Transcription
Hi. So, uh, in the last video, we've covered a third type of model that we are going to use in this class. So now that we know how to establish the equilibrium in the models we [Music] we have chosen previously, we can move one step forward, and we are moving to comparative statics. Comparative statics is a fancy name for comparison of two equilibrium states.
So now that we know how to find equilibrium, the next task that we need to engage in is c is to see how this equilibrium will change if one of the underlying assumptions, parameters, or constants is going to change. And we're going to start with a very familiar to us already market model. Okay. So before we can move on to comparative static, we need to find equilibrium in this form. We already know how to do it, so so let's let's do it right away. Okay. So the model is given by these three equations, the same ones we had previously. We know that in equilibrium, quantity demanded and wanted to be supplied are the same, so we can reduce this model to just two equations and solve them uh using reverse rule. Okay. First thing we do, we of course put uh and put all the endogenous variables to the left-hand side. Okay. Let's let's do this step. We've got plus b b equals to a and q minus d p equals to negative c, right. Okay. Then take this model and put it into matrix notation. I don't even need this matrix to be that may [Music] c. Okay. As as usual, we start by calculating the determinant. Determinant here is negative minus b or simply negative b plus d because we know that those two parameters of the model are actually positive, so we know that this is negative, and we know that the model has solutions. Okay. Then we should calculate first aq. Okay. So we replace the first column a minus c, then we've got b minus t, and we get out of this that this is negative a d plus b c or negative a d minus b c, which of course in this case, depending on the values of the parameters, can be equal to zero, but we are assuming uh that a b is bigger than bc, just like we did before. Okay. And then we calculate ap. This is again gonna be very easy because we've got one one a minus c, uh so we've got that this is negative a plus c, right. Okay. And from that uh we are getting that equilibrium price which is equal to ap over a is equal to a plus c over l over b plus d and q [Music] in equilibrium is equal to a d minus bc over b plus d. Okay. And look, those are exactly the same results we hit previously, right. So actually nothing new is happening over here. So what we did this far is that we found the equilibrium in this model just like we did during the first class. However, now we want to get a little bit more information than we had just before. We want to know what happens when one of the parameters that we see over here changes its topic. Okay. And the best way of course to see it is if we're gonna draw uh if we're gonna draw this on the ground. Okay. So look, we are starting uh we are starting just by drawing the equilibrium. We put, remember we put p over here, q here. We start a and then we move downwards, or we remember that this point is given by a over b and we start over here with negative c. Right. This is our demand, this is our supply, and of course we remember [Music] that this point is c over d, and look at the intercept of the of demand and supply. Of course, we've got equilibrium price and equilibrium. Okay. So what do we want to learn now? Okay. What if one of the parameters that we've got in our model is going to change? Okay. Let's start, for example, with parameter a. Okay. Parameter a is in the demand function. Look, we call it like a catch at all variable because it captures different determinants of demand. For example, we can think about a US customers income, right. But of course we could put instead of a different interpretations associated with some psychological uh valuations of goods. We could put here prices of substitutes, prices of complements, number of substitutes, number of compliments. However, it's best to think about it as customers income. Of course, we know as long as the good is normal or luxury, the higher is the income of customer, the more customer wants to spend on a given good. Okay. So look what we get is that if customers income goes up, we should have that a is going up. Okay. And look, this a appears in our equilibrium solutions, right? We've got it over here and over here. So given the parameters of the model, given the assumptions, this is the equilibrium price and actually you wanted, but now because this a enters equilibrium price and quantity, I can directly see what's going to happen to equilibrium price and quantity as a is changing. Now what do I need to do to find this effect? Well, actually you already know what we need to use because we've already been using it very often in mathematics, and look all we've got to do is to calculate partial derivative. Okay. Let's start with price. Look what I'm going to do now is to see what's going to happen to price if customers income is going up. How can I evaluate it? Well, I can calculate derivative, partial derivative of equilibrium price with respect to n. So look, this expression will tell me what happens to p, equilibrium price, as a increases by a very very small amount, but of course because I can calculate it for any point, I will get more general tendency from this result. Okay. So what do I need to do? Look, I need to calculate partial derivative of this expression, right. But look, this is a fraction, right? So I can easily divide it into two parts, right. So as you see this and this expression are exactly the same. Okay. Now let's figure one more thing out. Look, I can even take this a out of this fraction and put it in a bag, right? Still this expression and this and this are the same expressions. Okay. Let me remind you the idea about calculating partial derivative. When we calculate partial derivative, we are assuming that one variable is changing. In our case, this is going to be a, while everything else is treated as constant. And