I am assuming that you generally are familiar with the material in chapters
2, 3, and 5.  I will cover a few sections in these chapters, but you will
be mostly responsible for them on your own.

All non-chapter readings are in the library. Although the papers are interesting in their own right, their general pupose is to demonstrate how the various mechanics and ideas can be used to create economic arguments. For example, the Solow paper provides a nice example of how to use Euler's theorem.  I'll try to let you know when we will go through a paper in detail, and when we will be focusing
in on a particualar part. To help you a little bit, please note that a `*' indicates more relevant reading.

Week 1: Maximization

* a. Silberberg, Ch. 1, Section 2.5, Section 3.6, Ch. 4.

* b. Alchian ``Evolution, Uncertainty, and Economic Theory'' Journal of Political Economy 1950.

Week 2:  Traditional Comparative Static Methodology and the Envelope Theorem

* a. Silberberg, Ch. 4, Ch. 6.

Week 3: Cost Functions

* a. Silberberg, Ch. 7,  Ch. 8.

b. Akerlof and Yellen. ``Can Small Deviations from Rationality Make Significant Differences to Economic Equilibria?'' AER Sept. 1985.

c. Alchian ``Cost'' in  Economic Forces At Work

Week 4: Cost Functions: Special Topics

* a. Silberberg, Ch. 8, Ch. 9.

b. Alchian ``Cost and Output'' in Economic Forces At Work

Week 5: Midterm 1.

Week 6:  Demand Theory

* a. Silberberg Ch. 10

* b. Barzel, ``The Testability of the Law of Demand'' in Sharpe and Cootner (eds) Financial Economics: Essays in Honor of Paul Cootner 1982.

* c. Barzel and Suen ``The Demand Curves for Giffen Goods Are Downward
Sloping''. EJ 1992.

Week 7:  Demand: Special Topics

Consumer's Surplus

* a. Silberberg Ch. 11.5.

All-or-nothing Demands:

* a. Suen, ``Statistical Models of Consumer Behavior with Heterogeneous
Values and Constraints'' EI, 1990.
* b. Barzel, ``Rationing by Waiting''  Journal of Law and Econ.
1974.

Week 8:  Special Topics, cont.

Household Production:

* a. Silberberg, Chapter 11.4.
b. Becker, ``A Theory of Allocation of Time,'' Economic Journal
1965
c. Pollak and Wachter, ``The Relevance of the Household Production
Function and Its Implications for the Allocation of Time,'' JPE 1975

Quality and Shipping the Good Apples:

* a. Leffler ``Ambiguous Changes in Product Quality'' AER 1982.
* b. Silberberg Chapter 11.3.
c. Umbeck ``Shipping the Good Apples Out: Some Ambiguities in the Interpretation of `Fixed Charge'{''}, JPE 1980, 88 no. 1.
d. Kaempfer and Brastow, ``The Effect of Unit Fees on the Consumption
of Quality'', EI 1985.
e. Bertonazzi, et al. ``Some Evidence on the Alchian and Allen Theorem: The Third Law of Demand?'' EI July 1993.
 

Week 9: Midterm 2.

Week 10: Intertemporal Choice

* a. Silberberg, Chapter 12.

Week 11: Expected Utility Theory

* a. Silberberg Chap. 13.

Week 12: Contracts and Incentives

* a.Silberberg Chap. 15.

This package contains old problems and exam questions that I've asked over the years that are relevant to 802. The purpose of these questions is to familiarize you to the type of question I might ask you on an exam, and to give you some practice in both technical and intuitive questions. I won't hand out the answers so don't ask me. I suggest you form a study group and work on them each week.

I will announce which questions are due for assignments throughout the semester.

1)

It is a ``well known fact'' that the correlation between investor return and education level for stockbrokers is negative - better brokers usually have less education. Is this consistent with the notion of maximization or a refutation of it?

2)

Consider the following two models of a discriminating monopolist subject to a tax in one market:

In model one, cost is a function only of total output, whereas in (ii), cost is a more complicated (and general) function of each separate output. The tax rate t is a parameter.

Demonstrate any (potentially) observable similarities and differences between these two models.

3)

From question (2), and using model (i), compare the effect on total output y = y₁+y₂ from:

a.

a per unit tax t on y₁ alone, versus

b.

a per unit tax t on y.

4)

Consider the profit maximization model with two factor inputs, x₁, x₂.

a.

Derive the choice functions for this model.

b.

Show that the factor demand functions are homogeneous. Of what degree? In what parameters?

c.

Show that the elasticities of x^*_i(w₁,w₂,p) with respect to w₁, w₂, p must sum to zero.

d.

Define x₂ as an inferior factor when

¶x₂^*/ ¶p < 0. Does this make sense as a definition of inferiority?

e.

If x₂ is an inferior factor, then show that ¶x^*₂/w₁> 0 .

f.

Continue to assume that x₂ is an inferior factor. Derive the expression for the response of demand for x₁ to changes in the output price. What do you know about the sign of this derivative?

g.

Now suppose that x₂ is held constant (at its previously profit maximizing level). How does the response of x₁ to changes in the output price compare with the case where x₂ is allowed to vary?

h.

How would your answer to (g) change if x₂ were normal rather than inferior, and x₁ were inferior?

5)

Consider the utility function of U(x₁, x₂), where U₁, U₂ > 0. For a given level of utility, x₂ is a function of x₁; that is,

a.

Show that this condition is true, if and only if the determinant:

6)

Consider the direct utility function U = x₁³x₂³.

a.

