Transcription
Hello and welcome to another lecture of this course, Mathematics for Economics Part-I. Today, what we shall do, we shall do the tutorial; and in this tutorial, we are going to cover some of the topics of optimization. That is, we have talked about how to find a maximum and minimum of a function and the economic applications of such problems of maximization and minimization. So today, we shall discuss some of the problems related to that.
So, as you can see on your screen, this is due to real second part. So this batch of tutorials is focused towards optimization and other topics. So we start with the topic of optimization. So this is the first question that you can see on your screen: A producer has a cost function, C(q) = aq² + bq + c. a, b, and c are small letters. Under what conditions is the function convex? This is the first part. The second part: the function is concave; and the third part: both convex and concave to the quantity axis.
Just to clarify what is meant by the quantity axis, if you recall, this is how a cost function might look like. So q, that is quantity of output, is represented along the horizontal axis; and C(q), that is cost associated with the different quantities, are represented along the vertical axis. So this function might look like this, or it could be something like this. So it might have different kinds of shapes. We want to find out, given this particular form, C(q) = aq² + bq + c—this is a polynomial function—under what condition this polynomial function represents a convex cost function; that is Part 1. In Part 2, under what conditions does this polynomial function represent a concave function; and in the third part, both convex and concave? So the function has to be specified as convex or concave with respect to the horizontal axis, that is the quantity axis.
All right, so this is the function that is given to us: C(q) = aq² + bq + c. Now, from this, I can find out the derivative of the function with respect to q. The first derivative gives us 2aq + b. I am just using the power rule. A and b, remember, are constants; and from the first derivative, I can differentiate this once more, and I will get the second derivative, d²C(q)/dq², and that will simply be twice a. So this is something we have found out. Now the conclusion will come. Under what conditions is the function convex? Now here we use the property that a function is convex if the second derivative is positive. The second derivative, d²C(q)/dq², is greater than 0. Now, when will this second derivative be greater than 0? It will be greater than 0 if 2a > 0, and 2a > 0 if a > 0. So this is the required condition. The function is convex if a (small a) is positive.
Second part: concave. The function is concave, and here, like before, we use the property that the function is concave if the second derivative, d²C(q)/dq², is negative; and like before, here d²C(q)/dq² has been found to be 2a. So 2a < 0 if a < 0; that is what I have written here. So the second part is done. The third part: both convex and concave. So this is a bit tricky, but it is not actually a big deal because a function is both convex and concave if the second derivative is equal to 0. In that case, it can be both convex and concave. The idea that is being applied here is that in this case, what I have written here is strict concavity; so in case of weak convexity, it is ≥ 0, and for weak concavity, it is ≤ 0. So if we have weak convexity and weak concavity, then the relationship of the second derivative with respect to 0 is also weak. So, in that case, the function is both convex and concave in a weak sense if the second derivative is equal to 0. So actually, this should be the answer: the function is convex if a ≥ 0, the function is concave if a ≤ 0, and the function is both convex and concave if a = 0. So we are not talking about strict convexity or concavity, but convexity and concavity in a weak sense, and therefore we are getting these results.
The average revenue function of a firm is given to be this AR(q). It is given to be (1 + 3q − q²) / 3. Find the marginal revenue function and verify that it is concave to the q axis. And what is its maximum point? So the revenue function is given to us. The average revenue function is given to us. We have to figure out the marginal revenue function. Average revenue function is AR. Let us suppose AR = (1 + 3q − q²) / 3. From average revenue, you can go to total revenue; let us call that TR. Like average revenue, total revenue will also be a function of quantity q, and that is found out by multiplying the average revenue with q (quantity), and it turns out to be (q + 3q² − q³) / 3. So this is the total revenue function, and it is a function of quantity. From this, I can easily find out the marginal revenue; the marginal revenue, which we are calling as MR, MR as a function of q, can be found out from the total revenue by taking the first derivative. So it is calculated to be 1 + 6q − q². 3q²/3 will be simply q², so you have −q² here. So we have applied the power rule in this case. Marginal revenue, therefore, is 1 + 6q − q².
