Math Review 1. Graph and label the following points on the Cartesian plane (x-y coordinate plane) (0,5) a b=(4,0) c=(3,3) d = (2,3) 2. Graph the following functions on the Cartesian plane (a) y=3+4x (b) y=10-2r 3. Differentiate the following functions with respect to z Do not simplify (a) f(x)=2x +3ln(z)-x+x/2 - 100z (b) f(x)= In(x)27 (c) f(x) = 2+3 10+1/3 (d) f(x) = (In(x))-1/3 4. Differentiate the following functions with respect to z and y Do not simplify (a) f(x,y)= 3x + 4y-zy (b) f(x, y) = (zy) + (2x-7y) (ln(y)-x-) (c) f(x,y) = n() (d) f(x,y) = (zin(y))/2 5. Constrained Optimization Bill gets utility (satisfaction) from two goods, z and y, according to the utility function. U(x,y)=In(a)+In(y). While Bill would like to consume as much as possible he is limited by his income, which is 100 dollars. Good z costs $2 per unit and good y costs $4 per unit. Maximize Bill's utility subject to the budget constraint using the Lagrangean method. Budget Constraints (Ch 2) 6. Liz has a monthly income of $200. She only buys two things: rice and beans. Suppose rice cost $4 per pound and beans $2 per pound. (a) What is Liz's opportunity cost for one more pound of rice? (b) What is Liz's opportunity cost for one more pound of beans? (c) Write the expression and graphically illustrate Liz's budget constraint. 1 (d) Draw Liz's original budget line and a new budget line with an income of $300. (e) Draw Liz's original budget line and a new budget line that represents the price of a pound of beans increasing to $6. (f) There is a special at the Winn Dixie. The price of beans is $2/pound and if one buys 20 pounds, they get 10 pounds for free. (This only applies to THE FIRST 20 pounds.) Graph this new budget constraint (Liz has income of $200). 7. Suppose Max has $100 to spend on chocolate bars and diet cokes. The price of chocolate bars is $1. The first 10 diet cokes cost $2 each but every diet coke after that is only $0.50 each. (a) Draw the budget set. (b) In total, Max can purchase 170 diet cokes. What is the opportunity cost for the i. first diet coke? ii. 10th diet coke? iii. 11th diet coke? iv. 170th diet coke? 8. Ann consumes only cups of coffee, z, and teaspoons of sugar, za. Ann has an income of $200. A cup of coffee costs $2 and a unit of sugar costs $1. (a) Draw Ann's budget line (graph ry on the horizontal axis and 2 on the vertical axis; label all important points and intercepts). (b) Suppose that congress eliminates all agriculture subsidies from the federal budget. Therefore, the price of teaspoon of sugar increases to 84. Draw Ann's budget line (graph z, on the horizontal axis and r on the vertical axis; label all important points and intercepts). (c) Ignore the information in part b. Instead, of eliminating the agriculture subsidies, congress decides to balance the budget through an increase on taxes. Ann now pays an income tax of 80%. Draw Ann's budget line (graph z, on the horizontal axis and z on the vertical axis; label all important points and intercepts). Preferences (Ch 3) 9. In Professor X's class, the first midterm exam and the second midterm exam are weighted equally toward the final grade. With the first midterm's score on the horizontal axis, and the second midterm's score on the vertical axis, draw three indifference curves for a student in Professor X's class, 10. Barb only consumes beefy double cheese burritos (b) and big macs (m). Barb is always indifferent between two beefy double cheese burritos and one big mac. (a) With beefy double cheese burritos on the horizontal axis and big macs on the ver- tical axis, draw two of Barb's indifference curves. Label Barb's highest preferred indifference curve In and her lowest preferred indifference curve It. (b) Write down the marginal rate of substitution. 11. Jim only consumes beefy double cheese burritos (b) and big macs (m). Jim is always indifferent between three beefy double cheese burritos and two big macs. (a) With beefy double cheese burritos on the horizontal axis and big macs on the vertical axis, draw two of Jim's indifference curves. Label Jim's highest preferred indifference curve ly and his lowest preferred indifference curve IL- (b) Write down the marginal rate of substitution. 2