So let us check that the conditions of IFT are met. So we apply the conditions to the first order conditions provided by this equation. What we need to check? In terms of differentiability and continuity of the functions involved we see that since demand function is twice continuous differentiable this equation. Depending on y and c is differentiable with respect to both variables y and c. This is a second order condition which we'll use a bit later. Now, let's check whether this condition. Remember from IFT, we need to check that the derivative with respect to y is not zero where y is exactly this equation. So let us differentiate with respect to y. So when we differentiate we get exactly pi double prime. So this is pi double prime and a condition is set that them sign over this second over derivative should be negative, so this is a negative. So the condition claiming that such a derivative shouldn't be zero. This condition is met. So the implicit function ym, the output of the monopolist exists as a function of c the value of the marginal cost. Moreover, we can find derivative we're looking for. According to IFT we use the formula where in the denominator replace exactly the f with the y which equals p double prime y plus 2 p prime. And what goes up there in the numerator we need to put the derivative of this equation with respect to C which is simply negative one. So then by crossing out the minuses together we get 1 over. And this is negative because denominator is negative. So what's the idea behind it. When the marginal costs of the monopoly increases their output of this monopoly goes down. And if we would like to find another derivative derivative of the monopoly price with respect to module cause we simply apply the chain rule. So we need to differentiate our demand function at this value and multiply by. So all in all. We have the formula. Well, both derivatives in the denominator and y value are taken at ym of course. Now, we can get an interesting conclusion. We can draw conclusion from this formula. Let's suppose that the government imposes an excise tax on this monopoly. So the value of this text is t tax is imposed on the output. And it can be interpreted as the growth of the marginal costs. So initially there. Of course function was c times y. And now one is becomes greater earlier, so we add up all the value of that imposed text. How will it affect the final price. A charge by the monopolist. It's hard to tell without knowledge of the demand function, but quite often. The demand function is a linear function. So let's suppose that this is exactly the case, so this is a linear function where a and b are positive numbers and we plug into this final formula in order to calculate the derivative Since, it's linear its second derivative equals 0. So we can cross out this term and when we cross it out as a result we get 1 over 2. It's interesting it means that if a monopoly charges firstly. Some price which is pm then after the government imposes a tax the monopoly price grows by the amount of t over 2. So initially it was n. And this prize becomes BM plus t over 2 because this derivative gives rise to the price in exactly this amount. [SOUND]