Now find the marginal revenue function and verify that it is concave to the q axis. We have found out the marginal revenue function. Now, how to show that it is concave to the quantity axis? Well, that we have seen in the previous problem. I take the second derivative. Here, what is the first derivative? The first derivative is d/dq(MR), and that will be, if you differentiate this with respect to quantity, it becomes 6 − 2q. And you take the second derivative, so differentiating this with respect to quantity, −2. So that is what I have written here. You can see that the second derivative is −2, which is a negative quantity. Hence, the marginal revenue function is indeed concave. This is the property that the second derivative being negative means the function is concave. And the last part: what is its maximum point? So if it is a concave function, so it might have a maximum point. So how to find that? I take the first derivative of the marginal revenue function, set that equal to 0, and from that I will get the stationary point; and the first derivative I have already found out, it is 6 − 2q. So 6 − 2q = 0; that is what I am getting from this condition; and if I solve that for q, I will get q = 3. And this point is actually at the maximum point. It is not the minimum point or an inflection point because I have already seen that the MR function is concave.
Here is a problem from profit maximization, and we are talking about a firm which is in a perfectly competitive market. This firm faces a price p given by 10. So, as it is known in a perfect competition market, the price is constant. So the producer cannot influence the price; it is given. The cost of production function is given by c(q) = 4q + q². Question is: what is the output level of the firm if it wants to maximize its profit? Secondly, what is the magnitude of maximum profit? So that is the question. What is given to us? The price is given; price is 10. So from this, I can figure out the total revenue, TR. TR will be price multiplied by quantity, so this is 10q; and therefore, the profit will be total revenue minus the cost; cost is known to us, 4q + q². So the profit function has been found out. I have to now find out the output level where the profit is maximized. To do that, I have to depend on the first and the second-order conditions. The first-order necessary condition of profit maximization is the first derivative of the profit with respect to quantity is equal to 0; so that was down to this right: d/dq(10q − 4q − q²) = 0, and this is actually 6q − q² = 0. And if I take the derivative of 6q − q², I get 6 − 2q, and that gives us the value of q. q = 3. But we now also need to check the second-order condition. So here the second derivative of the profit function that we have to check it turns out to be −2. This was the first derivative of the profit function; if you differentiate this with respect to q, you get −2; so that is our second-order condition, and it is being satisfied. So the maximum profit is going to occur at output level 3. And what is the maximum profit level? So that was the second part. For that, I have to plug in this q, that is the maximum point, in the profit function. What is the profit function? It is 6q − q²; so I just have to put q = 3 here. So if I do that, I will get the maximum profit to be 9, so that is the answer.
Another problem related to perfect competition. A firm in a perfectly competitive market faces the price 5. It has a cost of production given by this: C(q) = 3q² − 595q + 40,000 if q > 0. And this C(q) = 0 if q = 0. So this is the scenario. The cost function is actually not—it is not a smooth function because if you put q = 0 in this form, then you get C(0) = 40,000. But actually, it is specified that if q = 0, C(q) is not 40,000; it is 0. So you can see that the cost function is having a jump at q = 0. It is not a continuous function. Now the question is this: what is the output level of the firm if it wants to maximize its profit? What is the magnitude of maximum profit? So, in that sense, as far as the questions are concerned, the questions are like they were in the previous problem. However, the form of the cost function is very different in this problem. So we adopt the strategy that we had adopted earlier. The price is given. Total revenue can be found out by multiplying the price with quantity, so it is 5q. So the profit will be total revenue minus total cost, this; and the first-order condition of profit maximization, as we know, I differentiate the profit function with respect to q, set that equal to 0. And here, if I do this calculation, it comes out to be 5 − 6q + 595 = 0. And if I solve that, I get 6q = 600, implying q = 100. We also need to check the second-order condition. So I have to differentiate this part once again with respect to q, and that will give me −6. So the second derivative is −6; thus, the profit is indeed maximized at q = 100 because the profit function is concave. What about the maximized profit? So that is an important question, and as we shall see, this is going to be a very critical question: when the producer has maximized his profit, then what is the volume of that maximized profit? Here, the maximum profit is occurring at q = 100. So I put the q = 100 in the profit function, so this is the profit function: 500 − 3(100)² − 40,000. So I just solve this; I just simplified this, and it comes out to be −10,000. So what it basically means is that if the producer has to produce some good, if the producer produces q > 0, then the highest amount of profit that he can get is −10,000. It is negative. If the firm, instead of producing anything at all, shuts down, then what happens? If the firm shuts down, it does not produce anything. So total revenue is equal to 0. Remember, q = 0; cost is not −40,000; it is 0 actually. That is specified in the question; so therefore, profit will be 0 − 0 = 0. Therefore, the optimal output will be 0, and as a result, optimal profit will also be 0. So that is why I said that we need to figure out what is the maximized profit. If it is turning out to be negative, as it is happening here, then there is no point of the producer to produce anything at all. It will produce 0 amount of output.