Verify that this utility function is an increasing monotonic transformation of the Cobb-Douglas utility function V = x₁^.5x₂^.5. Does this imply that U is a well behaved utility function?

b.

Show that U exhibits increasing marginal utility for both goods. What is the moral of this?

7)

Now consider the utility function U = -.5[V(x)]² + V(x) where V(x) = x₁² + x₂² -2.

a.

Show that in a neighborhood of the point (1,1), U_i(1,1) > 0 and U_ii(1,1) < 0, so that the utility function exhibits positive but diminishing marginal utility.

b.

Show that U is an increasing monotonic transformation of V provided that the x's are close to the point (1,1). That is, show that the function f(z) = -.5z²+z is monotonically increasing in the neighborhood of z = 0.

c.

Show that U is not a well-behaved utility function. (Hint: draw the indifference curves). What is the moral of this utility function?

8)

Consider an economic model whose general mathematical structure is:

a.

How are the choice functions, x_i^*(a) derived?

b.

How is F(a) derived?

c.

Show that the rates of change of F and h with respect to a are the same, for choices consistent with the model.

d.

Show that at such points, the rate of change of the slope of F is greater than the corresponding rate for h.

e.

Illustrate (b), (c), and (d) graphically, labeling your graph carefully.

f.

Show that h_1a ¶x^*_i/¶a > 0 for any a. Explain why this is the source of the comparative static results.

9)

Consider the standard utility maximization model

a.

Show that U^*_pi = -l^*x^*_i.

b.

Show that ¶(l^*x^*_i)/¶p_j = ¶(l^*x^*_j)/¶p_i

c.

Suppose now that U^*(p₁ ... p_n, M) is additively separable in the prices, (ie U^*_pipj = 0). Show that

10)

Let f(x) be a constant returns to scale production function. Show that if every factor x₁ is paid the value of its marginal product pf_i then profits are zero.

11)

Let y = f(x₁,x₂) be a production function that is homogeneous of degree one. Show that if the average product of input 1 is increasing then the marginal product of input 2 must be negative.

12)

Consider y = Ax₁^a x₂^b where a+ b = 1.

a.

Show that this production function exhibits CRS.

b.

For a two factor crs production function, the elasticity of substitution is given by:

c.

Assuming price taking in both markets, calculate the cost minimizing input demand functions.

d.

Can you calculate the profit maximizing demands? Why not?

13)

Consider the utility function U(x,y) = x² + y².

a.

Do these preferences satisfy quasi-concavity?

b.

Derive the ordinary demand curves x^* and y^* and the utility constant demand curves.

c.

What is the expenditure function for these preferences? Fix p_y = 1 and U = 1 and draw M(p_x,p_y,U) as a function of p_x.

14)

A demand study results in the following data:

15)

The Lagrange multiplier on the budget constraint in a standard utility maximization problem is often referred to as the marginal utility of money. The reciprocal of this multiplier is given by the partial derivative of the expenditure function with respect to utility. Using this, show that there exists a normalization of U(x) such that l is independent of U if preferences are homothetic.

16)

Suppose that a consumer is endowed with goods X_i rather than money income. The demands that result from such a model are often called ``Slutsky'' demands, and we can denote them as X_i = X^s(p, X⁰) (where p is the price vector and X⁰ is the endowment). If we endowed the individual with dollars, rather than goods we would get a Marshallian demand X_i = X^m(p,M), and of course, we can also define a Hicksian demand X_i = X^u(p,U⁰). Assume throughout this question that X_i is inferior, but not Giffen.

a.

Explain the meaning of the following identity:

b.

At the point of identity, what is true about the slopes of each demand curve?

c.

When you are not at the point of identity what is true about the slopes of the three demand curves?

d.

Now draw the demands for good i.

e.

Explain why the demands are different, being as intuitive as possible.

17)

Consider the Cobb-Douglas production function:

a.

Is this function homogeneous? Of what degree? Under what conditions will it be concave?

b.

Let v be the rental rate on captital, and w be the wage for labor. Derive the conditional factor demand functions and the cost function. Verify the properties of the cost function.

18)

Consider the production function:

a.

Is this function homogeneous? Of what degree? Is it concave?

b.

Find the conditional factor demands and the cost function. Verify the properties of the cost function.

19)

Consider the following production function:

a.

Is this function homogeneous? Of what degree?

b.

What are the conditional demand functions?

20)

In class we defined a CES production function. Find the associated cost function.

21)

Suppose a company offers its employees free parking that would rent for $200 per month. Employees are not allowed to sublet the spots. If the company decides to charge $200 per month for the spaces, would we expect any change in the quality of cars parked in the lot? (If it makes it easier, assume there are only two kinds of cars: good cars which cost $500 per month to rent, and poor cars which cost $100 per month to rent.)

22)

An economist at UCLA did the following test. He watched coffee drinkers at the school's cafeteria and counted how many that had coffee-to-go purchased large cups vs. small cups, and repeated the procedure for people consuming coffee within the cafeteria. He claimed that consumer theory predicts that people having coffee-to-go will consume the larger cups, relative to those remaining. Why would he predict this? Do you agree?

23)

Consider the following set of preferences:

a.

Show that the demand for x is independent of income.

b.

Let the price of x fall from 1 to .25. Calculate (a) the consumer surplus (b) the compensating variation (c) the equivalent variation associated with this fall in p_x.

c.

In general CV and EV will not be equal to one another. In light of your answer to this question explain why not.

 

	Px	Qx	Py	Qy
Time 1	10	20	5	20
Time 2	8	15	8	?

