( ) is just a function of x, with

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1 II. MULTIVARIATE CALCULUS The first ecture covered functions where a singe input goes in, and a singe output comes out. Most economic appications aren t so simpe. In most cases, a number of variabes infuence a decision, or a number of factors are required to produce some good. These are functions of man variabes. If f is a function of X and Y with range Z, this is written: z f x, f: X Y Z ( ) Sometimes the domains X and Y wi be the same, so this might be written: f: X Z f: X X Z Of course, f coud be a function of more than two inputs. A production function might have capita, abor, and materia inputs: Y F( K, L, M) Utiit might be a function of goods (which are consumed in nonnegative amounts), taing on an rea vaue: U:R + R Demand for a particuar good x is a function of the price of that good, the price of other goods (et s sa just good ) and the person s weath: x x p, p, w ( x ) In undergrad micro, ou probab taed about how quantit demanded changes when something ese, ie the price of, changes, hoding everthing ese constant or ceteris paribus. This transates into the idea of a partia derivative, denoted b: x p or sometimes: xp px, p, w ( ) or x ( px, p, w) The price of good x is hed constant, and weath is hed constant though the person s weath ma change if he is a arge producer of good, we re not concerned about those possibe effects. A subscripted variabe denotes the partia with respect to the variabe; a subscripted number means the partia derivative with respect to the second argument of the function. The margina product of abor might be denoted b FL ( K, L, M); the margina utiit of consuming good z woud be denoted b U z. ( ) is just a function of x, with Taing a partia derivative f x is ie pretending f x, constant. Because of this, the addition rue, the product rue, and the quotient rue wor the same as in the univariate case. Summer 00 math cass notes, page 4

2 Considering that other factors ma change is the notion of a tota derivative. Using the previous exampe, consider that demand for x is a composite function of prices and weath, which is itsef a function of prices: ( ) x x p, p, w p, p x ( x ) Then the tota derivative of x with respect to the price of is: xp ( x, p, w) xpx p w w dx dp + (,, ) p w p The first term on the right-hand side is the partia derivative of x with respect to price of. This tes ou the effect hoding everthing ese constant. For the tota derivative, we aso add in these other effects in this case, that the person s weath changes when the price of changes, and that his demand for x changes when his weath changes. The second term on the right-hand side comes from apping the chain rue. At this point, et s tr some probems. Simiar to derivatives are differentias. Remember that, given a change in the run of a function (bac to one variabe) ou can use the sope at that point to approximate the change in the rise: f ( x) x When the x get rea, rea tin this is a good approximation. When the get infinite tin or infinitesima, this turns out to be exact correct, not an approximation at a. The convention in cacuus is, of course, to use dx to denote this sma change: d f ( x) dx This is nown as a differentia. If ou divide both sides b dx, ou get the derivative. In mutivariate cacuus, a function and its differentia might oo ie: f w f( x,, z) dw x dx + f d + f z dz The wa that I figure out differentias is b writing down a ist of a the variabes (dependent or independent) and then going through the function. Whenever a term shows up that has one of these variabes, I tae the partia derivative of that term and I tac on a dx or whatever the variabe is. If the term has mutipe variabes in it, I do this for each of them. Consider the reationship: 3 w f( x,, z) 7x + x + 6z z The differentia of this function is: dw 4 xdx 36x dx 4 xd ( ) + ( ) + ( ) + z d dz z ( ) 6 dz Summer 00 math cass notes, page 5

3 If we sa and z don t change, and we want to see what the effect of a sma change in x on w is, we set ddz0. Not surprising, the resut is exact the same as the partia derivative of w with respect to x. Additiona, we might be interested in changing both x and a sma bit, and see the resut on w. We can aso do this using the differentia. Differentias are more versatie than just taing the derivative. What if we are a firm, and we want to eep production the same, and see how much more of one factor we need if we reduce another? Let s sa we have the production function: Y F( K, L, M) where capita, abor, and materias are our inputs. The differentia of this production function is: F dy K dk + F L dl + F M dm Our output wi remain the same, and et s eep materias the same. If we increase abor a sma bit, how much does our capita requirement change? Setting dydm0, we have that: F 0 + K dk F L dl With some rearranging, this produces the foowing resut: dk dl F L F K This is change in capita necessar to eep production the same. In economics, it is interpreted as the margina rate of technica substitution. If ou remembered from previous casses that the MRTS of abor for capita is the ratio of their margina products, ou ve now seen where that resut comes from. Another interesting question woud is how much of one good a person woud require to offset in utiit changes from a change in another good. Let s sa there are goods, abeed x through x, in the person s utiit function: U U x, x, x, K, x ( 3 ) The differentia of this function is: U du x dx + U x dx + + U x dx K We want to now how much of good the person woud trade for good m, eeping his utiit the same. He is not trading another other goods. Then: Summer 00 math cass notes, page 6