look, anything that is treated in cons as constant is gone in differentiation. So what's gonna happen over here is that this part does not contain a, so it's gone when we differentiate, and look here we've got a more complex expression with a. Look, calculating it is extremely easy, why? Because look, you can think about this expression simply as a coefficient on a. Look, imagine that you have something like this: y equals uh uh I don't know e, not maybe p is not the best choice, h times x, right? Look, this is our x and our h, and a is our x, right? So if I calculate, in this case we don't even need to part partial derivative, but let's do it like that, I just get h, right? So in this case it's the same story. If I differentiate this and then with respect to a, this is my coefficient, so what I'm getting as a result is 1 over b plus d. What can I say about this? This expression is definitely positive, and this is a very important information because it tells us that as a [Music] increases, price goes up as well. By how much? Well, by this much, we don't know how much exactly it is, but we definitely know that this expression is positive. And look, even though this expression might seem confusing at the beginning, I hope we can see that calculating this derivative is really easy. And look, if I'm going to repeat this exercise now for quantity, I can use the same tricks to calculate this derivative very easily. So we've derivative with respect to a of a d [Music] minus b c where we assume that a b minus b c is positive over b plus, and look again I need to differentiate this with respect to a, and look I can use exactly the same trick as I did here. I can divide this expression into part a d over b plus d minus b c over b plus, but look this is again not the end. What I can do is to take this a up front, and look again this is going to be gone for differentiation because it's written is constant, and this part it's really the coefficient on a, so if I differentiate with respect to a, I'm getting d over b plus t. How much is it? Well, I can definitely tell you that this is something between [Music] zero and one. Why? Well, we've got b plus uh d a d divided by b plus d where all of them are positive numbers, so here we are dividing a smaller number by a bigger number, so this will definitely be somewhere between zero and one. However, it is not that important to us. For us, important thing here is that it is clearly bigger than zero because look now we get the full scope of results. What do I mean by that? We learn that as customers income goes up, we expect equilibrium price to go up and equilibrium quantity to go up. Now how do we see this on the graph? Well, look, now our a is getting bigger, right? So a goes up, it's going somewhere, let's just say here. So now you see that the main function starts higher, and it cuts through the b axis lower because here we will have some burger a prime over b, but look we can also see this increase in demand is associated with higher equilibrium price and higher equilibrium quantity. Okay. So we went through the first case. Let's now continue with demand and let's see what happens if b increases. Okay. So we need to start by drawing uh another graph. So we've got some a a over b and this is our q b. We've got some negative c or q s here. We've got c over d. Here is our equilibrium price and our equilibrium [Music] quantity. Okay. So here is u, here is of course. Okay. So what do we need to do now? Well, we need to now examine the impact of b, right? So what is going to happen if b is changing? Okay. What is b in our context? Look, we know that this coefficient negative b gives us the negative slope of demand, right? The more the higher the price, the lower the quantity demanded, right? Makes perfect sense. But what does influence this uh what does it influence this b? Well, this b informs us about the slope of demand, right? And in your microeconomics class, you've actually made a connection between the slope of demand function and elasticity of demand. So definitely b impacts overall elasticity of demand function. Look, we are going to discuss the last message in detail in the future as a separate a separate topic. So for now, let's refrain from that. However, let's remember that actually this parameter b [Music] has a key impact on the degree of the elasticity of demand, right? Of course, we know that the flatter the demand, the more elastic it is. What does it mean that demand demand is more elastic? It means that it's more responsive to changes in price level. Okay. So look, just as we did with a, we need to give some interpretation to b. So let's just say that our b again just like here is associated with demand elasticity. However, remember b is associated, so increase in b [Music] then we've got uh our in our model means, for example, that the good the new let's just say that it means that new substitutes appeared on the market. Again, why does appearance of substitute should impact elasticity? Okay. Look, if you have just uh one good on a market that has no substitutes, then even if the price is going up, you still need to use and you want to get this item, you still got to buy it, right? Because you cannot substitute it with anything. However, if substitute appears and at the beginning it has the same price and you still prefer the one that you bought at the beginning. However, if the price of the good you were buying is going up, you can move to substitute, right? If price of cola is going up, you can move to a substitute like Pepsi. And because of that, demand becomes more elastic, right? Because you have more options, so customers are can are becoming more responsive to changes in price in prices of a certain good simply because they have the possibility to substitute this good with some other good when price goes up, and this impacts on the elasticity of this good. Okay. Look, b just like a appears in the solution for the price, equilibrium solution for the price air for quantity. And look, the thing that's going to give us a little trouble is that it appears