1 / 69

4. Models with Multiple Explanatory Variables

4. Models with Multiple Explanatory Variables. Chapter 2 assumed that the dependent variable (Y) is affected by only ONE explanatory variable (X). Sometimes this is the case. Example: Age = Days Alive/365.25 Usually , this is not the case. Example: midterm mark depends on:

brewster-neumann
Download Presentation

4. Models with Multiple Explanatory Variables

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 4. Models with MultipleExplanatory Variables Chapter 2 assumed that the dependent variable (Y) is affected by only ONE explanatory variable (X). Sometimes this is the case. Example:Age = Days Alive/365.25 Usually, this is not the case. Example: midterm mark depends on: • how much you study • how well you study • intelligence, etc

  2. 4. Multi Variable Examples: Demand = f( price of good, price of substitutes, income, price of compliments) Consumption = f( income, tastes, wages) Graduation rates = f( tuition, school quality, student quality) Christmas present satisfaction = f (cost, timing, knowledge of person, presence of card, age, etc.)

  3. 4. The Partial Derivative It is often impossible analyze ONE variable’s impact if ALL variables are changing. Instead, we analyze one variable’s impact, assuming ALL OTHER VARIABLES REMAIN CONSTANT We do this through the partial derivative. This chapter uses the partial derivative to expand the topics introduced in chapter 2.

  4. 4. Calculus and Applications involving More than One Variable 4.1 Derivatives of Functions of More Than One Variable 4.2 Applications Using Partial Derivatives 4.3 Partial and Total Derivatives 4.4 Unconstrained Optimization 4.5 Constrained Optimization

  5. 4.1 Partial Derivatives Consider the function z=f(x,y). As this function takes into account 3 variables, it must be graphed on a 3-dimensional graph. A partial derivative calculates the slope of a 2-dimensional “slice” of this 3-dimensional graph. The partial derivative ∂z/∂x asks how x affects z while y is held constant (ceteris paribus).

  6. 4.1 Partial Derivatives In taking the partial derivative, all other variables are kept constant and hence treated as constants (the derivative of a constant is 0). There are a variety of ways to indicate the partial derivative: • ∂y/∂x • ∂f(x,z)/∂x • fx(x,z) Note: dy=dx is equivalent to ∂y/∂x if y=f(x); ie: if y only has x as an explanatory variable. (Therefore often these are used interchangeably in economic shorthand)

  7. 4.1 Partial Derivatives Let y = 2x2+3xz+8z2 ∂y/ ∂x = 4x+3z+0 ∂y/ ∂z = 0+3x+16z (0’s are dropped) Let y = xln(zx) ∂ y/ ∂ x = ln(zx) + zx/zx = ln(zx) + 1 ∂ y/ ∂ z = x(1/zx)x =x/z

  8. 4.1 Partial Derivatives Let y = 3x2z+xz3-3z/x2 ∂ y/ ∂ z=3x2+3xz2-3/x2 ∂ y/ ∂ x=6xz+z3+6z/x3 Try these: z=ln(2y+x3) Expenses=sin(a2-ab)+cos(b2-ab)

  9. 4.1.1 Higher Partial Derivatives Higher order partial derivates are evaluated exactly like normal higher order derivatives. It is important, however, to note what variable to differentiate with respect to: From before: Let y = 3x2z+xz3-3z/x2 ∂ y/ ∂ z=3x2+3xz2-3/x2 ∂2y/ ∂ z2=6xz ∂2y/ ∂ z ∂ x=6x+3z2+6/x3

  10. 4.1.1 Young’s Theorem From before: Let y = 3x2z+xz3-3z/x2 ∂ y/ ∂ x=6xz+z3+6z/x3 ∂2y/ ∂ x2=6z-18z/x4 ∂2y/ ∂ x ∂ z=6x+3z2+6/x3 Notice that d2y/dxdz=d2y/dzdx This is reflected by YOUNG’S THEOREM: order of differentiation doesn’t matter for higher order partial derivatives

  11. 4.2 Applications using Partial Derivatives As many real-world situations involve many variables, Partial Derivatives can be used to analyze our world, using tools including: • Interpreting coefficients • Partial Elasticities • Marginal Products