4 U 0 + x dx U x dx m m Rearranging produces the margina rate of substitution: dx U xm dx U x m Again, ou ma reca that the MRS equas the ratio of margina utiities. It shoud aso be the sope of a ine tangent to the indifference curve at that point. Thin about this in a two-good word draw some pictures in x xm space. If we start with a particuar point and name a others points which do not change the utiit that is, the set of a points such that du0 we ve identified an indifference curve. Its sope at this particuar point woud be described b dx dxm, which is exact what we found here. Indifference curves and technoog frontiers are both exampes of impicit functions. An expicit function is what we usua thin of as a function: f( x) An impicit function oos ie: c f( x, ) where c is some constant (or at east a constant, as far as we re concerned). Sometimes we get a reationship between x and that s virtua impossibe to sove for one in terms of the other. For instance, tr soving this for : 9 f( x, ) x 4 x Given that ou can t isoate, is there an hopes of figuring out d dx? Yes, using the differentia as before: 4 x d 6 ( ) + ( 6 dx 4 x d 4 dx d 9 0 x ) ( ) ( ) ( 6 4dx x ) x x d 8 9 d dx 6 4 x 6 4x 4 x The point is that even if ou can t sove expicit in terms of x, ou can sti evauate its derivative. When ou re using a generic utiit functions or production function, ou can t sove expicit for one variabe in terms of another. You can sti oo at margina effects, as we did earier. These are a appications of m favorite principe in mathematics, the impicit function theorem. Summer 00 math cass notes, page 7

5 ( ) be a function which is continuous differentiabe in the Theorem: Let f x, neighborhood of a particuar point ( x, ). Suppose that f( x, ) c, and f 0 at ( x, ). Then there exists a function φ such that φ ( x) and: d f x φ ( x) dx f in this neighborhood of ( x, ). This is ver usefu in economics. For instance, consider a person who gets utiit from consumption when oung and when od. The first of these equas w s, weath when oung minus savings, and the second is w + ( + r) s, weath when od pus the return on savings. The utiit function is: Ucc (, ) Uw sw, + ( + rs ) ( ) We usua find that a utiit maximizing individua sets the margina rate of substitution of consumption when od for consumption when oung equa to the gross interest rate: ( ( ) ) ( ( ) ) Uw sw, + + rs c Uw sw, + + rs c ( ) + r How does a change in the interest rate affect savings? One side of the equation needs to be a constant, which can amost awas be achieved b simp subtracting one side off. It aso oos ie we can mae things simper b mutiping through b the denominator. Let s give this impicit function the name Q: Uw ( sw, + ( + rs ) ) Uw sw, rs Qw (, w, sr, ) + ( + ( + ) ) ( r) c c The impicit function theorem tes how to find a of these effects; for instance, ds dr Q r Q s So far, the functions covered have given a singe rea number as their output. A vector-vaued function returns a rea number in each of severa dimensions. In a simpe production formua, a x amount of crude oi and amount of some chemica and a refiner of size z wi produce w gaons of gas. Essentia, we re taing about 3 a production function mapping three inputs into a singe output, F:R+ R +. However, refining oi isn t so simpe. There are mutipe outputs, different grades of gasoine, erosene, diese and other refined ois. We might describe this with the 3 function F:R+ R+, where is the number of outputs of production. The factors described above produce w gaons of diese, w of high-octane gas, a the wa through w pounds of jet fue. The coection w ( w, w, w3, K, w ) is a - dimensiona vector. Summer 00 math cass notes, page 8

6 Demand functions for goods are often written as a singe -dimensiona vector vaued demand function. Rather than having to write a bunch of functions of the same variabes: x x p, p, K, p, w ( ) (,, K,, ) x x p p p w M x x p, p, K, p, w ( ) Simp write one: x x p, p, K, p, w ( ) Mae it cear that this is vector vaued. Some peope use bars above or beow, x or x, a variabe to denote a vector, or ese the use bodface tpe: x or x. Somewhat unconventiona (in economics), I use an arrow x. v Aternative, writing a statement ie x:r+ R R+ (demand is an -dimensiona vector-vaued function of positive prices and weath) or simp xpw (, ) R + is aso cear. Summer 00 math cass notes, page 9

7 Derivatives are often represented in vector or matrix form as we. When f is a function of x v R, the (technica, transposed) vector Df x v v v v v f( x) f( x) f( x),, K, ( ) or Df x v x ( ) ( x x x ) denotes its partia derivatives with respect to x i. (Both notations are unambiguous when f is a function of x v on; but if f is aso a function of other variabes, the second is better.) Simiar, second derivatives (and cross-partias) are denoted b the matrix: v v Df x ( ) or Df x v ( x) The matrix of first derivatives is caed the Jacobian matrix of a fuction; whereas the matrix of second derivatives is the Hessian matrix. References: Simon and Bume: Chapters 3-4. Sdsæter, Strøm, and Berc: Chapter4. Saas and Hie: Chapters 5-6. Summer 00 math cass notes, page 0