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Direct and Indirect Effects in Total Derivatives Suppose We Have This Lesson 28-1: Direct and Indirect Effects in Total Derivatives Suppose we have this situation where z is our dependent variable and z is a function of x and y. = ( , ) Now in most instances, x and y are independent푧 푓 of푥 each푦 other. They’re not functionally related in most cases. But there are certain situations where y will be a function of x. So, suppose we had this: z is a function of x and y, and y is also a function of x. These situations occur sometimes where these two variables here are functionally related to each other. In this type of set-up, you will have what’s called a total derivative of z with respect to x. Because, if you substitute in for y here, notice y is just a function of x. = ( ) So, we could write this as f equals a function푦 of x,푔 g푥 of x. = ( ) = ( , ( )) So, in essence, g can be expressed as 푧a function푓 푥 of푓 x 푥alone.푔 푥 And you can find the derivative of z with respect to x. But let me show how to do this in a slightly different way. Let’s take this expression here – z is a function of x and y – and let’s just forget about this part. I don’t want to erase this. We’re still assuming y is a function of x. Let’s take the total differential of z. The total differential of z will be exactly the same as before. What will the total differential be? [Student comment] Either partial z or partial f. Since we said z, I’ll put partial z. [Student comment] Generally, you’d do this one first because x comes first. Whatever comes first is what you’d generally do first. = + 휕푧 휕푧 푑푧 푑푥 푑푦 [Student comment] 휕푥 휕푦 That’s the same expression. It’s still true whether x and y are functionally related or not. But, in this case here, we know that y is a function of x, so we could express z as a function of x. So, we might be interested in dz/dx. In order to get dz/dx, the derivative of z with respect to x, notice this dz is a total differential. But if we want the derivative of z with respect to x, all we have to do is divide both sides of this equation by dx. Doing so, we’d get dz/dx equals partial z partial x. Notice by dividing dx by dx, that’s just one. Plus partial z partial y dy/dx. = + 푑푧 휕푧 휕푧 푑푦 And this equation right here is called the푑 푥total 휕derivative푥 휕푦 푑 푥of z with respect to x. Notice this total derivative has two components. Partial z partial x, this term right here is referred to as the direct effect. Direct effect: 휕푧 And clearly, it’s the direct effect of x on z, holding휕푥 y constant. It’s just the partial derivative. And that’s called the direct effect. This term here is the indirect effect. Indirect effect: 휕푧 푑푦 It’s the indirect effect of x on z. Which is just휕 the푦 푑 effect푥 of x on y multiplied times the effect of y on z. And it’s called the indirect effect. And this thing over here is called the total derivative. Total derivative: 푑푧 푑푥 Lesson 28-2: Channel Map Illustration of the Total Derivative = ( ) = ( , ( )) 푧 푓 푥 푓 푥 푔 푥 = + 푑푧 휕푧 휕푧 푑푦 x is sort of the ultimate source of change푑 푥in this휕 푥model.휕푦 푑 So,푥 notice x affects y. So, let’s draw an arrow: x affects y. That comes from this function right here: y is a function of x. So this arrow right here represents the effect of x on y. And I’ll just write: “via the g function.” Now, y in turn affects z via the f function. The value of y affects the value of z through that f function. So, we’ll draw another arrow from y to z. So, I’ll say this arrow is via the f function. And, there’s one more arrow here because x also has a direct effect on z via the f function. So, I have to draw one more arrow here, this one right here. That is the direct effect via the f function. So, this is the direct effect. It’s the direct effect of x on z through the f function. And this set of two arrows right here is the indirect effect. Does that make sense? [Student comment] The indirect effect is both of these arrows combined. Because, see, x affects y and y affects z. So, actually, mathematically, this