Here is a somewhat different question. Suppose the production function of a firm is given by this: f(L) = ALα, where α lies between 0 and 1. A is positive. The price of the good produced is given by p (small p). The wage rate of labor is given by w (small w). What is the employment level of a profit-maximizing firm? And let us call that employment level L*. Comment on ∂L*/∂w, ∂L*/∂A, ∂L*/∂p; so these are the 3 partial derivatives that we have to find out. Secondly, suppose the government announces a policy to give a wage subsidy of (1 − β)w per unit of labor employed, where β lies between 0 and 1. What is the effect on the employment level? That is, what is the effect of this wage subsidy—this is called wage subsidy—on the employment level? Earlier, the employment level was L* without subsidy. So how is that going to change with subsidy? So that is the question.
Now we are going to follow the basic rule. We first find out the profit function and try to see at what level of—here employment L—so the L is the decision variable; at what level of employment is the producer going to maximize the profit? So here is the profit function: p × q − wL. q is nothing but the production function. What is cost here? There is only 1 input that we can see, so cost is −wL. The wage rate multiplied by the labor employment. The firm will decide the employment by satisfying the necessary and sufficient conditions of profit maximization. This is standard. The necessary condition is this: dπ/dL = 0, so π is this: p × ALα − wL. I take the derivative of this, so I get pαALα−1 − w = 0. This can be actually rewritten as pαALα−1 = w. Just adding w to both sides, and I will get this. And this gives me this: I am expressing L in terms of the rest of the parameters, and so L = (w/(pαA))^(1/(α−1)). Now we know α is actually less than 1, so α − 1 will be less than 0; so therefore, I have taken the reciprocal of this so that the power here is positive, so this is the positive number. So what I am getting is L = 1/(w/(pαA))^(1/(1−α)). And let us call this L*. This is the optimal employment level of the producer; as you can see, it is a function of many parameters, principally w, A, and p. So these are the parameters of interest. But is the second-order condition satisfied? That we need to check. And it is actually satisfied because if I take the derivative of this part—this was the first derivative—I take the derivative of the first derivative, so I will get the second derivative; the second derivative is turning out to be this: pα(α − 1)ALα−2. And this is less than 0 because α − 1 is negative and α − 1 is appearing here. So this is negative, and that is the second-order condition that the profit function should be concave; and the profit function here is concave if you have the second derivative to be negative. So we have found out L*, the optimal labor employment. Now I have to talk about the partial derivative of the L* with respect to these parameters, w, A, p, and this is simply manipulation. I have to take the derivative with respect to w of this function, this function; and that, after some steps, it turns out to be this: −1/(1 − α)(pαA)(w/(pαA))^(2−α)/(1−α). What about the sign of this entire thing? It is negative because 1 − α is positive and the rest of the terms are also positive. So ∂L*/∂w < 0; that is my conclusion. What about the partial derivative of L* with respect to A? And similarly, I am not specifying the steps involved here. Here it is turning out to be (w/(1−α)(pαA²))(w/(pαA))^(2−α)/(1−α); and as you can see by inspection that this is a positive quantity because α < 1. So ∂L*/∂A > 0. Similarly, ∂L*/∂p is found out to be (w/(1−α)(p²αA))(w/(pαA))^(2−α)/(1−α); and this is also positive. So this is negative, this is positive, this is positive. What it means is that as the wage rate rises, it reduces the employment level. This employment level, L*, is going down. As productivity level rises or price rises, employment level rises. A (capital A) can be thought of as a parameter representing the productivity because look at this here; A (capital A) is occurring. And if A (capital A) is high, that means the same amount of labor will be capable of producing more output. So that is why one can think of A (capital A) as a marker of productivity. So as productivity rises, or as price rises—p (small p) is the price as we can see—L* goes on rising because the partial derivatives are positive.