24)

Answer the following based on the table above. X and Y are the only two goods. The consumer has stable, ``normal'' (convex, transitive, etc.) preferences. For what values of Q_y in Time 2 can we conclude that:

a.

the consumer is better off in Time 1 than Time 2.

b.

the consumer is better off in Time 2 than Time 1.

c.

x is inferior.

d.

y is inferior.

25)

An individual consumes two commodities x₁ and x₂ which are sold at prices p₁, p₂. His expenditure function is given by:

a.

Use the Envelope theorem to derive the compensated demand functions for the two goods. What is the own price substitution effect for each of these commodities? From this, what can you infer about the shape of the indifference curves for the preferences that generated this expenditure function?

b.

The government intends to place a fixed sales tax (t) on commodity 1 so that the post tax price of x₁ will be p₁+t. At the same time the government wants to leave the individuals's welfare unchanged by handing out a lump sum transfer of S. For the expenditure function above, derive an expression for the subsidy that maintains the consumer's utility unchanged. In principal S could depend on (p₁, p₂, t, u). Does it depend on all of these in this case? If not explain why not.

26)

Determine whether each of the following is a legitimate expenditure function:

a.

M = p₁p₂U.

b.

M = p₁^a p₂^1-aU for 0 < a < 1.

c.

M = (p₂/p₁)U.

27)

An individual has preferences for current and future consumption which can be represented by the utility function

a.

Does the individual exhibit a ``time preference'' for current consumption? Find the optimal C₀ as a function of r, Y₀, and Y₁, and express net borrowing, (C₀ -Y₀) as a function of the same parameters. How does net borrowing vary with r? What happens if Y₁ = 0?

b.

Suppose an economy consists of 200 people with these preferences. 100 of these individuals has an endowment of 10 in the current period and nothing in the future, and the remaining 100 have an endowment of nothing now and 9 in the future. What is the equilibrium interest rate?

28)

Assume that an individual has an intertemporal utility function of the following form:

a.

An increase in the interest rate at all dates.

b.

Income in a single period rises.

c.

An income tax is imposed that taxes comprehensive income (ie. Y_t plus any interest income).

d.

A public pension scheme is introduced in which persons contribute a fixed amount every year for years 1 through t and then receive benefits from years t through T. The scheme is actuarily sound in that the present value of contributions equals the present value of benefits.

29)

Consider the following pair of prospects:

A.

($1000; 1), a certainty of receiving $1000, vs.

B.

($5000, $1000, 0; .1,.89, .01)

Where the $ figures are payoffs and the numbers following the semi-colon are the probabilities attached to each payoff respectively.

Choose which of A or B you would prefer.

Now consider a second pair of prospects:

C.

($1000, 0; .1, .9)

D.

($1000, 0; .11,.89)

Choose which of C or D you would prefer.

The independence axiom of the expected utility hypothesis states: ``a risky prospect A is weakly preferred to a risky prospect B if a p, 1-p chance of A or C respectively is weakly preferred to a p, 1-p chance of B or C for arbitrary positive probability p and risky prospects A, B and C.''

a.

If an individual were indifferent between A and B, and if his preferences conformed to the independence axiom, describe his indifference curves for varying probabilities (A, B; p, 1-p) of receiving A or B.

b.

If an individual preferred B to A, what ordering by the individual of C and D would violate the independence axiom? Explain.

c.

If the independence axiom were omitted, how would expected utility analysis be complicated?

30)

Consider the utility function u(y) = 100 + 200y - y².

a.

Over what range is that utility function monotonically increasing?

b.

Over this acceptable range, what is the Arrow-Pratt measure of absolute risk aversion? Does it rise or fall with income?

c.

What are the consequences for the demand price of a risky asset (4, 2; .5, .5) as certain income rises from 40 to 60 for this utility function. Relate your answer in (c) to (b).

31)

Consider a situation where consumption decisions are being made for the next period. There are two states, a and b, and the subjective probabilities held by all individuals are p_a = .5, p_b = .5.

There is one group of individuals with utility function v = C₁^.5 and another with the utility function v = lnC₁. Each group has the same number of individuals with the same endowments (y_1a, y_1b) = (400, 100). To simplify, consider trade (price taking) between a representative individual from each group with the demands of the representative individuals together equal to their endowments in equilibrium. What is the relative price of a claim on consumption in state `a' to that of a claim in state `b'?

32)

Consider an expenditure function M(p₁,p₂,U⁰).

a.

Prove that M^*_p1 = x₁. What is x₁?

b.

Show that the marginal cost of utiltiy falls with an increase in price of good 1 if good 1 is an iferior good.

33)

Consider the third model of Barzel and Suen (if you didn't read that far, don't dispair, the mechanics are identical to the first model). Suppose that there are only two goods per period, and that the utility function within each period is additively separable and has diminishing marginal utility for each good.

a.

What is the income effect for both goods within a given period?

b.

Does the Barzel and Suen argument imply that normal goods will have upward sloping demands? That is, if the price of good 1 increases in a future period, do people transfer income to that period, and therefore have the possibility of having the income effect offset the substitution effect?

c.

If it is possible to have inferior inputs in production, why are the profit maximizing demand curves for inputs always downward sloping. How does your argument relate to the paper by Barzel and Suen?

34)

You've won a trip to the CEA meetings in PEI this summer! The university will pay for one economy ticket, but if you want to go first class, you'll have to pay the difference yourself. An economy ticket normally costs $500 while a first class ticket normally costs $1000.

