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4. Models with Multiple Explanatory Variables Up until this point, we have assumed that our dependent variable (Y) is affected by only ONE explanatory variable (X). Sometimes this is the case. Example: The amount of dollars in your bank account is equal to the number of cents in your bank account/100. Usually, this is not the case. Example: your mark on the midterm depends on how well 1 you study and your innate intelligence. 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.) 2 4. The Partial Derivative It is often impossible to analyze all various movements of explanatory variables and their impact on the dependent variable. 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. 3 4. Calculusand 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 4 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). 5 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: 1) ∂y/∂x 2) ∂f(x,z)/∂x 3) 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 6 in economic shorthand 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 7 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(x2-xz)+cos(z2-xz) 8 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 9 4.1.1 Higher Partial Derivatives From before: Let y = 3x2z+xz3-3z/x2 ∂ y/ ∂ x=6xz+z3+6z/x3 ∂ 2y/ ∂ x2=6-18z/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 10 4.2 Applications using Partial Derivatives As many real-world situations involve many variables, Partial Derivatives can be used to analyze these situations, using tools including: Interpreting coefficients Partial Elasticities Marginal Products 11 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) 12 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 starving. 13 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 14 4.2.1 Corny Example Intercept = 500 -if it doesn’t rain, there are no scarecrows and no fertilizer, the farmer will harvest 500 bushels dcorn/drain=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 (∂ 2corn/ ∂ rain2=-2<0, concave) 15 4.2.1 Corny Example dcorn/dscare=50Fertilizer -More scarecrows will increase the harvest 50 for every tonne of fertilizer -if no fertilizer is used, scarecrows are useless (∂ 2corn/ ∂ scare2=0, straight line, no curvature) dcorn/dfertilizer=50Scare -More fertilizer will increase the harvest 50 for every scarecrow -if no scarecrows are used, fertilizer is useless (∂ 2corn/ ∂ fertilizer2=0, straight line, no curvature) 16 4.2.1 Demand Example Consider the demand formula: Q = β1 + β2 Pown + β3 Psub + β4 INC Where 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 17 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) 18 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) 19 4.2.2 Cobb-Douglas University Consider a production function for university degrees: Q=aLbKcCf Where L=Labour, K=Capital (excluding computes), and C=Computers More simply, labour can reflect professors, capital can reflect classrooms, and computers can reflect computers. 20 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 elasticity gives us the MARGINAL PRODUCT of labour 21 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 product of professors is 12.5 22 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/ ∂ L=Marginal Product of Other (MPO) 23 4.2.2 Cobb-Douglas Elasticities Since the “Professor Elasticity of Demand” (or PED…an arbitrary name) is expressed: PED = ∂ Q/ ∂ L(L/Q) We can find that PED =b(Q/L)(L/Q) =b The partial elasticity with respect to labour is b. Find the remaining derivatives and elasticities. 24 4.2.2 Logs and Cobbs We can highlight elasticities easier by using logs: Q=aLbKcCf Ln(Q)=ln(a)+bln(L)+cln(k)+fln(C) We now find that: PED= ∂ ln(Q)/ ∂ ln(L)=b Therefore using logs, elasticities more apparent. 25 4.2.2 Logs and Demand Reverting to 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 26 4.2.2 ilogs Considering the demand for the ipad, assume: Ln(Qdipad)=2.7 -1ln(Pipad)+4 ln(Pnetbook)+0.1 ln(I) We now find that: Own Price Elasticity = -1, demand is unit elastic Cross-Price Elasticity = β3, 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 27

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