Examples of Problems by Difficulty:

One thing to note is that “difficulty” is subjective. If you find the hard ones easy and the easy ones hard, don’t feel like there is something wrong. This was just a pass at assessing difficulty based on the amount of time it takes to answer in my experience.

Easy – Moderately Easy

  1. Define the Marginal Rate of Substitution.
  2. MRS of y for x is the instantaneous rate of exchange of y for additional x to keep utility constant.
  3. Give an example of preferences over three goods, x y and z, which are complete.
  4. If the utility function is U(x,y) = 2x + 3y, find the MRS.
  5. MRS = -2/3
  6. Illustrate an example of a contour map for perfect substitutes utility function.
  8. What is the general form of a utility function for perfect complements?
  9. Give an example of a Cobb-Douglass utility function.
  10. Define what a neutral good is.
  11. A neutral good is a type of good where more does not lead to more preferred bundles. We are indifferent between bundles with any amount of this type of good.
  12. If A is weakly preferred to B and B is weakly preferred to C, then what is the preference relation between A and C if preferences are transitive.
  13. If the price of x is 10, the price of y is 20, and income is 100, write the function for the budget constraint.
  14. If income increases and quantity demanded of x decreases, then is x a normal or inferior good?
  15. Inferior because we know that meaning that we are consuming less x when we have more income.
  16. If the demand for a good is , what is the quantity demanded of x when the price of x is 2 and income is 20?
  1. What is the general form of the Lagrangian for some utility function U(x,y) and budget constraint ?

Moderate

  1. Show an example of preference relations which satisfy completeness and reflexivity but not transitivity.
  2. Solve the following maximization problem for
  3. First note, this is a Cobb-Douglass utility function… we can use the Lagrangian
  4. Using (i) and (ii), we can reduce this to two equations with two unknowns x and y
  5. Divide (i) by (ii):
  6. We can now use this with (iii) to solve for x:
  1. If the demand for good x is

    and the original price of good x is and , what is the compensated income needed to keep the original bundle affordable after a change in price to .
  2. First need to find original x:
  3. Now we can find the change in income which would keep this bundle affordable after the price change:
  4. The compensated income is just
  5. Explain the concept of “diminishing marginal rate of substitution”. Illustrate an example of an indifference curve which demonstrates this concept.
  6. Diminishing MRS is when we move along an indifference curve and substitute x for y, the rate at which which we substitute diminishes
  7. Explain what a Giffen good is.
  8. A Giffen good is a type of good where as it becomes more expensive, the utility maximizing quantity demanded of the good increases.
  1. Illustrate an income offer curve when x and y are normal goods.
  3. Solve the following maximization problem for
  4. First note, this is a Cobb-Douglass utility function… we can use the Lagrangian
  5. Using (i) and (ii), we can reduce this to two equations with two unknowns x and y
  6. Divide (i) by (ii):
  7. We can now use this with (iii) to solve for x:
  1. Illustrate graphically the optimization problem when X and Y are perfect complements.

  1. Illustrate graphically the optimization problem when X and Y perfect substitutes and where the
  3. True or False: Regardless of utility function, the best bundle is where the marginal rate of substitution is equal to the price ratio.
  4. False, if, for example, our utility function is linear (perfect substitutes), then the best bundle is either consuming only x (if |MRS|>px/py) or only y (if |MRS|< px/py).
  1. Illustrate graphically the income and substitution effect of a decrease in the price of x. Based on your illustration, is x a normal or inferior good?
  2. This can take many forms. Here is one example where x is a normal good
  4. We know it’s a normal good because the income effect (x3-x2) is positive when income goes up

More Difficult

  1. Illustrate an indifference curve which satisfies monotonicity but not convexity.

Hint: one example is a special case we’ve discussed in class

  1. These are perfect substitutes. I know they are monotonic because holding x constant and adding more y leads to higher indifference curves
  2. They are not convex as taking the average of two bundles we are indifferent between leads to a bundle on the same indifference curve. (we don’t strictly prefer the average)
  1. If the marginal utility of x is 10, the marginal utility of y is 20, the price of x is 5, the price of y is 20, and income is 100. Find the optimal bundle