35)

Suppose U(x₁,x₂) has vertically parallel indifference curves if you put x₁ on the horizontal axis.

a.

Sow graphically what the income effect is for good 1. Bonu points if you can show this algerbraically as well (this isn't hard, it just takes a bit of time)]

b.

What does this imply about the demand for good 2?

c.

What will be true about the ordinary demand (x^m) and the Hicksian demand (x^u)?

36)

How long will you wait for an elevator, given that you've decided to take it? Why?

37)

Consider the model of firms in long-run competitive equilibrium, where competition forces each firm to minimize

a.

Holding w₂ fixed at some w₂⁰ throughout, and letting x_i⁰ = x^*(w₁⁰, w₂⁰) sketch AC and AC^*. Explain why your graph looks the way it does, and in particular show geometrically that AC^* is concave in w₁.

b.

Show that

c.

Show that

38)

Farmer Willy has a two period time horizon, and he grow and exclusively eats turnips. His utility function is

a.

If there is no capital market, how many turnips will Willy (i) consume this period? (ii) plant this period? (iii) consume next period?

b.

Now suppose there is a capital market in which Willy can either borrow against his future harvest or invest his current harvest, and let r denote the interest rate. How much will Willy (i) consume this period? (ii) plant this period? (iii) harvest next period? (iv) consume next period?

39)

Ransford, that wild and crazy economist, and famous Nevada gambler, says that he would prefer having $90 with certainty to having a 1/3 chance of winning $90, but that he would prefer the 1/3 chance of winning $90 to receiving $20 with certainty. Moreover, receiving $20 with certainty is preferable to a 1/2 chance of winning $20, but the 1/2 chance of $20 dominates a 1/4 chance of winning $90. Are Ransford's preferences consistent with maximizing expected utility? Explain.

40)

One day, Karim the economist was waiting in a line for almost an hour. When the person in front of him got to the head of the line he had to fill out a form which took about five minutes. The person complained that the time spent in line would be much less if the office gave the forms to those in line before they reached the window so they could fill them out while waiting. The clerk thought this was a good idea, and was about to tell the supervisor the plan when Karim said ``Excuse me''. What did Karim say?

41)

Guomin has to decide how much of his $100 to put into bonds and how much to hold as money. The interest rate r = 5%, and bonds earn interest. If state a occurs bonds have a capital gain of g = 10, but if sate b occurs they have a capital loss of l = 10, so that his state budget line passes through the point where W_a = W_b = $105 with a slope of -1. Suppose Guomin's utility is given by V(W_a, W_b) = p_aW_a + (1-p_a)W_b.

a.

What must be true about p_a in order for Guomin to hold some bonds?

42)

Now suppose that at the beginning of period 0 his portfolio is initially entirely in money, and that in order to buy bonds or sell them short he must pay now out of his $100 a brokerage fee of 2% of the current value of any bonds in which he trades.

b.

Draw the budget line in W_a, W_b space.

c.

What value of p_a is now necessary in order to induce the investor to hold any bonds?

43)

Why do North Americans have a reputation for wasting food, buy keeping appointments?

44)

Last week when I was at the gym, I overheard the following conversation.

Person 1:

``Boy that was a hard squash game''

Person 2:

``Yeah, but you didn't play for very long.''

Person 1:

``True. I just reached the point of diminishing returns, and quit.''

Is this person an economist? That is, is it true that people stop doing things when they reach the point of diminishing returns?

45)

In the standard two good model, if good 1 is a Giffen good, what does that say about whether good 2 is a gross complement or substitute?

46)

In class we saw that ¶x_i^* /p was ambiguous in sign. That is, when the output price increased, more or less of an input might be used. Using the fact that the supply curve of a profit maximizing firm is upward sloping, show that both inputs cannot decrease with an increase in p.

47)

Consider the following model:

a.

F_a = f_a.

b.

F_aa = f_aa.

c.

Using the Conjugate Pairs theorem, find ¶x₁^* / ¶a.

48)

Suppose that l^m = l^m(M) That is, suppose it did not depend on prices.

a.

Show that

b.

Using the above result, and the Slutsky equation, find the relationship between the income effects for x₁ and x₂.

c.

A homothetic utility map implies that

49)

A former graduate student at SFU wanted to test the Alchian and Allen proposition. He collected data on tea from Sri Lanka and England. He, indeed, found that the relative price of the high quality tea fell when it was shipped to England. To his horror, though, the quantity of high quality tea consumed was higher in Sri Lanka! In his paper he concluded that the demand for tea was upward sloping. Can you think of something obvious that the student overlooked with respect to the Alchian and Allen proposition?

50)

Suppose you had a stock that just fell in price. How could you tell if this was due to a fall in the expected net profits, or due to a rise in real interest rates? (Hint: could you think of another market you could look at?)

51)

The paper by Pollak and Wachter said that unless there were CRS and no joint production, we couldn't say anything about the demands for commodities. However, in class I showed you two cases where there was joint production and we could still say something (remember the case of productive consumption - ``dressing for success'').

a.

Why were we able to get a result, P & W notwithstanding?

b.

A lot of consumption behavior is counter productive. For example the consumption of cigarettes and drugs can reduce rather than increase earnings. Suppose we think of this problem as:

c.

Assuming, as with the productive consumption case, that there is a similar effect with respect to the slope of the demand curve as what you found in (b) would you predict alcoholics to have more or less elastic demands for alcohol? Does this make sense?

52)

The quantity of timber in a growing forest is f(t) after t years. The price of a unit of lumber is constant over time and is denoted by p. There is only a fixed costs, c, of harvesting this forest. The rate of interest is r, and interest is compounded continuously. Consider only the case of a single rotation.

a.

