 So, in last module we discussed the profit function, profit is basically the difference between total revenue and total cost. And since we are discussing the short run and we are keeping capital stock and technology as constant. So, our choice variable is just n that is the number of workers. So, with respect to n, we will maximize it. So, in calculus we know that there are two conditions for maximization, first order condition and the second order condition. First order condition is that we take with respect to the choice variable first derivative let us say equal to 0 put or second order condition the second derivative let us say and it should be negative. So, first when you take the first order derivative with respect to n. So, you can see that we have done n over kali pi over kali n is equal to. So, when you take this with respect to n derivative since p is independent of n. So, it will appear as it is or second job f of n hand that is the production function with respect to n derivative. And second term with respect to n derivative the answer will be w. So, derivatives simple rules they are familiar. So, it is simple exercise. So, you first order derivative and we are putting it equal to 0. And just in last row you can see we just by rearranging it we are taking the w on the other side. So, on the left hand side we have p multiplied by kali f over kali n. Kali f over kali n is simply the marginal product of labor. So, when you multiply this with p. So, this will give you the marginal revenue product of labor m r p n. So, this is basically the money value of the marginal product of labor and it should be equal to the w n and w is your nominal wage. So, this is the marginal product of labor and w over p is the real wage. w was the nominal wage p is the price level. So, if we can see nominal variable divided by price level then it becomes your real variable and we denote real variable as small w. So, you remember that capital W we are using for the nominal wage and small w we are using for the real wage. So, here we are saying that the marginal product of labor should be equal to the real wage. So, there are two ways to do this. First way we have said that the money value of the marginal product m r p n should be equal to the nominal wage and now we are saying that marginal product of labor should be equal to the real wage. We will see once again that the marginal revenue product of labor m r p n stands for marginal revenue product of labor is equal to p times m p n. m p n what was the marginal product of labor when we multiply it with the price of the output. So, you will get the marginal product of labor in terms of money and m p n is the first derivative of the output with respect to labor or you can see here m this is the small w. So, marginal product of labor should be equal to the small w this is the real wage. So, marginal product of labor real wage equal to the marginal revenue product of labor should be equal to the nominal wage and this is what we are saying that marginal product of labor should be equal to the real wage and second order condition is that the second derivative should be negative. So, when you take the second derivative of the profit function then it should be negative. In the second derivative again you will see that p as it is there and in the second derivative we square it should be negative or p is the price of the output and it cannot be negative. And second part what is this this is the second derivative of output with respect to labor or by assumption we have discussed that marginal product of labor is positive but diminishing rate. So, this will make it negative. So, second order condition we say automatically under this assumption will satisfy you. Now, if we want to graphically look at this graph we have taken n is the units of worker or vertical axis for p y p y this is the nominal value of the output y is the output p is the price of p times y is the money value of the income and w n n is the units of labor w is the wage rate. So, when w n multiply then this will become your total cost total payment through the labor. So, you have graph looking straight line w n since w is constant by assumption we have discussed that firm is working in perfect competition input market where firm is not able to influence the wage and change the wage and overall demand and supply is determined. So, w is constant for the firm. So, w is constant that is why we have the straight line w times n. So, this is showing the amount of profit. So, this is showing the amount of profit. So, positive here because this curve is positively slope, but since the slope is gradually decreasing. So, this indicates that margin product of labor is diminishing. So, in both of the difference in both of the curves difference here this is showing the amount of profit. So, this amount here you have total revenue or for example, if you employ so many units of labor. So, on this number of units of labor this is your total cost according to the straight line and this is your total revenue and the difference is your profit. So, different points where you can see that this is the different amount of profit at different units of labor. So, firm's objective is firm wants to select the number of workers on which profit maximizes. So, where these two curves have vertical distance, the distance where the maximum will be that will be firm's profit maximizing output level and