Group-Clicker Question (1 pt): Solve for x1 and x2 .Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Group-Clicker Question (P): Find x1 and x2 that maximizesGroup-Clicker Question (P): Find x1 and x2 that maximizesSlide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Group-Clicker Question (1 pt): Solve for x1 and x2 .1. x1=1, x2=02. x1=1, x2=1/23. x1=3, x2=-24. No solution exists0%0%0%0%“Optimization hinders evolution.” -Alan J. PerlisECON 410Math Review – Partial Derivatives & OptimizationPlease take a moment to make sure your clicker is working properly. Select an answer. You should see a green light. If it’s a green/red blinking light, no light at all, or you see the Ghostbuster’s sign in your LCD, please come to the front of the class and we’ll try to fix it before class begins.Please take a moment to make sure your clicker is working properly. Select an answer. You should see a green light. If it’s a green/red blinking light, no light at all, or you see the Ghostbuster’s sign in your LCD, please come to the front of the class and we’ll try to fix it before class begins.+ =+ =1 21 212 2 3xx xx2Class 3 – Math ReviewHW1 is due on Tuesday. Make sure to submit your answers in Sakai.HW1 is due on Tuesday. Make sure to submit your answers in Sakai.Most information I post to our Facebook group page. Please join if you have not.Most information I post to our Facebook group page. Please join if you have not.3Class 3 – Math ReviewAlgebraAlgebra•Working with Variables•Solving a single equation for a single variable•Solving multiple equations for multiple variables•Working with Variables•Solving a single equation for a single variable•Solving multiple equations for multiple variablesDerivativesDerivatives•Basic Derivative Rules•Chain Rule•Partial Derivatives•Basic Derivative Rules•Chain Rule•Partial DerivativesOptimizationOptimization•First Order Conditions•Second Order Conditions•First Order Conditions•Second Order Conditions4Class 3 – Math ReviewPartial DerivativesPartial DerivativesWhat is a derivative?Hours of Practice per weekGrade1 2 3 475808590How much will my function increase (or decrease) if I increase my variable by a little bit.5Class 3 – Math Review6Class 3 – Math Review7Class 3 – Math ReviewPartial DerivativesPartial DerivativesWhat is a partial derivative?Hours of Practice per weekGrade1 2 3 475808590How much will my function increase (or decrease) if I increase my variable by a little bit in a specific direction.8Class 3 – Math ReviewPartial DerivativesPartial DerivativesPartial Derivative Notation:The partial derivatives of a function, f(x1,x2,x3,…), are 21 3, , ,...df df dfdx dx dx2 311 2 1 2 1 2( , ,...), ( , ,...), ( , ,...),...x x xx x x x x xff f1 1 2 2 1 2 3 1 2( , ,...), ( , ,...), ( , ,...),...x x x x xff fx9Class 3 – Math ReviewPartial Derivative Rule:When we are taking the derivative in a specific direction, all other directions are fixed. Thus, to take a partial derivative with respect to a specific direction (variable), treat all other directions (variables) as constant.10Class 3 – Math ReviewExample: Find the partial derivatives of f(x,y)=xyfx(x,y):f(x,y)=xy fx(x,y)=yfy(x,y):f(x,y)=xy fy(x,y)=xconstantGroup-Clicker Question (P): Find all partial derivatives of f(x1,x2,x3)=x1x2x31. f1=f2=f3=12. f1=f2=f3=x1x2x33. f1=x2x3, f2=x1x3, f3=x1x24. f1=x1, f2=x2, f3=x35. f1=f2=f3=x2x3+x1x3+x1x2Group-Clicker Question (P): Find all partial derivatives of f(x,y)=x2-y21. df/dx=2x, df/dy=2y2. df/dx=2x, df/dy=-2y3. df/dx=2x-2y, df/dy=2x-2y4. df/dx=-2y, df/dy=2x5. df/dx=2y, df/dy=2xGroup-Clicker Question (1 pt): Find all partial derivatives of f(x,y,z)=x2-ln(z)y21. df/dx=2x, df/dy=-2y, df/dz=1/z2. df/dx=2x, df/dy=-2y, df/dz=-y23. df/dx=2x-ln(z)y2, df/dy=-2yln(z), df/dz=-y2/z4. df/dx=2x, df/dy=-2yln(z), df/dz=-1/z5. df/dx=2x, df/dy=-2yln(z), df/dz=-y2/z14Marginal Effects:The change to a value that occurs when another value changes. In other words, the derivative.Class 3 – Math ReviewExamples:Marginal Product of Labor (MPL): How much production changes with an extra employee Marginal Utility with respect to Good 1 (MU1): How much utility changes with an extra unit of Good 1Marginal Revenue (MR): How much extra revenue is generated by producing (and selling) another unit of a good.��( , )f K LL��1 21( , )U x xx��( )R qq15Higher-Order Derivatives:As the derivative of a function is, itself, a function, we can take derivatives of derivatives, etc.Class 3 – Math ReviewHow much output I can produce with a given level of Capital (K) and Labor (L)How much extra output I can produce if I currently have K units of capital, L units of labor, and I increase my capital by 1 unitHow much the rate at which I can gain extra output using extra capital will change if I currently have K units of capital, L units of labor, and I increase my capital by 1 unit=1/2 1/3( , )f K L K L-��=1 1 3/2 /( , ) 12Kf K LLK-=-��213/2 /32( , ) 14Kf K LLK16Class 3 – Math Review17Class 3 – Optimization ReviewSomething behaves according to a functional specificationWhat is the best choice??18Class 3 – Optimization Review19Class 3 – Optimization ReviewEthiopia's poverty-stricken economy is based on agriculture, accounting for almost 46% of GDP, and 85% of total employment. The agricultural sector suffers from frequent drought and poor cultivation practices.Population below $1.25 per day: 29.2%Life expectancy: 60 years20Class 3 – Optimization Review0 100 200 300 400-0.8-0.6-0.4-0.200.20.40.6Ethiopian Herd Growth RateHerd SizeGrowth Rate “Stochastic Wealth Dynamics and Risk Management Among a Poor Population”; Travis Lybbert, Christopher Barrett, Solomon Desta, and D. Layne Coppock; The Economic Journal, October 2004Group-Clicker Question (1 pt): Assume I told you “If you study for an extra hour, your midterm grade will go up”. Is it possible that your current level of studying will maximize your score?1. Yes2. No0%0%Group-Clicker Question (1 pt): Assume I told you “You’re burned out. Your midterm grade would have increased if you had stopped studying and gone to sleep an hour