  12. 4.2.1 Interpreting Coefficients Given a function a=f(b,c,d), the dependent variable a is determined by a variety of explanatory variables b, c, and d. If all dependent variables change at once, it is hard to determine if one dependent variables has a positive or negative effect on a. A partial derivative, such as ∂ a/ ∂ c, asks how one explanatory variable (c), affects the dependent variable, a, HOLDING ALL OTHER DEPENDENT VARIABLES CONSTANT (ceteris paribus)

  13. 4.2.1 Interpreting Coefficients A second derivative with respect to the same variable discusses curvature. A second cross partial derivative asks how the impact of one explanatory variable changes as another explanatory variable changes. Ie: If Happiness = f(food, tv), ∂2h/ ∂ f ∂tv asks how watching more tv affects food’s effect on happiness (or how food affects tv’s effect on happiness). For example, watching TV may not increase happiness if someone is hungry.

  14. 4.2.1 Corn Example Consider the following formula for corn production: Corn = 500+100Rain-Rain2+50Scare*Fertilizer Corn = bushels of corn Rain = centimeters of rain Scare=number of scarecrows Fertilizer = tonnes of fertilizer Explain this formula

  15. 4.2.1 Corny Example 1) Intercept = 500 -if it doesn’t rain, there are no scarecrows and no fertilizer, the farmer will harvest 500 bushels 2) ∂Corn/∂Rain=100-2Rain -positive until Rain=50, then negative -more rain increases the harvest at a decreasing rate until rain hits 50cm, then additional rain decreases the harvest at an increasing rate

  16. 4.2.1 Corny Example 3) ∂2Corn/∂Rain2=-2<0, (concave) -More rain has a DECREASING impact on the corn harvest OR -More rain DECREASES rain’s impact on the corn harvest by 2 4) ∂Corn/∂Scare=50Fertilizer -More scarecrows will increase the harvest 50 for every tonne of fertilizer -if no fertilizer is used, scarecrows are useless

  17. 4.2.1 Corny Example 5) ∂2Corn/∂Scare2=0 (straight line, no curvature) -Additional scarecrows have a CONSTANT impact on corn’s harvest 6) ∂2Corn/∂Scare∂Fertilizer=50 -Additional fertilizer increases scarecrow’s impact on the corn harvest by 50

  18. 4.2.1 Corny Example 7) ∂Corn/∂Fertilizer=50Scare -More fertilizer will increase the harvest 50 for every scarecrow -if no scarecrows are used, fertilizer is useless 8) ∂2Corn/∂Fertilizer2=0, (straight line) -Additional fertilizer has a CONSTANT impact on corn’s harvest

  19. 4.2.1 Corny Example 9) ∂2Corn/∂Fertilizer ∂Scare =50 -Additional scarecrows increase fertilizer’s impact on the corn harvest by 50

  20. 4.2.1 Demand Example Consider the demand formula: Q = β1 + β2 Pown + β3 Psub + β4 INC (Quantity demanded depends on a product’s own price, price of substitutes, and income.) Here ∂ Q/ ∂ Pown= β2 = the impact on quantity when the product’s price changes Here ∂ Q/ ∂ Psub= β3 = the impact on quantity when the substitute’s price changes Here ∂ Q/ ∂ INC= β4 = the impact on quantity when income changes

  21. 4.2.3 Partial Elasticities Furthermore, partial elasticities can also be calculated using partial derivatives: Own-Price Elasticity = ∂ Q/ ∂ Pown(Pown/Q) = β2(Pown/Q) Cross-Price Elasticity = ∂ Q/ ∂ Psub(Psub/Q) = β3(Psub/Q) Income Elasticity = ∂ Q/ ∂ INC(INC/Q) = β4(INC/Q)

  22. 4.2.2 Cobb-Douglas Production Function A favorite function of economists is the Cobb-Douglas Production Function of the form Q=aLbKcOf Where L=labour, K=Capital, and O=Other (education, technology, government, etc.) This is an attractive function because if b+c+f=1, the demand function is homogeneous of degree 1. (Doubling all inputs doubles outputs…a happy concept)

  23. 4.2.2 Cobb-Douglas University Consider a production function for university degrees: Q=aLbKcCf Where L=Labour (ie: professors), K=Capital (ie: classrooms)C=Computers