employment level. So, what we do in this? Now, we have seen mathematically and graphically that we will see that when the slope of both the curves will be equal, the vertical distance will be maximum. Straight line's slope is constant and the slope of this total revenue curve is changing. So, you can see at a point that we have drawn the tangent at a point that the tangent is parallel to this straight line. This indicates that both the slopes are equal. This is basically the same thing that we did in the first order condition. We also said that the marginal cost and marginal revenue should be equal to each other. The slope of the total revenue curve is showing the marginal revenue and the slope of the total cost is showing the marginal cost. So, the parallel between the two is that both are equal. In simple microeconomics, you have learnt these things in your elementary courses. So, this is the maximum. And we are also giving this in the lower graph. In the lower graph, what we have done is that we have drawn the curve. This shows the total profit. This is basically the vertical distance of both the curves that we have plotted. So, when this vertical distance will be maximum, then the profit will be maximum. So, we have drawn the same point. So, this indicates the profit is maximum. Obviously, the profit is 0 in the beginning and then it is increasing. This is the maximum on A and then it is decreasing. So, the profit becomes 0 again on B and if you increase further units of workers on B, then your profit will become negative. So, the profit is maximum on B to the left side. But we have to find the number of workers on which your profit is maximum. Similarly, if you want to discuss the marginal revenue and marginal cost reference, then this is the real wage. This is in fact your marginal cost. Since wage is constant, from the workers higher, lower or more, you can increase the wage on a constant wage. That is why this line is parallel to the x axis. This is the straight line. This is the real wage and the marginal product of labour is negatively sloped. So, where these two intersect, they intersect at the end star, then your marginal revenue and marginal cost will be equal to marginal product of labour and real wage. We can also understand this in another way. For example, suppose you are here, then what is happening on this amount of labour? What is happening is that this unit of your worker is giving you the marginal product this much, but the cost you have to bear on this is less than that. So, this worker is getting this much increase in your profit. So, this worker firm should hire because this worker will become the profit of the firm. If we add another worker, then the profit of the firm will increase and increase, but at this point, after this, if the worker will hire the firm, what is the situation now? That the worker has to pay more, this much, but the product given to the worker is less than that. So, this worker will be less in profit, so the worker will never hire this worker and come here at the end star and stop the firm. So, the end star is the point where the marginal product of labour is equal to the real wage, that firm will maximize the profit. So, here we will sum up how many workers will be hired. The simple analysis is that where marginal benefit is equal to the marginal cost. If you hire the worker, what is the firm earning and what is the firm paying? So, if the real wage is more than the marginal product of labour, then we have seen it graphically. This means that the worker is claiming more, but the worker is contributing less. So, the firm should not hire the worker. If the worker is paying more and contributing more, then the worker will increase in profit, that is why the worker will hire the worker. So, the final point is that how many workers will be hired? Where the marginal product of labour and the real wage will be equal. So, here we have summed up. You can see that in real terms, we are saying that marginal product of labour is equal to the real wage. When this situation happens, then the firm should increase the number of workers. If the situation is reversed, then the firm should reduce the number of workers. What are you seeing in this? It is just W. We have used small W. We have used W over P. This is just the definition of W. Small W is the real wage. Capital W is the nominal wage. So, it is just definition. We have used in real terms and the lower line is in nominal terms. What are you seeing in nominal terms? The marginal product of labour is multiplied by the price level. This is its nominal value. We will compare this with the nominal wage. If the marginal product of labour is more than the nominal wage, then the firm should increase the number of workers. So, labour demand curve. The relationship between the real wage rate and the quantity of labour demand. If the real wage is more than the marginal product of labour, then the firm should not increase the number of workers. If the firm is equal, then the firm should increase the number of workers. So, the real wage demand curve is the relationship between the real wage and the marginal product of labour. So, if you recall the marginal productivity theory, you must have studied the marginal microeconomics. You must have learnt that the requirement of the firm's profit maximization if we hire our workers to the limit where the marginal product will be equal to its cost.