in the bottom, which means that we will have to use quotient rule. So what we need to do now is to calculate comparative static derivatives because oh when we calculate derivatives of equilibrium values, we call them comparative static derivatives, right? Because we are trying to examine how changes in the underlying variables are impacting the equilibrium solution. So look, let's just do what we did over here uh but now we need to do that with b. Okay. So first let's calculate comparative static derivative of price. Okay. So look again we will be examining equilibrium price, right? And we are estimating and we are calculating the derivative of this equilibrium price, and look over here we cannot use the simple trick as before. We need to calculate this issue this uh and this derivative simply using the quotient like what was the quotient rule? Let me just remind you the derivative of f of x divided by g of x is equal to f prime of x times g of x minus f of x times g prime of x all divided by g square x. Okay. All we need to do now is to remember that and apply the formula over here. And now just remember a, c, and d are constant. The only thing that changes is b, right? So we start by taking the numerator and putting in squares, right? That was easy, right? Now derivative of the function upstairs right with respect to b, but look there is no b here, so this is just zero, right? We should now multiply it by b plus d, and we're just gonna write it. Look, it's not gonna change anything, right? It's just zero. So now you've got minus the first part now stays the same. Oh, I'm sorry, a plus c, and derivative of this with respect to b, right? But look, d string is constantly revealing of b, it's just what. Okay. So we get that this is negative a plus c over b plus d squared. Now how much is it? Well, look, expression in the bottom is definitely positive, right? And it's taken to the power of two. A and c are also positive, so this is positive, so what do we see over here? Look, it's always good to like denote that, for example, this we see is positive, right? This part is also positive. If I divide something positive by positive, I'm going to get positive, but that's minus a product, which tells us that the value of this expression is negative. This tells us that the higher is b, so the higher is the elasticity of demand, the lower should be equilibrium price. Okay. So now let's do the same thing but for quantity. Again, we will have to just use the same tracks. And look, once we know how to do it for the demand, when government are gonna be doing this for supply is gonna be just extremely extremely easy because but okay, let's go back. So we now need to calculate comparative static derivative of q with respect to v. So again we differentiate expression a b minus bc over b plus d, and we use the same trick, so we use the uh uh we use the motion rule, and what do we see? A times d again [Music] uh again derivative of this uh is zero. However, look what we've got here, here we've got my negative bc, so b appears both in the numerator and in denominator, right? We need to take this into account when we're going to be calculating the result. Okay. So what we've got, this is b plus d squared, right? Because we start with this part, and now derivative of this is negative c, this is negative c, right? Multiplied by b plus d, right? The function at the bottom, then minus minus derivative of this times expression at the bottom, so this is minus uh a b minus bc times one, right? Okay. Now we will have to work a little bit more to get this done. So we've got that this is negative c minus uh the negative cb uh minus cb minus a b a d, I'm sorry, and minus plus bc, right? So what do we see over here? Those two expression cons expressions cancel each other, and we can take d negative t out actually, and we get that this is negative d times a plus c over b plus d squared. Okay. That wasn't uh just as easy as the previous one, but still look there's not much actually of the new things here. We just applied the same role. We just needed to calculate a little bit more, and look we again see that this expression must be positive, this expression must be positive, e is positive, we've got minus in front, so we see that this is lower than zero. So what is the information that we get from this? Well, we get the information that the more elastic the demand function is becoming, the lower the equilibrium price and lower the equilibrium quantity. Now again, how can I see it over here? Look, if b is going up, right, we are dividing here a by a bigger number, right? Which means that now the demand function is more flat when you look from the perspective of q axis, and as a result we've got here new point a over b prime. Remember that this is an accident that I cut it through the same place, it can be any place, and we see that as a result equilibrium price went down and equilibrium quantity went down as well. Okay. So now that we know how we can do it uh for demand, doing it for supply is going to be just as easy. Actually, now that we know all the tricks, we can do it in just a couple of steps. Okay. So let's let's say that now we're gonna deal with c. What is c? C can be thought of as a cost of production or as taxes, so this is this would be probably the most the easiest interpretation like cost of it like. So let's treat c as either cost of materials or taxes. And now again we are interested what will happen to equilibrium price and equity quantity as taxes gold. Look, we see that c appears in the expression for one for price and for quantity, so there's not going to be a problem with calculating comparative static derivatives. So let's start with the price. Look. So we need to calculate comparative static derivative of a plus c over b plus d, but look we can use the same trick. Look, c here is the only thing changing, so part of a can be uh just forgotten. You don't need to worry about it; it's gonna be zero. Look, c has just coefficient one over one over b plus d, right? So actually the value of the derivative is just the