Quantitative Microeconomics

1. John has income of $200 each month to spend on day passes to the gym (x) and food (y). Assume both of these goods are continuously divisible. A day pass to the gym costs $10 and food costs $1 per unit. 
.  (a)  Draw a neat and well-labeled graph of John budget constraint.  
.  (b)  Suppose John health insurance company offers a wellness plan that pays 50% of gym expenditure up to $40 each month. On another neat and well-labeled graph, draw John’s new budget constraint.  
.  (c)  Assuming John spends his entire budget, the opportunity cost of an additional gym pass takes on two values at different points on the graph. What are those values?  
2. Willie is throwing a party where he will serve lobster meat and shrimp. Both of these are sold by the pound and are continuously divisble. A pound of lobster meat costs $30 and a pound of shrimp costs $15. Willie has $180 to spend on lobster meat and shrimp. 
.  (a)  The store where Willie is shopping offers a deal where anyone buying at least 8 pounds of shrimp receives an extra 3 pounds of shrimp for free. On a neat and well-labeled graph, draw Willie’s budget constraint. Place pounds of lobster on the horizontal axis and pounds of shrimp on the vertical axis.  
.  (b)  Suppose Willie has well-behaved preferences over bundles of lobster meat and shrimp. In three sentences or less, explain why Willie’s optimal bundle would never contain exactly 10 pounds of shrimp.  

3. Michael has preferences over two goods, x1 and x2, represented by the utility function u(x ,x )=x2/3×1/3. 
12 12 (a) Find the MRS12 associated with this utility function. 
.  (b)  Use the MRS12, the price ratio, and the budget constraint to find Michael’s optimal bundlewhenm=3000,p1 =50,andp2 =10.  
.  (c)  Find the equation of the indifference curve containing the optimal point and solve it for x2. (You will need to calculate the utility at the optimal point in order to do this.)  
.  (d)  Graph the budget line, the optimal point, and the indifference curve you just found on a single graph. Be sure to label the curves, the optimal point, and the axes.