Now we come to the second part: the government announced a subsidy of (1 − β)w per unit of labor employed. For the employer, the cost of labor after the subsidy is announced is given by this: w − (1 − β)w, because this part is given by the government. So the producer will only have to bear the wage that is paid to the worker minus the part that is provided by the government; and this is simply βw. Now this is the wage that is actually paid by the producer; and as we have seen earlier, the optimal employment level is decided by this rule, the first-order condition. This was the condition. The marginal product on the left-hand side: you have price multiplied by the marginal product; and on the right-hand side, the wage that is paid by the producer. Now what has changed now is that the left-hand side has remained the same. The right-hand side is no longer w. Right now, the producer is paying only the β part of the w. So the producer is paying βw. So the new condition of profit maximization is this: pαALα−1 = βw. From this, we can solve for L, optimal labor employment; and that, like before, is turning out to be 1/(βw/(pαA))^(1/(1−α)). And let us call this L′. Now I am going to claim that L′ is actually greater than L*. On what basis am I saying this? Remember, β < 1; that means βw < w. Hence, if I take the reciprocal, 1/(βw) > 1/w. And that helps us conclude that this is going to be correct. What you see here is that βw is occurring in the denominator, and here w is occurring in the denominator. And what is the left-hand side? Left-hand side is L′, the right-hand side is L*. So we have just proved that L′ > L*, which means that if the government provides wage subsidy to the producer, then the producer actually will raise the employment level. So this is the kind of policy that is often advocated by many people: that if the government wants to improve the employment level in the economy, then what the government can do is that it can tell the producers, “Look, out of the 100 rupees wage that has to be paid to the laborer, you pay maybe 70 percent of that, 70 rupees, and I will pay 30 percent; that I will pay 30 rupees.” In that case, the cost per labor to the producer in that case falls; it falls to 70 instead of 100; and the effect is going to be that the producers will raise the employment level.
Now we come to a different topic, which is: suppose you have a monopoly market, then what happens to profit maximization? A monopolist has the following cost function: C(q) = 200q + 40q². The inverse demand function is given: p = 1200 − 10q. First question: what is the profit-maximizing output level? Second question: what is the corresponding price? And third: what is the maximum profit? This is the case where you do not have perfect competition, but there is a monopolist; and the monopolist wants to maximize the profit, then how does he do so? Now, like before, the profit function is total revenue minus total cost. What is total revenue? p × q; cost is C(q). So pq, that is total revenue, can be found out by multiplying the quantity with price. Price is obtained from the inverse demand function, that is 1200 − 10q. We substitute that here minus the cost, and this is simplified as 1000q − 50q². This is the profit function. And as usual, we use the first-order necessary condition. We take the first derivative of the profit function, set that equal to 0; that is d/dq(1000q − 50q²) = 0. That gives me 1000 − 100q = 0; that is q = 10. What about the second-order condition? We have to have the second derivative of the profit function to be negative. Here, actually, it is negative because this is the first derivative; and if you take the second derivative, you just get −100, which is negative. So the second-order condition is indeed satisfied; hence, q = 10; so that is what we got from the first-order condition, q = 10. 10 is the level of output where profit is maximized. Now we check for the price. The price at which the producer is going to sell his goods. The profit-maximizing price can be found by plugging q = 10 in the inverse demand function. This is the inverse demand function: 1200 − 10q is the price; that is the inverse demand function. I put q = 10 here. I get p = 1100. So this is the optimal price. The maximized profit can be found by plugging q = 10 in the profit function. Profit function was π(q) = 1000q − 50q². I will put q = 10 here. And if I do so, it turns out to be 5000. So the profit is 5000 when the producer is operating optimally.
