(1.)  Evaluate the definite integrals of each of the following functions for the range X = 0 through X = 5


(2.)  Suppose capital investment I(t) in any time period t is: I(t) = 5t^1.25   for t = 0...T. 

1. Calculate the quantity of capital stock accumulated between t = 0 and t = 6 if the investments are made yearly in lump amounts [calculate I(0) through I(6) and sum]. 
2. Calculate the quantity of capital stock accumulated between t = 0 and t = 6 if investment is a continuous process [evaluate the definite integral of I(t) between t = 0 and t=6].

(3.)  The demand schedule for widgets is Q = 100 - 10P^0.7.  If P falls from $50 to $40 per unit, calculate the increase in consumer surplus. 

(4.)  Suppose you make a one-time deposit of $1000 (PV) in a savings account earning 6% annual interest (r = 0.06), and you just let the interest accumulate. 

1. If the interest is compounded annually, how much will you have in the account after t = 20 years?  Use the formula FV = PV(1+r)^t
2. How much will you have if the 6% annual interest is compounded monthly (r' = 0.06/12 = 0.005 per month over 20x12 = 240 months)?
3. How much will you have if the 6% annual interest is compounded daily (r" = 0.06/365.25 = 0.00016427 per day over 20x365.25 = 7,305 days)?
4. How much will you have if the 6% annual interest is compounded continuously?  Use the formula FV = PVe^rt .

(5.)  Suppose you open a savings account and add $1000 to it every year, and let the interest accumulate.

1. If the 6% annual interest is compounded annually, how much will you have after 10 years?  [Calculate the FV of each annual deposit, then sum.]
   
2. If the 6% annual interest is compounded continuously, how much will you have after 10 years?  [Evaluate the integral of FV = PVe^rt over the range t = 0 to t = 10 -- much easier!]

(6.)  Suppose you are bidding for a Treasury bond that will have a redemption value (FV) of $10,000 exactly 10 years from today (there are no coupons or other intermediate interest payments).   Using either formula,  PV = FV(1+r)^-t  or  PV = FVe^-rt ...

1. calculate the maximum (PV) you would bid for a 5% annual rate of return.
2. calculate the maximum you would bid for an 8% annual rate of return.

(7.)  The lottery is advertising a current jackpot of $10 million!  Tickets are only $1, and each ticket has a 1-in-20 million chance of winning!  If you win, you will receive $500,000 per year for 20 years!  Of course you will have to pay 40% each year in taxes, so your actual take will be $300,000 per year.

1. Using a discount rate r = 0.06, calculate the present value of a 20-year stream of $300,000 payments.
2. Multiply the actual present value of this jackpot by the 1 in 20 million odds of winning to obtain the statistical value of a $1 ticket.  (Now you see why some economists consider the lottery to be "a voluntary tax on stupidity!")
   

(8.)  Investment A will cost you $20,000 today and will be worth $35,000 in 12 years.  Investment B will cost you $10,000 today and will be worth $20,000 in 15 years.  Which investment has the higher implicit rate of return (r)?  (Solve either discounting formula for r as a function of PV, FV and t.) 

(9.)  The inverse demand for widgets is P_d = 100 - 6Q^0.5  and the inverse supply is P_s = 10 + 3Q^0.5.

1. Calculate the equilibrium price and quantity in this market.
2. Solve the integrals of these inverse functions for the range Q=0 to Q=Q_eq, and then calculate the consumer and producer surpluses.
3. Suppose the government imposes a tax of $18 per widget sold.  Calculate the new equilibrium Q where P_d = P_s + 18.  How much will the government collect in tax revenues from this market?  What is P_s?  What is P_d?
   
4. Calculate the new consumer surplus; how much CS is lost because of this tax?  Calculate the new producer surplus; how much PS is lost because of this tax?
   
5. Calculate the economic waste (deadweight loss) caused by this tax (lost CS + lost PS - tax revenues). 

(10.)  A consumer has $1000 to spend over two time periods.  Her discount rate between time periods is r = 0.10, and her utility function in time period i is U_i = X_i^0.5 where X_i is consumption in time period i.  So she wants to maximize U₀ + U₁/(1+r) = X₀^0.5 + (X₁^0.5)/(1+r) subject to the constraint 1000 - X₀ - X₁ = 0. 

1. Set up the Lagrangean and solve for the optimal values of X₀ and X_1. 
2. What is the present value of the combined utilities? 
3. How much less utility would she get from simply consuming $500 in each time period? 
4. How much less utility would she get from consuming all $1000 in time period 0 and nothing in time perio 1?