Derive an expression which characterizes the optimal harvest time for this lumber given that you want to maximize net present value. What are the second order conditions for this optimization problem?

b.

How does an increase in the harvest cost affect the optimal t? Verify your answer.

53)

Consider the following data about the two objective characteristics of three homogeneously divisible products:

 

	X1	X2	X3
Units of char. 1 per lb.	30	30	70
Units of char. 2 per lb.	70	30	30
Price per lb.	30		30

54)

A firm exists over a sequence of periods. Its production function for period t is

55)

Let

a.

Graph p^* and p⁰ against p.

b.

What comparative statics are obvious from the graph?

c.

Explain the result in part (b) in plain English.

56)

Suppose that l^m = l^m(p₁), where l^m is the marginal utility of money income, and comes from our standard two good utility maximization problem.

a.

Using the Envelope theorem (without proving it) find the income effects for x₁^m and x₂^m.

b.

Again using the Envelope theorem, show that the cross effects are equal. (ie. ¶x₁^m / ¶p₂ = ¶x₂^m /¶p₁)

c.

Could good 1 be a Giffen good? Why or Why not?

57)

My wife belongs to a fitness club. The other day I went to meet her there and I commented on how few patrons there were. She said ``well it doesn't matter that much, because the owners of the club also own the building''. How can you tell my wife is an accountant at heart?

58)

An economist, using faculty and industry wages, estimated that the value of a ``faculty lifestyle'' was worth about $30,000 per year. That is, the sacrifice in income that a Phd made by entering academics rather than private industry was about $30,000. Following the Suen argument, would this be an accurate estimate, or an under or over estimate, assuming that individuals with Phd's are heterogenous with respect to valuing leisure?

59)

A university (where parking is priced above the market clearing price) recently increased the price of parking in its two lots by $15 per term. Lot A is unambiguously better than lot B, and the prices reflect this at $60 and $30 respectively. What would you predict will happen in terms of the usage of the two lots? (Ignore income effects).

60)

``The Fisher Separation Theorem implies that firms maximize the net present value of investment.'' True or False? (In your answer, assume that a firm lasts two periods. In the first period an investment `I' is made, and in the next period some output f(I) is realized.)

61)

Explain the following two observations using Becker's theory of household production. Feel obligated to also spell out any assumptions you must make regarding incomes, wage rates, etc.

a.

Despite years of ``gender education'', if you walk into any toy store you will see that most of the ``girl'' toys are Barbie dolls, kitchen supplies, and other sundry household items, while the bulk of ``boy'' toys are cars, swords, and anything to do with Batman.

b.

Estimates of the elasticities of gasoline and haircuts show that at a given price, the elasticity of gas is greater than the elasticity of haircuts.

62)

Assume that an individual has an intertemporal utility function of the following form:

a.

Using math, show what would happen to the absolute levels of consumption if there was an unanticipated fall in the interest rate. (That is, I want you to grind through the comparative statics. Recall that for a constrained max problem the m ×m BPPM have sign (-1)^m-r where r is the number of constraints.)

b.

Show your answer to (a) graphically. (If you were unable to do part (a), you should still be able to do this.)

c.

Suppose there were more than two periods involved. Could this type of utility function allow for ``ratchet effects'' in consumption? That is, some people claim that once you get used to high consumption level it is harder to go back to a low consumption level than if you had never had a high consumption period.

63)

Briefly evaluate the following statements:

a.

``Strong (additive) separability and homogeneous separability are mutually exclusive concepts''.

b.

``Strong (additive) separability rules out Giffen goods, but homogeneous separability does not.''

64)

If the utility function is strongly separable, then ¶X^m_i/¶P_j = 0. True/False/ Uncertain. Explain.

65)

Suppose U = x₁+x₂^a where a < 0.

a.

Is this a well behaved utility function? Explain.

b.

Graph it with x₂ on the horizontal axis.

c.

Is consumer's surplus well defined for x₂?

66)

The price of beer is $.50 each and you buy 6. The price falls to $.25 each and you buy 10. If beer is a normal good, both the compensating and the equivalent variation measures of consumer's surplus will lie in the range of $1.50 to $2.50. True/False/Uncertain. Explain.

67)

Consider Barzel's model of rationing by waiting.

a.

What happens to the total waiting time if the total amount (N) given away increases? What happens to the waiting time per unit?

b.

Answer the same question, except this time the batch size (k) increases?

68)

From Wing's model of heterogeneous consumers, what happens to the value of using a price mechanism (as opposed to random allocation) when the size of the market increases? (ie. there are more of every type of consumer.) (No math is required, a graph and intuitive explanation will do.)

69)

Assume the following utility function

70)

Empirical studies have noted that when education increases within a family, the number of children, television sets, and pounds of food within a household falls. At the same time, the dollars spent per child, per TV, and per pound all increase. Why might this happen?

71)

Norm, a risk averse individual, is given the opportunity to make a drawing from one of two density functions, f(x) or g(x), where x is a cash prize. The means of the distributions are equal and the variance of f(x) is larger than the variance of g(x). Norm will choose to draw from g(x). True or False, explain (Five (5) bonus points for drawing your answer correctly.)

72)

Assuming everyone in the class is in an intertemporal competitive equilibrium, we all have the same marginal rate of time preference. True or False, explain.

73)

If Rob calculated a value of Mike's coefficient of absolute risk aversion to be -4.

a.

What does this mean?

b.

