Question 1: Analyzing Economic Problems

Question 1. A firm produces the newest gadget on the market. For production of the gadget it uses E machine-hours and L person-hours of labor to produce a quantity Q of the gadget. The production function is given by

Q =√(EL).

The firm must pay pE for each machine-hour and wage pL for each hour of labor. The production manager is told to produce a total amount of Q¯ units of the gadget by upper management. The production manager wants to produce this amount with the lowest cost C possible.

a) Which are the endogenous and exogenous variables of the problem?
b) What is the firm's cost function?
c) Set up the optimization problem (do not solve it).

Question 2. Think about a constrained optimization problem from your personal life and set it up mathematically. Points are given for uniqueness and correct mathematical representation. Do not solve the problem, just set up the objective function and the constraint.

Question 3. Your demand for chocolate is given by Q^d = 50 -1/a*P, where a is your craving for chocolate. For now assume a = 1. Your room mate has a stack of chocolate which she doesn't like to share, but she will sell it to you according to the supply function Q^s = D + 5*P. Assume for now that D = -10. Think of P as the number of times you have to do the dishes per week.

a) Sketch the supply and demand curve, label the axes and intercepts
b) Make and example of what D could represent in terms of the scenario. What does the negative value of D imply?
c) Calculate equilibrium quantity and price
d) Show graphically and analytically what happens when your craving for chocolate increases from 1 to 4.
e) What would a decrease in D mean in terms of what you answered to b)? Show graphically and analytically what happens when D decreases from -10 to -15 (with a = 1)

2 Demand Elasticities

Consider the following demand curve:

P = 15 - 1/20 Q^d + 4 P²y +1/5√I

where Q is quantity demanded, P is the good's own price, P_y is the price of another good, and I is income.

Question 4. Price Elasticity (Assume Py = 1/2 I =100)

a) Sketch the curve in the appropriate graph for price elasticity; label axes and intercepts.
b) What is the slope (?Q/?P) of this function?
c) At a price of P = 9, what is the price elasticity? What kind of good is this demand for (and why)?
d) Make a (unique) example for a good of this kind.

Question 5. Income Elasticity (Assume Py = 1/2)

a) Sketch the curve in the appropriate graph for income elasticity; label axes and intercepts.
b) What is the (marginal) slope (?Q/?I) of this function?
c) At a price of p = 4 and I = 100, what is the income elasticity? What kind of good is this demand for (and why)?

Question 6.Cross-Price Elasticity (Assume I = 100)
a) Sketch the curve in the appropriate graph for cross price elasticity; label axes and intercepts.
b) What is the (marginal) slope (?Q/?Py) of this function?
c) At a price of p = 2 and py = 2 ,what is the cross price elasticity? What is the relation between the two goods (and why)?
d) Make a (unique) example for two goods with this relationship.
3 Preferences and Utility

Question 7. Calculate marginal utility for both goods and the marginal rate of substitution for the following utility functions:
a) U(x,y) = 5x + 2y
b) U(x,y) = x + 2 ∗√y
c) U(x,y) = x1/4 ∗y3/4
d) U(x,y) = min(3x,7y)
4 Consumer Choice

Question 8. A consumer has the following utility function and budget constraint:
U(x,y) = x ∗y
w = p_xx + p_xy,
where w is the consumer's wage and px and py are the prices of good x and y, respectively.
Assume w = 50,p_x = 2,p_y = 10
a) Draw the budget line and sketch the indifference curves. Do you expect an interior solution? Why?
b) Solve for the optimal consumption bundle using the substitution method we learnt in class.
c) Solve for the optimal consumption bundle by equating the MRS with the slope of the budget line. Do you get the same solution as in b)
d) Now assume the government introduces a 20% wage tax (0.20∗w has to be paid to the government and will never be seen again). Show graphically and analytically how this changes your answer.

Question 9. A consumer has the following utility function and budget constraint:
U(x,y) = 3x + 5y
I = p_xx + p_xy,
Assume I = 10,p_x = 4,p_y = 5
a) Draw the budget line and sketch indifference curves. Do you expect an interior solution? Why?
b) Find the optimal consumption bundle
c) There is now inflation in the economy. All prices as well as your income increase by 10%. How does this change your answer?

Question 10. [Consider the following internet plan. The first 2 gigabyte of data download cost a lump sum of $50, every additional 0.1 gigabyte costs an additional $1 dollar. But if you go over 3 gigabyte you will have to pay 5$ per 0.1 gigabyte.

a) Draw the budget line (to scale) as a choice between gigabyte download bandwidth on the x-axis and a composite good (money) on the y-axis. What is the maximum amount of bandwidth you can afford? Your budget is $100?

b) Draw a second plan where you have to pay $60 for unlimited internet.

c) Show and explain with the help of sketching different kinds of indifference curves which plan would be preferred under which circumstances. How much bandwidth would be consumed in these cases?