Another problem related to a monopoly market, but this has something to do with the government as well. A monopolist has the following inverse demand function: p =
Equal to 100 minus 2q. The cost function is C(q) = 60q + F, if q > 0 and is equal to 0 if q = 0, where F denotes the fixed cost of production. What is the profit-maximizing output of the monopolist? Is it dependent on F? What is the corresponding price, and what is its profit? So this is the first part. You can see that there is a fixed cost involved here. So something similar to this we have seen before where fixed cost was involved.
Now we come to the second part: if a lump-sum tax of capital T is imposed, then how does it affect the monopolist's behavior? Monopolist's behavior means what is the quantity that it is going to produce, price it is going to charge, etc. If instead of a lump-sum tax a fixed percentage tax on profit is imposed, then how does the behavior change? How does the behavior change if an excise tax (that is, tax on output) is imposed? So it is quite a long question. We have to do it carefully and slowly to get the answer. But it is not very difficult. Let us first get the expression of the profit function. So this is the standard method.
The profit function is π(q) = total revenue - total cost. We know the inverse demand function; we put it here, that multiplied by q minus the cost function, which is simply 60q + F. So this is the profit function: 40q - 2q² - F. We know the first-order necessary condition is that the derivative of this profit function should be equal to 0, and that will give me 40 - 4q = 0 or q = 10. Second-order condition: I take the derivative of this once again; I will get -4, which is negative. So indeed, the profit is getting maximized here at q = 10.
Now what is the amount of profit? The profit is given by π, which is a function of 10 now. So I put q = 10 here; that actually simplifies to be 200 - F. This is the maximized profit. Now, as we have seen earlier, this maximized profit, if it is less than 0, then the producer will better stop production because if he does not produce, then the cost is 0 and the revenue is also 0, so the profit will be 0 if he does not produce. So the maximized profit, if it is less than 0, then it is better to stop production. Thus, we can write the optimal output as follows: it is q = 10 if F < 200 because what is the maximized profit? It is this 200 - F. So as long as F < 200, then the maximized profit is positive. So in that case, it is worthwhile to produce. On the other hand, if F = 200, then the producer will get 0 profit whether he produces 10 or he produces 0. So both are optimal in this case. And finally, if the fixed cost is more than 200, then the maximized profit is actually negative. In that case, it is best not to produce, so q = 0. So the summary is this: if the fixed cost is high, it is preferable to shut down.
If the production is positive (that is, F < 200), then we can find the corresponding price and profit as follows. So price will be just plugging in q = 10 in the inverse demand function, so this comes out to be 100 - 2 * 10. So this is 80. So this is the optimal price, and as we have seen earlier, the maximized profit is 200 - F.