If Mike faced a gamble which had a standard deviation of 3, what would his risk premium be?

c.

If Mike wakes up tomorrow, and his utility function has changed to U = 40+160v(c) where v(c) is his utility function today, and he faced the same gamble, what would his new risk premium be?

74)

What will have a larger influence in revising your annual consumption, a gift of $1000 or an anticipated salary increase of $50/month if the real rate of interest is 10% per year, and you will work for 40 more years. (Ignore compounding within a given year.)

75)

Christy has an initial wealth level of W₀ and must decide how much to invest in a risky asset earning a random rate of return, r, and how much to invest in a riskless asset whose rate of return is 5%. final wealth will be:

a.

What are Christy's absolute and relative risk aversion functions?

b.

Suppose r = .8 with probability .6, and r = -.45 with probability .4. Derive the optimal A^* as a function of W₀.

c.

How does the optimal A^* vary as W₀ increases? Is this consistent with what you know about Christy's absolute risk aversion function?

76)

Consider the model

a.

f_a = f_a

b.

f_aa > f_aa > 0.

c.

Prove (a) algebraically

d.

prove (b) algebraically

e.

From part (d), prove that ¶x₁^*/¶a > 0.

77)

Consider a firm with production function y = f(x₁,x₂) which maximizes sincerity minus taxes, where sincerity equals

78)

A friend of mine was looking at a house for sale with plans of renting it out to a tenant (the house sold for $200,000). On driving past the house, I remarked to the person beside me ``I wonder what rent he has to charge to break even?'' To which the said individual responded ``that would depend on how much of a down payment was made.'' Is this true? Briefly explain.

79)

Consider the standard cost minimization model:

a.

Write out the first and second order conditions for this problem. Is diminishing marginal product of any input implied?

b.

Prove that the factor demand functions are homogeneous of degree zero in prices.

c.

Using the Conjugate Pairs Theorem, show that these demand functions are downward sloping in their own prices.

d.

Explain how you would arrive at the cost function. What does it depend on? Explain the meaning of this function.

e.

Define S_ij = ¶x_i/¶p_j. Explain why these are pure substitution effects.

f.

In Value and Capital, Hicks, in gory detail, proved the following two results.

Can you prove them using modern methods in a couple of lines?

g.

Defining substitutes as S_ij > 0, and complements as S_ij < 0, prove that it is not possible for all goods to be complements to each other.

80)

In a two good world, where consumers have fixed endowments of goods rather than money income, a good can have ¶x₁/¶p_i> 0 without being inferior with respect to income. T/F Explain.

81)

If a consumer has a utility function of two goods, U(x₁, x₂) = u¹(x₁) + u²(x₂) where both u¹, u² are strictly concave, then the ordinary demand for both goods are downward sloping. T/F Explain.

82)

Consider the production function

83)

Suppose that the indirect utility function is of the form

a.

Find the ordinary demand for x₁.

b.

Find the elasticity of demand for x^m₁.

c.

Find the expenditure function M^*(p,U)

d.

Find the Hicksian demand for x₁.

84)

``An increase in the diversity of consumers' tastes will increase the cost of using allocation mechanisms other than the price system.'' T/F Explain.

85)

Suppose a company offers its employees free parking that would rent for $200 per month. Unlike the question on the problem set, suppose the company does allow you to sublet the spot. If the company suddenly decides to charge $200 per month for the spaces, would we expect any change in the quality of cars parked in the lot? (If it makes it easier, assume there are only two kinds of cars: good cars which cost $500 per month to rent, and poor cars which costs $100 per month to rent.)

86)

Maria's utility function is U = min(x,y). Maria has $150 and the price of both goods is $1. Maria's boss is thinking of sending her to another town where the price of x is still $1, but the price of y is $2. Maria, familiar with the concepts of EV and CV learned in one of her favorite courses, pouts and complains bitterly. She claims that although she doesn't mind moving for its own sake and the new town is just as nice as Vancouver, the move is as bad as a pay cut of $A. She also says she wouldn't mind moving if when she moved she got a raise of $B. What are A and B equal to?

87)

A government subsidy program reduces the price of aerobic classes from $50 per unit to $40 per unit. As a result, Adel increases his rate of consumption of classes from 50 units to 60 units. Adel's gain from the subsidy program is at least $500 per period, but no more than $600 per period. T/F/U

88)

``To draw an analogy with the intertemporal problem, the relationship between the subjective rate of time-preference r and the market rate of interest r corresponds exactly to the relationship between the ratio of subjective probabilities p_a/p_b and the slope of the asset line l/g.'' T/F/U

89)

Economists always come up with different names for the same thing (eg. MV=MRS), ``utility of expected value'' and ``expected utility'' is another example. T/F/U

90)

If the income elasticity of current and future consumption is positive, a decrease in the market rate of interest will have an unambiguous effect upon the quantity of savings in a two-period model. T/F/U

91)

Diminishing marginal utility in all goods implies that the marginal utility of money income is also diminishing with respect to income. T/F/U

92)

A monopolist producer of light bulbs, will design his bulbs so that they will have to be replaced more frequently than competitive producers would. T/F/U

93)

Let p be the price vector and let E be the minimum expenditure needed to achieve a utility of V⁰

a.

Prove that the rate at which E increases with p_i can be expressed as

b.

Hence, or otherwise, derive the Slutsky equation for ¶x_i/¶p_j.

c.

How must this equation be modified if the individual has an endowment of x⁰_j units of commodity j?

d.