Hint: this is a case of perfect substitutes

  1. If marginal utility is constant, then we know we have MRS that is constant. That means we are dealing with perfect substitutes
  2. We know |MRS| = (10/20) =(1/2) > (1/4) = (5/20) = (px/py)
  3. Since the indifference curves are steeper than the slope of the budget constraint, we would only consume x, so y*0
  4. Knowing that the optimal bundle is on the budget constraint:
  5. Thus
  1. Solve the following maximization problem for
  2. Note: this is a linear utility function. We are dealing with perfect substitutes and any attempt to use the Lagrangian will end in dire failure
  3. However… We know we will have a corner solution (only consume x or y)
  4. We consume only x when a/b>px/py, and only y if a/b<px/py
  5. a/b = 2/2=1 > 1/3 = px/py
  6. Thus we consume only x and y*=0
  7. We know the best bundle is on the budget constraint:
  8. (x*,y*) = (6,0)
  9. If the utility function is , derive the demand function for x and y.

Hint: you cannot use the Lagrangian

  1. This is the case of perfect complements
  2. The best bundle is on the joint where 2x=y
  3. The best bundle is on the budget constraint also, so we can substitute:
  4. Solving for x we get:
  5. Substituting into (b) to find y:
  1. Illustrate the Income-Offer Curve when x is an inferior good and y is a normal good.
  3. Derive the demand functions for x and y if our problem is to maximize subject to a budget constraint.
  4. First note, this is a Cobb-Douglass utility function… we can use the Lagrangian
  5. Using (i) and (ii), we can reduce this to two equations with two unknowns x and y
  6. Divide (i) by (ii):
  7. We can now use this with (iii) to solve for x and y:
  1. True or False: If and , then the consumer would only consume x if

Explain your answer.

  1. TRUE
  2. The |MRS| =10/10 = 1 > px/py. When this happens, we want to consume only x (more utility per dollar spent on x, steeper indifference curves than budget constraint)
  1. True or False: If the price of x increases and the demand for x decreases, then x is an inferior good. Explain.
  2. FALSE, its impossible to determine from this information
  3. It is possible that the substitution effect and income effects are both negative, which would happen if the good was normal. If x is inferior, the income effect could be positive but still end up with a total effect which is negative.
  4. Suppose the demand function for x is

The original price of x is and income is . Find the income and substitution effects of increasing price to .

  1. We need m’, x(px,m), x(px’,m’), and x(px’,m)
  2. X(px,m) = 10/1 = 10
  1. Illustrate graphically the income, substitution, and total effects of an increase in the price of x when x is a normal good.
  2. I omitted the indifference curves, but A is the best bundle on budget constraint B1. When prices go up for x but we compensate income to keep A affordable, the new best bundle is bundle B. The substitution effect is x2-x1. When we allow purchasing power to go down ,the we move from budget constraint B1’ to B2. The income effect is x3-x2. The total effect is x3-x1
  3. We know this is a normal good because when purchasing power decreased (decrease in m), we wanted less of the good.
  1. Cassie likes to eat cookies and milk in a 2 to 1 ratio (2 cookies for one cup of milk). For example: if she has 3 cookies and 1 cup of milk, she would throw away the third cookie. Find a utility function which could represent her preferences over cookies and milk.

Hint: it is a special case we have discussed in class

  1. This is the case of perfect complements because she likes to consume things in a specific ratio
  2. We know the utility function will take the form
  3. What we need to decide is what a and b are.
  4. One thing to note, at each joint,
  5. Each joint will be reflective of the ratio that we want. So we know at least one joint will have 2 cookies (x) and 1 milk (y), bundle (2,1)
  6. Substituting into :
  7. So b = 2a.
  8. Now we can choose a value for a and then find b
  9. This says that the bundles on the joint have x = 2y. Is that right? If we want (2,1), then ax=1(2) = 2(1) = by.
  10. Really, as long as b=2a, then any utility function is okay.
  11. Another easy way to get at this is to note that (2,1) is on a joint, and the value of x is exactly 2 times the value of y, so x=2y. a=1 and b=2.