Consider a competitive firm with a total cost function given by TC(q)=q^2/1000. Suppose that, in order to incentivize higher production, the government decides to refund firms for their cost of producing the FIRST q units produced, up to 1000 of them.
What is the LARGEST market price at which the policy actually has an impact on the amount supplied by the firm (in comparison to the free-trade situation in which no such refund is present)?
Well, well, well, it seems the government wants to spice things up in the competitive market! Let's see how this refund policy affects our clownish firm.
Now, in the absence of any refund, our firm would maximize its profit by producing the quantity where marginal cost (MC) equals the market price (P). We can find MC by taking the derivative of the cost function, which in this case is TC(q) = q^2/1000.
Taking the derivative, we get MC = (2q)/1000.
In the refunded scenario, the firm essentially has a lower cost of production for the first q units. The refund covers the cost for these units, making their effective marginal cost zero. So, the firm will produce the quantity q where MC = P for the first q units.
To determine the largest market price where the refund has an impact, we need to find the quantity q that maximizes the firm's profit under the refund policy.
Now, under the refund policy, the firm will produce q units at zero cost and then produce any additional units at the regular cost function. Hence, the total cost for the refund scenario is TC_Refund(q) = q^2/1000 - (q - 1000)^2/1000.
To figure out the profit, we need to subtract the total cost function from the total revenue (TR). Since the market price P is the same for both scenarios (refund and free-trade), TR can be represented as Pq.
Profit for the refund scenario is Profit_Refund(q) = TR_Refund(q) - TC_Refund(q) = Pq - (q^2/1000 - (q - 1000)^2/1000).
Now, the largest market price at which the policy has an impact is the price that maximizes the profit under the refund policy in comparison to the free-trade situation.
Take a deep breath because here comes the punchline: The impact of the refund policy on the quantity supplied by the firm ends when the profit is the same in both scenarios. This is when Profit_Refund(q) = Profit_FreeTrade(q), which simplifies to q = 1000.
So, my dear questioner, the largest market price at which the policy actually has an impact on the amount supplied by the firm is precisely when the quantity supplied is 1000.
Now, don't let this clown bot do all the work for you. Take a spin on the data and equations I've given you, and you'll uncover the world of economics with a few laughs along the way! Hope this helps, and happy clowning around in the competitive market!
To determine the largest market price at which the policy has an impact on the amount supplied by the firm, we need to compare the cost of production with and without the refund.
In the free-trade situation (without the refund policy), the firm would produce the quantity that maximizes its profit, which is where the market price equals the marginal cost of production. In this case, the total cost function is given by TC(q) = q^2/1000.
The marginal cost function is the derivative of the total cost function with respect to quantity (q), so let's find the marginal cost (MC):
MC = d(TC)/dq = d(q^2/1000)/dq = (2q)/1000 = q/500.
Without the refund policy, the firm would produce where MC(q) = P, where P is the market price. So, in the free-trade situation, the firm would produce q = 500P.
With the refund policy, the government refunds the cost of producing the first q units up to 1000 units. This means that the marginal cost is effectively zero for the first 1000 units produced, after which it becomes MC(q) = q/500.
Let's consider two scenarios:
1. Market price is lower than the cost of production for the first 1000 units:
If the market price is lower than the cost of producing the first 1000 units (P < TC(q) for q ≤ 1000), the refund policy will have no impact because the firm would produce less than or equal to 1000 units in both situations. Thus, the policy does not incentivize higher production.
2. Market price is higher than the cost of production for the first 1000 units:
If the market price is higher than the cost of producing the first 1000 units (P > TC(q) for q ≤ 1000), the refund policy will incentivize the firm to produce more than 1000 units. The point at which the firm switches from producing 1000 units to a higher quantity is determined by where the market price equals the marginal cost.
Setting P = MC(q), we have:
MC(q) = q/500 = P.
Solving for q, we get:
q = 500P.
Now, let's solve for the largest market price at which the policy has an impact. Since the government refunds the cost of producing the first q units up to 1000 units:
For q ≤ 1000, the firm would produce q units.
For q > 1000, the firm would produce 1000 units because it is no longer receiving a refund.
Thus, the largest market price at which the policy has an impact on the amount supplied by the firm is P = TC(1000), which means the market price should be equal to the cost of producing the first 1000 units.
Substituting q = 1000 into the total cost function, we have:
TC(1000) = (1000^2)/1000 = 1000.
Therefore, the largest market price at which the policy has an impact on the amount supplied by the firm is $1000.
To determine the largest market price at which the policy has an impact on the amount supplied by the firm, we need to compare the cost of producing the first q units under the policy to the cost of producing q units without the refund.
Under the policy, the government refunds firms for their cost of producing the first q units, up to 1000 units. So, the cost of producing the first q units under the policy is 0.
Without the refund, the total cost function is TC(q) = q^2/1000.
To find the amount supplied by the firm under the policy, we equate the cost of producing q units without the refund to the cost of producing the first q units with the refund:
q^2/1000 = 0
Simplifying, we get:
q^2 = 0
q = 0
So, under the policy, the firm will produce 0 units irrespective of the market price.
Therefore, the largest market price at which the policy has an impact on the amount supplied by the firm is 0.