indirect effect is the product of these two derivatives: dy/dx times partial z partial y. Mathematically, it’s the product of these two effects. Lesson 28-3: Direct and Indirect Effects in Total Revenue Let me give you a simple economic illustration of this. And I’m going to take the simplest possible example I can think of. Consider a firm’s revenue function. A firm that sells a single product. Revenue will be a function of price and output. Revenue depends on what the price per unit of the good is and the number of units you sell. And, in fact, this is a simple function. If the firm only produces one product, then its revenue is just P times Q. = ( , ) 푅 =푓 푃 푄 The quantity sold, Q, is going to be affected by푅 price.푃푄 So, you have Q is going to be some function g of price: = ( ) So, this fits exactly into the framework of the푄 total푔 derivative.푃 And, if we wanted to get the total derivative of revenue with respect to price, let’s again do it step by step. What is dR in this case? The total differential. [Student comment] Let’s go back to the formula version first. We’ll go step by step. [Student comment] Partial R partial P times dP. [Student comment] = + 휕푅 휕푅 푑푅 푑푃 푑푄 Now, given that the function f is just P times휕 푃Q, what 휕is푄 partial R partial P? Q. Just Q. If we take the derivative of PQ with respect to P, then it’s just Q. Q is the coefficient on P. So, it’s Q dP. And what is partial R partial Q? Just P. = 휕푅 푄 휕푃 = 휕푅 푃 So, the total differential of revenue is QdP plus휕푄 PdQ. Sounds kind of nice, doesn’t it? Sounds kind of funny. Q dP times P dQ. Okay. Now, I said in this case, since Q is a function of P, we can get the total derivative of R with respect to P. And from this formula right here, how would we get dR dP? = + 휕푅 휕푅 푑푅 푑푃 푑푄 [student comment] 휕푃 휕푄 Just divide both sides of this equation here by dP. = + 푑푅 푄푑푃 푃푑푄 = + 푑푅 푑푃 푑푄 푄 푃 And you get dR/dP equals Q. dP/dP will푑푃 be what?푑푃 One.푑 푃 So, it will be just Q plus P times dQ/dP: = + 푑푅 푑푄 푄 푃 And there’s your expression for the total 푑derivative푃 of푑 revenue푃 with respect to price. It’s just Q plus P times dQ/dP. Lesson 28-4 Price Elasticity in the Total Revenue Function = + 푑푅 푑푄 푄 푃 Actually, there’s something we can do with푑푃 this. Let’s푑 푃play with this a little bit more. Oftentimes, we’re interested in the sign of dR/dP, whether it’s zero, positive or negative. The firm would like to know this. If it raises price, what’ll happen to revenue? Will total revenue go up or will total revenue stay the same? Will total revenue go down? That would be a nice thing to know. The answer to this question about what the sign of dR/dP is depends on the price elasticity of demand. And let’s show that. We have this expression for dR/dP. dR/dP is equal to Q plus P times dQ/dP. = + 푑푅 푑푄 푄 푃 Now this expression here is very close to푑 what?푃 푑푃 [Student comment] dQ/dP times P. If I wanted to convert that into the price elasticity of demand, what would I have to do? [Student comment] Just divide by, not P, Q. So, I can do this. I can divide by Q so long as I put another Q up here n the numerator. Because Q over Q is just one. So, I can put a Q down here as long as I put another one up here in the numerator. Then, this expression right here becomes Q plus E, the price elasticity of demand of Q with respect to P times Q. = + = + , 푑푅 푃 푑푄 푄 푄 푄 퐸푄 푃푄 푑푃 푄 푑푃 And now I can factor out my Q and rewrite this expression as Q times 1 plus the price elasticity of demand: = + , = (1 + , ) 푑푅 푄 퐸푄 푃푄 푄 퐸푄 푃 푑푃 So, this formula is actually quite useful in microeconomics. If you know what the price elasticity of demand is, you could figure out whether dR/dP is positive, zero or negative. For example, suppose the price elasticity of demand is exactly minus one. Then what is the sign of dR/dP? If this price elasticity of demand right here is exactly minus one, what is the sign of dR/dP? [Student comment] Zero. Because you have one minus one, this is a zero times a Q is equal to dR/dP.
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