.Given the marginal cost function of a producer as MC=3Q+4Q-4. Find the output level that minimizes the cost of the producer and sketch the total cost curve and the marginal cost curve of the producer.
The total revenue function of a producer who produce Rice is expressed as TR=Q3-Q2-4Q+4. Find the output level that maximize revenue of the producer and sketch the total revenue and the marginal revenue curve of the producer.
Solution to the question
To find the output level that minimizes the cost of the producer, we need to find the point where the marginal cost (MC) is equal to zero. Setting the marginal cost function equal to zero, we can solve for the output level, which we'll call Q*.
MC = 3Q^2 + 4Q - 4
0 = 3Q^2 + 4Q - 4
To solve this quadratic equation, we can use the quadratic formula:
Q* = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 3, b = 4, and c = -4. Plugging in these values, we get:
Q* = (-4 ± √(4^2 - 4(3)(-4))) / (2(3))
= (-4 ± √(16 + 48)) / 6
= (-4 ± √64) / 6
= (-4 ± 8) / 6
Simplifying further, we have:
Q* = (4 + 8) / 6 = 2
Q* = (4 - 8) / 6 = -2/3
Since negative output levels don't make sense in this context, we can ignore Q* = -2/3. Therefore, the output level that minimizes the cost of the producer is Q* = 2.
Now let's sketch the total cost curve and the marginal cost curve.
The total cost (TC) curve represents the total cost of production for different levels of output. It is the integral of the marginal cost curve.
To sketch the total cost curve, we need the total cost function (TC). To find it, we integrate the marginal cost function with respect to Q.
TC = ∫(MC)dQ = ∫(3Q^2 + 4Q - 4)dQ
Integrating each term, we get:
TC = Q^3 + 2Q^2 - 4Q + C
Where C is the constant of integration. To determine C, we need additional information, such as the cost at a specific output level. Let's assume that at Q = 0, the total cost is 0. In that case, C = 0. Therefore, the total cost function is:
TC = Q^3 + 2Q^2 - 4Q
Now, let's plot the total cost curve on a graph with Q on the x-axis and TC on the y-axis. Use the output levels from 0 to 4 to plot the curve.
For the marginal cost curve, we simply plot the marginal cost (MC) function on a graph with Q on the x-axis and MC on the y-axis. Use the same output levels from 0 to 4 to plot the curve.
Remember, the output level that minimizes the cost of the producer is Q* = 2. Therefore, it is important to include this point on both the total cost curve and the marginal cost curve.
Once you have plotted both curves, you should have a visual representation of the total cost and marginal cost of the producer.