Now we come to the tax part. So the government has imposed a lump-sum tax. What is a lump-sum tax? It basically pushed up the cost side. So the producer has to pay a fixed amount of money to the government; that is called a lump-sum tax. And that amount of money is given by capital T. So in this case, the profit function has changed. It has changed to this: the total revenue is pq minus the cost, and you can see in the cost there is an additional term. It is +T. So earlier it was just this much 60q + F. Now it is 60q + F + T. And I can simplify this to this expression: 40q - 2q² - F - T. Now what is to be noted is neither the first-order nor the second-order condition is affected by the inclusion of the lump-sum tax. The profit-maximizing output remains the same. Why am I saying this? This is that if you differentiate the profit function with respect to quantity, that will give you the first-order condition, and if you differentiate it twice, then it will get the second-order condition, but if you differentiate it once, then these terms will simply drop out because these are constants. So, therefore, the first-order and the second-order conditions are not going to be affected by the inclusion of this tax. However, it affects if the producer produces at all or not. So since the first-order and second-order conditions are not going to be affected, therefore the optimal quantity and the optimal price are going to remain the same as long as q > 0. But whether the producer produces at all or not, that is going to be affected by the inclusion of this lump-sum tax. And here is the reason: in the new circumstance, if q = 10 (that is the optimal quantity that has remained the same), the profit is given by this: 200 - F - T. So this has to be greater than 0 that we have seen. The maximized profit or the optimal profit cannot be negative, and that boils down to 200 - F > T. Earlier it was 200 - F > 0, although T does not affect the output price profit if anything is produced; if it is high (greater than 200 - F), output drops to 0. Actually, this is not correct. Profit is going to be affected. The maximized profit is going to be affected if the producer is producing anything positive because, as you can see, the maximized profit has this -T component. The optimal output and price remain the same in case of a lump-sum tax provided the producer is producing something positive, but if the producer is producing anything positive or not, that is going to be affected by this lump-sum tax.
The second question is tax on profit. If a fixed percentage of profit is taxed away, then the post-tax profit changes. So suppose small t be the proportion of profit that is tax. The profit of the producer, therefore, is equal to—let us suppose it is π(q)—it is this: (1 - t) multiplied by the profit expression that we had before. Here there is no lump-sum tax. Fixed cost is there. However, a portion of the profit (that is, t percentage of the profit) has been taxed, so therefore the producer is left with (1 - t) multiplied by the profit. And this is simply this: (1 - t)(40q - 2q² - F). You might recall that this expression was the profit expression earlier. Earlier means when there was no tax. The first-order condition will give the same solution of q as before as long as the tax rate t is a constant. The second-order condition will be like before. So what is being said is that if you set the derivative of this with respect to q = 0, then that will give you just the derivative of this with respect to q = 0 because this is just a constant. t is a constant, so this is a constant. So you are going to get the same first-order condition that you got when there was no tax. And similar to the second-order condition, it will be like before. It will be satisfied. Therefore, the optimal output and price remain the same. There is no change in the behavior of the producer. The optimal price and output we got earlier were what? q = 10 and p was, I think, = 80. So 10 and 80 optimal quantity and price. Those things will hold here also, but there was—remember a condition—whether the producer will be producing at all or not that depends on what about the profit. It has to be positive. So this is the maximized profit: (1 - t)(200 - F), and that is actually the same condition because (1 - t) is just a constant and this is positive, so what matters is this part. This has to be greater than 0. So in case of tax on profit also, the behavior of the producer does not change due to the tax. He will demand the same price. He will be producing the same output as before the tax. One thing will change: that the maximized profit will be less because a part of the profit has been taxed away.
Now we come to the third category of taxes: the excise tax. If there is a tax of t per unit on output produced, then the profit turns out to be this. So this is the revenue part, and this is the cost part, as you can see that this is a new term here: -tq. t is the rate of tax, and in case of excise tax, the quantity is being taxed. How much quantity the producer is producing, based on that the producer has to pay taxes to the government. So the total amount of tax it has to pay, which is tq, it is a product of the tax rate multiplied by the quantity it has decided to produce. So this is the cost function, the new cost function. Now again, I apply the first-order condition, that is, the derivative of the profit function with respect to quantity, set that equal to 0, and that gives me this: 40 - 4q - t = 0. And you get the solution of q as 10 - t/4. Just for comparison, earlier, when there was no tax, no excise tax, the quantity, the optimal quantity, was simply 10. Now that optimal quantity is 10 - t/4. So the optimal output changes after the imposition of tax. It falls by a bit. The second-order condition will be satisfied like before because if you take the derivative of this with respect to q, then this part simply drops out, so you get -4, so the second-order condition is satisfied. Correspondingly, the optimal price also changes. So I put the optimal quantity in the inverse demand function, so 100 - 2 multiplied by the quantity, so you get 80 + t/2. So the price actually has gone up due to the tax. Earlier it was 80; now it is 80 + t/2. The optimal profit will decline due to the excise tax, and as we know, if the profit goes below 0, then the optimal output is 0. Optimal profit will decline because of the tax because the quantity is being produced less, the price is charged more, so obviously the profit will decline. What is the conclusion? Unlike lump-sum and proportional profit tax, the excise tax affects the behavior of the producer. In lump-sum tax or proportional profit tax, there was no effect on the optimal quantity the producer was producing or the optimal price, except for the fact that whether the producing is going to be producing at all or not that was getting affected by the tax. Other than that, there was no change on the optimal quantity and price, but here, in case of excise tax, the optimal quantity and price are getting affected.