``In an endowment economy, if an individual's final consumption bundle is close to his endowment point, own price effects are negative.'' T/F/U. Explain.

94)

Consider a profit-maximizing firm with production function y = f(x₁, x₂), which sells its output in a competitive market at price p. The firm obtains inputs at competitively determined wages w₁ and w₂, but the factors interact with each other so that an additional term, pkx₁x₂, is present in total cost. The objective function of the firm is therefore:

a.

Derive the first and second order conditions. Explain the derivation of the factor demand equations.

b.

Is the expansion path for this firm the same as for a firm whose costs do not include the interactive term? Also, at the profit maximizing point, does the law of diminishing returns hold for each factor?

c.

Derive the comparative statics relations for w₁. Is there a refutable implication?

d.

Are the factor demand functions homogeneous of some degree in some or all of the parameters?

95)

Two vendors have shops right next to each other. One sells hamburgers and one sells french fries. The demand for hamburgers and fries in the two stores are not independent; that is, greater sales of hamburgers imply greater demand for french fries and vice versa. In fact, the demand curve for the two stores can be characterized as

a.

Find the Cournot solution for output and prices for both venders.

b.

Suppose the firms merge, and this has no effect on cost. Suppose also the merged firm decides to give french fries away for free. What price should they charge for the burgers to maximize profits?

c.

In which case are profits higher? Why?

96)

The following question can be done with only graphs and intuitive answers. Suppose that risk averse individuals face an uncertain state of the world. They have $100 that they can invest in money and earn a return (1+r) for certain, or they can invest in bonds that yield a gain (1+g) in state 1, and a loss (1-l) in state 2.

a.

Draw the state-space graph and show the case where the individual is holding some positive amount of bonds. Make sure you identify the equilibrium holdings of bonds and money.

b.

Suppose that r increases. On a new graph, show the new equilibrium. What happens to the amount of bonds held?

c.

Going back to the original situation again, suppose that the return g increases. On a new graph, show what happens to the amount of bonds held now.

97)

Suppose Jorge is more risk averse than Ramesh. Suppose they both have the same beliefs about the various states of the world that might exist tomorrow, and they both face the same prices for state claims. Furthermore, they both have the same risky state endowment. Given this, Jorge will choose a less risky consumption bundle. T/F/U. Explain (Hint: a graph will help.)

98)

Vin has a VM utility function v(c) = ln c and must choose a state-contingent consumption bundle (c₁, ... , c_s). The price of a state-s claim is p_s and the Vin's initial endowment has a value of [`C].

a.

Solve for Vin's optimum consumption in each state.

b.

Show that for any pair of states s and s¢:

c.

What condition defines the state in which consumption is greatest? Least?

d.

Is the rule derived in (c) true for any concave function v(c)?

99)

David is a a risk lover, therefore, he will always accept a fair bet. T/F/U. Explain.

100)

``The Fisherian model of intertemporal choice is inconsistent with a Keynesian consumption function C_t = a+bY_t, where C_t,Y_t are the levels of consumption and income in a given period.'' T/F/U. Explain.

101)

Milton Friedman has a saying that ``All goods are perfect substitutes at the margin.'' What does he mean?

102)

Flora has earnings of $200 at the beginning of period 0 and expects earnings of $250 at the beginning of period 1. Bernie has earnings of $150 in both periods. Neither of them has any other assets, but they can consume less or more than their earnings in a given period by lending to or borrowing from the other consumer. The market, in which they do this will have a single rate of interest r which neither of them influence. Neither of them cares about what happens after period 1.

If C_F(0) is Flora's consumption in period zero, and so on, then the utility functions for Flora and Bernie are:

a.

Write down the present value of each individual's wealth, and their budget constraints.

b.

Write down each person's demand function for consumption in period 0.

c.

Equate the demand and supply curves and calculate the equilibrium rate of interest.

d.

In equilibrium, who borrows how much from whom?

103)

Will elasticities of demand be greater for temporary or permanent price changes?

104)

When Edward is faced with prices p₁ = 9, p₂ = 12 he consumes at some point x⁰, where x₁ = 4, x₂ = 7, U(x⁰) = 10. When p₁ is lowered to p₁ = 8 Edward would move to point x¹, where x₁ = 6, x₂ = 6, U(x¹) = 15. From these data, estimate the following values:

a.

How much would Edward be willing to pay to face the lower price of x₁?

b.

How much would Edward, if initially at x¹, have to be paid to accept the higher price of x₁ voluntarily?

c.

Are your answers to (a) and (b) exact, or approximations? If the latter, what is the direction of the bias?

105)

The other day on the CBC morning news, there was a story on the new ``super bacterias''. These are bacteria that are immune to the current anti-biotics. On the report an expert said ``these bacteria have figured out how to get around our current medical arsenal.'' What would Alchian say about that statement?

106)

Quite often, whenever there is a natural disaster, we hear statements about how much the disaster cost. Suppose there was a large comet heading straight for earth. Does it literally make sense to ask the question: ``What is the cost of a comet hitting the earth''? Why or why not?

107)

Suppose a firm uses two inputs, and has the following rule for selecting a level of x₂, namely x₂ = (w₁x₁)/w₂, for a given level of output y⁰ = f(x₁,x₂).

a.

What would be the parameters of the cost function for a firm that used this decision rule?

b.

What would be the sign of ¶x_i/¶w_i? A simple isoquant graph is sufficient to make your point.

c.

Is the cost function that would result from such a rule cost minimizing? If not, are there any conditions under which it would be?

108)

``Marginal cost is the cost of the last unit produced''. True or false, explain.

