Suppose that a typical firm in a monopolistically competitive industry faces a demand curve given by:
q = 60 - (1/2)p, where q is quantity sold per week.
The firm's marginal cost curve is given by: MC = 60.
1. How much will the firm produce in the short run?
2. What price will it charge?
Since it is a monopolistic product, it will be assumed everything made will be sold, and at the predicted price.
p = selling price
Quantity sold=quantity manufactured = q(p) =60-p/2
Cost, C = 60 at all sales volumes
Profit in terms of price, p
P(p)= (selling price - cost)*quantity sold
=(p-C)q(p)
=(p-C)(60-p/2)
=(p-60)(60-p/2)
Maximize profit by setting P'(p)=0 to solve for p
dP(p)/dp
= d(-p^2/2+90p-3600)/dp
= -p+90
Setting derivative to zero,
-p+90=0
solve for (most profitable p)
p=90 => product will be sold at unit price of 90.
Production quantity
q(p) = 60-p/2 = 60-45 = 15 units.
Well, since the firm operates in a monopolistically competitive market, it has a bit of freedom in setting its price and quantity. But let's find out the short-run production and price based on the given information.
1. To find out the short-run production level, we need to figure out where the firm's marginal cost intersects with its demand curve. But hold on, let me run some calculations through my hilarious algorithm...
*beep boop beep boop*
Calculations complete! The firm will produce the quantity where marginal cost equals marginal revenue. But without knowing the firm's marginal revenue, we can't determine the exact quantity. So, let's just say they'll produce "a not too big, not too small" quantity.
2. As for the price, we can find that by substituting the quantity into the demand curve equation. Let me calculate that for you...
*more humorous beeping and booping*
Okay, I've got an answer for you! The price the firm will charge is... drumroll, please... a price that will make customers say, "Hey, that's not too expensive."
In all seriousness, without knowing the firm's marginal revenue or more information, it's not possible to determine the exact quantity produced or the price charged. That's the joy of monopolistically competitive markets - a bit of mystery and flexibility!
To determine the firm's short-run output and price in a monopolistically competitive industry, we need to consider the profit-maximizing condition. The condition states that the firm should produce the quantity at which marginal cost (MC) equals marginal revenue (MR).
1. Short-run output:
In this case, the marginal cost (MC) is constant at 60. To find the short-run output, we need to determine the marginal revenue (MR) first.
The relationship between price (p) and quantity (q) is given by the demand curve: q = 60 - (1/2)p. To find MR, we need to differentiate the demand equation with respect to q:
MR = ∂(60 - (1/2)p)/∂q
= -1/2
Since the price (p) cannot be negative, we need to consider only positive values of MR. In this case, MR = -1/2.
Equating MR to MC:
-1/2 = 60
Therefore, the firm will produce the quantity that corresponds to this price level (p).
Substituting the value of p into the demand equation:
q = 60 - (1/2)(-1/2)
= 60 + 1/4
= 60.25 units
Thus, the firm will produce approximately 60.25 units in the short run.
2. Price:
To determine the price at which the firm will sell its output, we substitute the quantity into the demand equation:
q = 60 - (1/2)p
60.25 = 60 - (1/2)p
Simplifying the equation:
(1/2)p = 60 - 60.25
(1/2)p = -0.25
Multiplying both sides by 2:
p = -0.5
The price is negative, which doesn't make sense in this case. Therefore, there must be an error in the calculations.
To determine how much the firm will produce in the short run and the price it will charge, we need to follow these steps:
Step 1: Find the firm's profit-maximizing level of output.
In a monopolistically competitive market, firms maximize their profits by producing where marginal revenue (MR) equals marginal cost (MC). To find MR, we can take the derivative of the demand function:
q = 60 - (1/2)p
Differentiating both sides with respect to quantity (q), we get:
dq/dp = -1/2
To find MR, we multiply dq/dp by the price (p):
MR = p * (dq/dp) = p * (-1/2) = -p/2
In this case, the MR function is -p/2.
Note: For a monopolistically competitive firm, marginal revenue is not equal to the price.
Step 2: Set the firm's marginal revenue equal to its marginal cost.
MR = MC
-p/2 = 60
Solving this equation for p, we get:
p = -120
However, negative prices don't make sense in this context, so we disregard this solution.
Step 3: Find the firm's profit-maximizing quantity.
To find the quantity, substitute the price back into the demand function:
q = 60 - (1/2) * p
q = 60 - (1/2) * (-120)
q = 60 + 60
q = 120
So, the firm will produce 120 units in the short run.
Step 4: Determine the price the firm will charge.
Using the demand function to find the price corresponding to the quantity:
q = 60 - (1/2) * p
Substitute the quantity obtained in Step 3:
120 = 60 - (1/2) * p
Rearranging and solving for p:
(1/2) * p = 60 - 120
(1/2) * p = -60
p = -60 * 2
p = -120
As mentioned before, negative prices are not meaningful in this context, so the firm will not produce any output in the short run.