  24. 4.2.2 Cobb-Douglas University Finding partial derivatives: ∂ Q/ ∂ L =abLb-1KcCf =b(aLbKcCf)/L =b(Q/L) =b* average product of labour -in other words, adding an additional professor will contribute a fraction of the average product of each current professor -this partial derivative gives us the MARGINAL PRODUCT of labour

  25. 4.2.2 Cobb-Douglas Professors For example, if 20 professors are employed by the department, and 500 students graduate yearly, and b=0.5: ∂ Q/ ∂ L =0.5(500/20) =12.5 Ie: Hiring another professor will graduate 12.5 more students. The marginal productof professors is 12.5

  26. 4.2.2 Marginal Product Consider the function Q=f(L,K,O) The partial derivative reveals the MARGINAL PRODUCT of a factor, or incremental effect on output that a factor can have when all other factors are held constant. ∂ Q/ ∂ L=Marginal Product of Labour (MPL) ∂ Q/ ∂ K=Marginal Product of Capital (MPK) ∂ Q/ ∂ O=Marginal Product of Other (MPO)

  27. 4.2.2 Cobb-Douglas Elasticities Since the “Professor Elasticity” (or PE) is defined as: PE = ∂ Q/ ∂ L(L/Q) We can find that PE =b(Q/L)(L/Q) =b The partial elasticity with respect to labour is b. The partial elasticity with respect to capital is c The partial elasticity with respect to other is f

  28. 4.2.2 Logs and Cobbs We can highlight elasticities by using logs: Q=aLbKcCf Converts to Ln(Q)=ln(a)+bln(L)+cln(k)+fln(C) We now find that: PE= ∂ ln(Q)/ ∂ ln(L)=b Using logs, elasticities more apparent.

  29. 4.2.2 Logs and Demand Consider a log-log demand example: Ln(Qdx)=ln(β1)+β2 ln(Px)+ β3 ln(Py)+ β4 ln(I) We now find that: Own Price Elasticity = β2 Cross-Price Elasticity = β3 Income Elasticity = β4

  30. 4.2.2 ilogs Considering the demand for the ipad, assume: Ln(Qdipad)=2.7-1ln(Pipad)+4ln(Pnetbook)+0.1ln(I) We now find that: Own Price Elasticity = -1, demand is unit elastic Cross-Price Elasticity = 4, a 1% increase in the price of netbooks causes a 4% increase in quantity demanded of ipads Income Elasticity = 0.1, a 1% increase in income causes a 0.1% increase in quantity demanded for ipads

  31. 4.3 Total Derivatives Often in econometrics, one variable is influenced by a variety of other variables. Ie: Happiness =f(sun, driving) Ie: Productivity = f(labor, effectiveness) Using TOTAL DERIVATIVES, we can examine how growth of one variable is caused by growth in all other variables The following formulae will combine x’s impact on y (dy/dx) with x’s impact on y, with other variables held constant (δy/δx)

  32. 4.3 Total Derivatives Assume you are increasing the square footage of a house where AREA = LENGTH X WIDTH A=LW If you increase the length, the change in area is equal to the increase in length times the current width: dL Length Width Area Notice that: δA/δL=W, (partial derivative, since width is constant) Therefore the increase in area is equal to: dA=(δA/δL)dL

  33. 4.3 Total Derivatives Length A=LW If you increase the width, the change in area is equal to the increase in width times the current length: Width Area dW Notice that: δA/δW=L, (partial derivative, since length is constant) Therefore the increase in area is equal to: dA=(δA/δW)dW Next we combine the two effects:

  34. 4.3 Total Derivatives Length A=LW An increase in both length and width has the following impact on area: Width Area dW dL Now we have: dA=(δA/δL)dL+(δA/δW)dW+(dW)dL But since derivatives always deal with instantaneous slope and small changes, (dW)dL is small and ignored, resulting in: dA=(δA/δL)dL+(δA/δW)dW

  35. 4.3 Total Derivatives Length dA=(δA/δL)dL+(δA/δW)dW Effectively, we see that change in the dependent variable (A), comes from changes in the independent variables (W and L). In general, given the function z=f(x,y) we have: Width Area dW dL

  36. 4.3 Total Derivative Example In a joke factory, QJokes=workers(funniness) You employ 500 workers, each of which can create 100 funny jokes an hour. How many more jokes could you create if you increase workers by 2 and their average funniness by 1 (perhaps by discovering any joke with an elephant in it is slightly more funny)?