Here is another problem from optimization. Let C(q) be the total cost of a firm producing q units of output. C(q) is differentiable. The average cost function is given by A(q) = C(q)/q. Prove that A(q) has a stationary point at q* > 0 if and only if the average and marginal costs are equal at q*. So as given in the question, A(q), the average cost, is equal to C(q)/q. C(q) is the cost function. Now if the derivative of the A(q), the average cost, is equal to 0 at q*, then I can call q* as a stationary point of the A(q) function. So I take the derivative of the A(q) function, the average cost function, and I apply the quotient rule, so I get q * C'(q) - C(q)/q². Now if we have a stationary point, then this has to be equal to 0. The derivative has to be equal to 0 for a stationary point, and that means that the numerator is equal to 0, which is equivalent to saying that q * C'(q) - C(q) = 0, and I can take C(q) to the right-hand side and divide both sides by q; that will give me C'(q) = C(q)/q. And what is C(q)/q? It is the average cost. So marginal cost is equal to average cost. This is what we are getting from this condition that the first derivative is equal to 0 of the average cost function. If at q*, marginal cost is equal to average cost, then at q*, A(q) has a stationary point since d/dq of A(q) will be = 0. So that is what we have just seen. If the marginal cost and average cost are equal at a particular quantity level q*, then that actually means that the first derivative of the average cost is equal to 0, which means that this is 0, and that actually means marginal cost is equal to average cost, hence the proof. So it is a basically "if and only if" kind of statement. If you have a stationary point, then marginal cost is equal to average cost. On the other hand, if marginal cost is equal to average cost, then you have a stationary point of the average cost function.
Here is another problem related to monopoly. A monopoly sells in the domestic market where the—it should be monopolist. So there is a monopolist firm which sells in the domestic market where the demand function is given by this: q₁ = 30 - (1/2)p₁. The cost of production is given by C(q₁) = 20q₁. Solve for the inverse demand function and hence obtain an expression for the profit function. So this is the first part. Now notice what we are given is not the inverse demand function. We are given the demand function where quantity is a function of the price. So the first thing to do is to—we have to find the inverse demand function, and from there, one can find the profit function. Now the demand function is given by this: q₁ = 30 - p₁/2. Now that is equivalent to 2q₁ = 60 - p₁. I have multiplied both sides by 2, and from here, I will get p₁ = 60 - 2q₁. Now actually, I have p₁ as a function of quantity; this is the inverse demand function. Now you might be wondering at this stage why we are talking about this subscript 1 all the time; that will become clear later on when we introduce another market. Now the inverse demand function has been found out. So, therefore, the profit function can be found out by using that profit = revenue - cost, and therefore I get (60 - 2q₁) * q₁ and minus the cost; cost is 20 multiplied by q₁. So this is simplifying to be 40q₁ - 2q₁², so this is the profit function π₁. Second part: is the profit function concave? Well, I can find that by taking the second derivative of the profit function, and the second derivative of the profit function is -4, which is negative, so the profit function is indeed concave. Third part: find the profit-maximizing output and price.