier are x, y, and z SO, p = 32 - 0.01x + y + z Now, Revenue for x is given by: Rx = px = 32 - 0.01x + y + zx = 32 - 0.01x - 0.01y - 0.01zx = 32x - 0.01x2 - 0.01xy - 0.01xz⇒Rx = 32x - 0.01x2 - 0.01xy - 0.01xz For maximize Revenue, Put dRxdx = 0 Now, dRxdx = ddx32x - 0.01x2 - 0.01xy - 0.01xz = 32 - 0.02x - 0.01y - 0.01z⇒dRxdx = 32 - 0.02x - 0.01y - 0.01zSo,dRxdx = 0⇒32 - 0.02x - 0.01y - 0.01z = 0⇒32 - 0.01x - 0.01x - 0.01y - 0.01z = 0⇒32 - 0.01x - 0.01x + y + z = 0⇒32 - 0.01x = 0.01x + y + z .......1 Similarly, Revenue for y is given by: Ry = py = 32 - 0.01x + y + zy = 32 - 0.01x - 0.01y - 0.01zy = 32y - 0.01xy - 0.01y2 - 0.01yz⇒Ry = 32y - 0.01xy - 0.01y2 - 0.01yz For maximize Revenue, Put dRydy = 0 Now, dRydy = ddy32y - 0.01xy - 0.01y2 - 0.01yz = 32 - 0.01x - 0.02y - 0.01z⇒dRydy = 32 - 0.01x - 0.02y - 0.01zSo,dRydy = 0⇒32 - 0.01x - 0.02y - 0.01z = 0⇒32 - 0.01y - 0.01y - 0.01x - 0.01z = 0⇒32 - 0.01y - 0.01x + y + z = 0⇒32 - 0.01y = 0.01x + y + z .......2 and Revenue for z is given by: Rz = pz = 32 - 0.01x + y + zz = 32 - 0.01x - 0.01y - 0.01zz = 32z - 0.01xz - 0.01yz - 0.01z2⇒Rz = 32z - 0.01xz - 0.01yz - 0.01z2 For maximize Revenue, Put dRzdz = 0 Now, dRzdz = ddz32z - 0.01xz - 0.01yz - 0.01z2 = 32 - 0.01x - 0.01y - 0.02z⇒dRzdz = 32 - 0.01x - 0.01y - 0.02zSo,dRzdz = 0⇒32 - 0.01x - 0.01y - 0.02z = 0⇒32 - 0.01x - 0.01y - 0.01z - 0.01z = 0⇒32 - 0.01z - 0.01x + y + z = 0⇒32 - 0.01z = 0.01x + y + z .......3Now, From equation (1), (2), and (3), we get 32 - 0.01x = 32 - 0.01y = 32 - 0.01z⇒x = y = z By putting the value in equation (1), we get 32 - 0.01x = 0.01x + x + x⇒32 - 0.01x = 0.013x⇒32 - 0.01x = 0.03x⇒32 = 0.04x⇒x = 800 i.e., we get x = y = z = 800 Therefore, p = 32 - 0.01800 + 800 + 800 = 32 - 0.012400 = 32 - 24 = 8⇒p = 8 and Rx = 32 - 0.01800 + 800 + 800800 = 32 - 0.012400800 = 32 - 24800 = 8800 = 6400⇒Rx = 6400 SO, Rx = Ry = Rz = 6400 Therefore, we get x = 800y = 800z = 800p = 8revenuex = 6400revenuey = 6400revenuez = 6400Answer: x = 1600p = 16R = 25600 x = 800y = 800z = 800p = 8revenuex = 6400revenuey = 6400revenuez = 6400