If firm has a short run demand and cost schedule of Q=200-5P and TC=400 +4Q, what is the price the firm should sell the product for?
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To find the price the firm should sell the product for, we need to find the quantity at which the firm maximizes its profit. This occurs when marginal cost (MC) equals marginal revenue (MR). The marginal cost is the derivative of the total cost function (TC) with respect to quantity (Q), and the marginal revenue is the derivative of the total revenue function (TR) with respect to quantity (Q).
Given:
Demand: Q = 200 - 5P
Cost: TC = 400 + 4Q
First, we need to find the total revenue (TR). TR is calculated by multiplying the quantity (Q) by the price (P):
TR = Q * P
Next, we can express TR in terms of P by substituting the given demand equation into the total revenue equation:
TR = (200 - 5P) * P
TR = 200P - 5P^2
To find MR, we take the derivative of TR with respect to Q (or P, in this case):
MR = d(TR)/d(Q)
MR = d(200P - 5P^2)/dQ
MR = 200 - 10P
Now, let's find MC. MC is the derivative of TC with respect to Q:
MC = d(TC)/d(Q)
MC = d(400 + 4Q)/dQ
MC = 4
To find the quantity at which MR equals MC, set MR equal to MC and solve for P:
200 - 10P = 4
-10P = 4 - 200
-10P = -196
P = (-196)/(-10)
P = 19.6
Therefore, the firm should sell the product for a price of $19.60.
To determine the price that the firm should sell the product for, we need to find the equilibrium point where the quantity demanded equals the quantity supplied. In other words, we need to find the price (P) at which the quantity (Q) demanded equals the quantity (Q) supplied.
Given that the demand equation is Q = 200 - 5P and the cost equation is TC = 400 + 4Q, let's solve for the equilibrium price:
Step 1: Set Q demanded equal to Q supplied.
200 - 5P = Q = Q supplied
Step 2: Substitute the expression for Q supplied into the cost equation.
TC = 400 + 4(200 - 5P)
Step 3: Solve for P.
TC = 400 + 800 - 20P
TC = 1200 - 20P
20P = 1200 - TC
P = (1200 - TC) / 20
To complete the calculation, we need to know the total cost (TC) of producing the product. Once we have that information, we can substitute it into the equation to find the equilibrium price (P) at which the quantity demanded equals the quantity supplied.