Explain why or why not Determine whether the following stateme...

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from A units to 2A units is greater than the cost of increasing production from 2A units to 3A units.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Let X be a marginal demand function that is positive and increasing, where X is the price of the product. Is the increase in demand from C units to 2 multiplied by C units less than the increase in demand from 2 multiplied by C units to 3 multiplied by C units. Note that C. 2 multiplied by C and 3 multiplied by C are price values. Awesome. So it appears for this particular problem, we actually to determined that is the increase from C units to 2 multiplied by C units less than the increase in demand from 2 multiplied by C units to 3 multiplied by C units. So now that we're trying to determine whether or not the statement is a yes or no, like true or false statement. Based on the information that is provided to us by the problem itself. So now that we know what we're trying to solve for, let's take a moment to read off our multiple choice answers to see what our final answer might be. A is yes, because DX is smaller on the closed interval of C2 multiplied by C than on the closed interval 2 multiplied by C 3 multiplied by C. B is no because DX is larger on the close interval of C2 multiplied by C than on the close interval of 2 multiplied by C 3 multiplied by C. Multiple choice answer C is yes, because DX is constant on both their intervals, and D is no because D X is increasing. OK. So, with that in mind, our first step is we need to let DX be greater than 0, and we need to note that D of X is increasing, so I'll put an arrow pointing upwards next to it to indicate that it's increasing. And we need to recall and note that the increase in demand from C to 2 multiplied by C will be the integral from C to 2 multiplied by C of D. Of XDX. And the increase in demand from 2 multiplied by C to 3 multiplied by C will be the interval of the integral from 2 multiplied by C to 3 multiplied by C of D X D X, and since D of X is increasing. It will be larger on the close interval of 2 multiplied by C3 multiplied by C, because it's higher values than on the close interval of C. 2 multiplied by C because it is smaller values. So let's make a note of that. So we have our closed interval of 2 multiplied by C 3 multiplied by C. And this is our higher value, which I'll call HV for short, whereas our close interval for C.2 multiplied by C is our smaller value, which I'll call SV for short. So therefore, the increase is less from C to 2 multiplied by C than from 2 multiplied by C to 3 multiplied by C. So with that in mind, let's take a moment to investigate each of our multiple choice answers individually to see which one is our correct and final answer. So starting with multiple choice answer letter A. Which states that yes, because D of X is smaller on closed interval C2 multiplied by C than on the close interval of 2 multiplied by C.3 C. This statement is indeed correct because based on the information we've collected together, this particular statement is spot on. Even though we found our correct and final answer, let's take a moment to investigate or other multiple choice answers to see why they are incorrect. So B, multiple choice answer letter B, no, because D of X is larger on the close interval of C2. Multiplied by C, then on the close interval of 2 multiplied by C.3 C. This statement is incorrect because DX is smaller on the closed interval of C2 multiplied by C than on the closed interval of 2 multiplied by C.3C. So that is why multiple choice sensor letter B is incorrect. Multiple choice answer letter C says yes because DX is constant on both intervals, and this is incorrect because once again we know that D X is smaller on the close interval of C2 multiplied by C than on the close interval of 2 multiplied by C.3 C. So that's why C is incorrect. And D states that no, because D of X is increasing, but we know that D Uh Because so we, OK, so D is incorrect because the statement, no, because D is increasing means that this is not correct because we know that D of X is smaller on the close interval of C2 multiplied by C rather than the the close interval of 2 multiplied by C.3C. And once again, since D of X is increasing, it will be larger on the close interval of indeed keep it simple, call it 2C, so it's gonna be once again increasing and it will be larger on the close interval of 2 C.3C because it has higher values than on the smaller value close interval of C.2C. So that will mean without a doubt that our final answer will have to be multiple choice answer letter A, which once again is yes because DX is smaller on the close interval of C. 2 multiplied by C than on the close interval of 2 multiplied by C3 multiplied by C. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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