  37. 4.3 Total Derivative Extension The key advantage of the total derivative is it takes variable interaction into account. The partial derivative (δz/δx) examines the effect of x on z if y doesn’t change. This is the DIRECT EFFECT. However, if x affects y which then affects z, we might want to measure this INDIRECT EFFECT. We can modify the total derivative to do this:

  38. 4.3 Total Derivative Extension • Here we see that x’s total impact on z is broken up into two parts: • x’s DIRECT impact on z (through the partial derivative) • x’s INDIRECT impact on z (through y) • Obviously, if x and y are unrelated, (δy/δx)=0, then the total derivative collapses to the partial derivative

  39. 4.3 Total Derivative Example Assume Happiness=Candy+3(Candy)Money+Money2 h=c+3cm+m2 Furthermore, Candy=3+Money/4 (c=3+m/4) The total derivative of happiness with regards to money:

  40. 4.3 Total Derivative and Elasticity Total derivatives can also give us the relationship between elasticity and revenue that we found in Chapter 2.2.3:

  41. 4.4 Unconstrained Optimization Unconstrained optimization falls into two categories: • Optimization using one variable (ie: changing wage to increase productivity, working conditions are constant) • Optimization using two (or more) variables (ie: changing wage and working conditions to maximize productivity)

  42. 4.4 Simple Unconstrained Optimization For a multivariable case where only one variable is controlled, optimization steps are easy: Consider the function z=f(x) 1) FOC: Determine where δz/δx=0 (necessary condition) 2) SOC: δ2z/δx2<0 is necessary for a maximum δ2z/δx2>0 is necessary for a minimum 3) Determine max/min point Substitute the point in (2) back into the original equation.

  43. 4.4 Simple Unconstrained Optimization Let productivity = -wage2+10wage(working conditions)2 P(w,c)=-w2+10wc2 If working conditions=2, find the wage that maximizes productivity P(w,c)=-w2+40w 1) FOC: δp/δw =-2w+40=0 w=20 2) SOC: δ2p/δw2= -2 < 0, a maximum exists

  44. 4.4 Simple Unconstrained Optimization P(w,c)=-w2+10wc2 w=20 (maximum confirmed) 3) Find Maximum P(20,4)=-202+10(20)(2)2 P(20,4)=-400+800 P(20,4)=400 Productivity is maximized at 400 when wage is 20.

  45. 4.4 Complex Unconstrained Optimization For a multivariable case where only two variable are controlled, optimization steps are more in-depth: Consider the function z=f(x,y) 1) FOC: Determine where δz/δx=0 (necessary condition) And Determine where δz/δy=0 (necessary condition)

  46. 4.4 Complex Unconstrained Optimization For a multivariable case where only two variable are controlled, optimization steps harder: Consider the function z=f(x,y) 2) SOC: δ2z/δx2<0 and δ2z/δy2<0 are necessary for a maximum δ2z/δx2>0 and δ2z/δy2>0 are necessary for a minimum Plus, the cross derivatives can’t be too large compared to the own second partial derivatives:

  47. 4.4 Complex Unconstrained Optimization If this third SOC requirement is not fulfilled, a SADDLE POINT occurs, where z is a maximum with regards to one variable but a minimum with regards to the other. (ie: wage maximizes productivity while working conditions minimizes it) Vaguely, even though both variables work to increase z, their interaction with each other outweighs this maximizing effect

  48. 4.4 Complex Unconstrained Optimization Let P(w,c)=-w2+wc-c2 +9c , maximize productivity 1) FOC: δp/δw =-2w+c=0 2w=c δp/δc=w-2c+9=0 w=2c-9 w=2(2w)-9 -3w=-9 w=3 2w=c 6=c

  49. 4.4 Complex Unconstrained Optimization P(w,c)=-w2+wc-c2 +9c δp/δw =-2w+c=0 δp/δc=w-2c+9=0 w=3, c=6 (possible max/min) 2) SOC: δ2p/δw2= -2 < 0 δ2p/δc2= -2 < 0, possible max Maximum confirmed

  50. 4.4 Complex Unconstrained Optimization P(w,c)=-w2+wc-c2 +9c w=3, c=6 (confirmed max) 3) Find productivity: Productivity is maximized at 27 when wage=3 and working conditions=6.

More Related