same as it was in the case of a, and we know that this is clearly a positive outcome. Uh okay. Then uh we get we do the same trick, but now we differentiate quantity with respect to c. So we've got a d minus b c over b plus d, and look this part is going to be gone. The coefficient on c here is negative b of course divided by b plus b, so we've got that this is negative b over b plus d. Again, we see that this expression is clearly negative. What interest? Look, again we see that this is b divided by b plus d, so this expression is actually somewhere between negative one and zero, but this is not that important to us. What we get from this is that as taxes or cost of materials is going up, production increases well uh quantity declines. Okay. So now that we've established this result, maybe it's a good idea to see this result on the graph. Okay. Just give me a second. Okay. So we start over here, and we've got [Music] [Music] we've got c negative c of course, and here we've got one day supply. Of course, here is c over d, and this gives us some equilibrium price and equilibrium. Now what happens when c is going up? Look, if c gets bigger, it means that this point minus c is going to go lower, right? Because c is bigger in the absolute value. Here we've got minus f, so now we see that this has moved that supply. Here we've got a new point c prime over d. Let's call that c prime, and what do we see over here? Price has increased and quantity has decreased, just as predicted by comparative static derivatives. Okay. And it makes perfect sense. The higher the taxes, the more companies will have to charge the customers, but again they will be willing to produce less, and you see over here, remember that was this the from this point a price covers the cost of production, right? So now we see that when cost of production increased as a result of fire taxation, this point moved to the right, and we see that this resulted in one hand in a higher price and on lower quantity on the other. Okay. So the only plus parameter that we haven't discussed here is of course d, and d uh is associated with the elasticity of supply. Okay. So let's just say that we've got a case like this: more companies enters the market. What will it do? This will make supply more elastic because now if one company is increasing the like look with higher competition, if you increase if you do not have much competition, if you increase the price, customers do not have where to go, right? However, if you've got a lot of competition and you increase your price, everybody goes to buy the product at somebody else's business, right? This is why we see that d is associated with elasticity of supply [Applause], and the bigger is the d, the higher is the elasticity of supply. So we see that let's say that nowadays increasing because more companies have entered the market. Okay. So again we're gonna be calculating comparative static derivatives just like we did previously. Okay. So you got first uh the comparative static derivative of price would expect with respect to d, so this is a plus c over b plus d. Again, we are using here the quotient rule, so we start by b plus d squared. Now we see that this expression here doesn't have d, so it's going to be zero. We can skip it, so we've got just minus this expression a plus c times derivative of this with respect to d, which is one, and we see that this expression is clearly negative. This is positive and squares this is positive. We've got minus in front, so we see that this is negative. The higher the elasticity of supply, the lower that price, of course, other things being equal, which is a hidden assumption every time when we calculate a partial derivative, not only comparative static partial directive. Okay. Then we do the same thing for quantity, and also we differentiate now, so we differentiate quantity [Music]. Again, we will have to resort to quotient rule, and here d appears over here, so we've got a times b plus d, and now we will have minus minus a b, I'm sorry, minus a d minus, so plus b c, so we see that those two expressions will cancel each other out, and at the end we are getting b times a plus c over b plus d squared. Okay. So we know that this expression positive in squares, this is positive, positive, so this is positive, and we get the final result that the more elastic is the supply curve, the higher the lower the equilibrium price and higher the equilibrium quantity, other things being heat. So finally, how will we see this on the graph? Well, this is not that bad, not that complicated to draw, uh especially this, we at this moment we are getting very very good at drawing this, so we are getting qd here, we've got a over b here, we've got a here, we've got negative c here, we've got here, we've got c over d, and look now d is getting bigger, right? Oh, of course we've got some equilibrium price and equilibrium point. Now d is getting bigger, so we divide c by a bigger number, so this bond will move somewhere here, right? We will have some point c over d prime, and here we see the result of uh of the supply.
Curve becoming more elastic. Right, you see that it's getting flatter when we look from the perspective of q. Here is just moving steeper. However, we clearly see that this leads to a lower equilibrium price and higher equilibrium quantity, just like you learned in your microeconomics class. Okay, so this is how we use comparative statics to analyze changes in the underlying model parameters on equilibrium values, including price and equilibrium quantity.
Okay. Now that we've dealt with that, we can move on and apply the tools that we already know—the derivatives—to more complicated—well, not more complicated issues—but we will be getting getting uh we'll be getting to more complex issues in the future. For however, next we're going to go to national income model. We've dealt with the market model, so of course next we're gonna analyze the impact of changes in the values of different parameters in national income.
Okay. Thank you for your attention, and see you in the next video.