109)

Consider the class of models

a.

Prove (don't just cite the Envelope Theorem) that f_b = l^*x₂^*, where l^* is the Lagrange multiplier.

b.

Derive and explain what comparative static results are forthcoming in this model, and which are not.

c.

Prove that

110)

A firm produces one output from two inputs according to the technology f(x) = x₁^1/3x₂^1/3. The firm is a price taker on both input and output markets. Find the firm's profit maximizing factor demand functions and the profit function.

111)

With w₁ on the vertical axis, and x₁ on the horizontal axis, draw the profit maximizing demand curve x^*, the cost minimizing demand curve x^c, and the long run zero profit demand curve x^L, when x₁ is an inferior good. Provide an intuitive explanation for the differences in elasticity.

112)

A bumper sticker reads: ``Real charity doesn't care if it is tax deductible''. True or false, comment.

113)

Some simple graphing questions to get you started.

a.

Draw a graph in utility space, with x₂ on the vertical axis and x₁ on the horizontal axis, a set of indifference curves where x₂ is inferior. Now drawing in budget constraints where the price of good two is changing, derive the Hicksian, Slutsky and Marshallian demands for x₂ in the second graph in price quantity space. Be sure to label everything carefully.

b.

In a completely separate graph, draw a Hicksian and Marshallian demand curve for x₁ when ¶x₁ / ¶M = 0, and the nominal price of x₁ is on the vertical axis. Now on another graph, graph them if the relative price of x₁ is on the vertical axis.

114)

Is the function y = x₁³x₂³ + x₁x₂² homogeneous? Is it homothetic? Explain

115)

Show that ¶l^u / ¶U⁰º ¶l^m /¶M ×l^c, where l^u is the compensated marginal utility of money, l^m is the ordinary marginal utility of money, and l^c is the marginal cost of utility.

116)

``If average costs are falling then the homogeneity of the production function must be greater than 1'' True, False, explain.

117)

``If people are going to stand in line, then we should make their wait as comfortable as possible. We should provide chairs, shelter, and make food available to them.'' The following statement was made by a Langley School Board member in the fall of 1996 as hundreds of parents spent up to two days waiting to register their children to a particular alternate school. In the fall of 1997, the advice was taken.

a.

What do you think happened to the line, and why?

b.

Were the parents in line any better off? (Be careful and explicit in your answer.)

118)

Consider the utility maximization problem, max U(x₁, x₂) subject to p₁x₁ +p₂x₂ = 1, where prices have been ``normalized'' by setting M=1. Let U^*(p₁,p₂) be the indirect utility function, and l be the Lagrange multiplier.

a.

Show that l^M = (¶U / ¶x₁)x₁^* +(¶U / ¶x₂)x₂^*

b.

Show that l^M = -[U^*_p1p₁ + U^*_p2p₂]

119)

In Wing's model of heterogeneous consumers, an increase in diversity of tastes could increase or decrease the demand for a good.

a.

Explain this using a graph.

b.

What would the effect of an increase in diversity have on the total value of a commodity? How does this relate to part (a)?

120)

The following is a quote from Silberberg, p. 346. in his discussion of two-stage budgets and separability. ``It is not the case, for example, that if the ``subutility'' functions f and g above [these are the subutility functions in the weakly separable stage 1 utility function] are homothetic that two-stage budgeting is possible.'' He then gives a function that results from a homogeneously separable utility map and states that ``clearly ¶x_i/ ¶p_j ¹ 0'', where goods x_i and x_j are in different subutility functions. Essentially Silberberg is saying that this inequality denies two-stage budgeting can take place with a homogeneously separable utility map. Is he right or wrong? Explain.

121)

In terms of our utility maximizing model,

a.

What is the difference between the ``ability to make a choice'', and ``the willingness to make a choice''?

b.

When people make statements like ``Joe Blow had no choice, he had to ... '' what are they really saying in terms of willingness versus ability?

c.

We design laws that presumably act as deterrents against crime. Is your answer in (a) and (b) consistent with this? Explain briefly.

122)

Economics, in the past 30 years, has been applied to such topics as religion, family life, and animal behavior. It seems silly to suggest that individuals (whether they are priests, fathers, or rats) are behaving according to the principle of maximization. What defense does Alchian offer in cases like these?

123)

Several years ago, a neighbor of mine was doing some ``funny'' things to his yard. He killed the trees (but left them standing), planted bamboo, and generally let the place go wild. My other neighbor was selling his place, and as I was talking to the real estate agent she said: ``that guy doesn't care what happens to his place because he inherited it.'' How can you tell the real estate agent is not an economist?

124)

Consider a profit maximizing firm with production function y = f(x₁, x₂), that sells its output at price p.The firm buys input x₁ at a fixed wage w₁, but the firm faces an upward sloping supply function for x₂, given by w₂ = w₂⁰+kx₂, where w₁,w₂⁰, p, and k are positive parameters.

a.

Derive the first and second order conditions and explain the derivation of the explicit choice functions implied in this model. Characterize each of these choice functions as a demand function, a supply function, or neither, and explain. Is the ``law of diminishing marginal product'' implied for each factor?

b.

Derive the comparative statics results available for the parameter w₁. Are there any refutable implications?

c.

How will the use of x₂ by this firm respond to an increase in k?

d.

Are the explicit choice functions homogeneous of some degree in some or all of the parameters? Show that they either are or are not. What relation, if any, does homogeneity of factor demand or other similarly derived functions have to the homogeneity of the production function?

e.

Find a ``reciprocity'' result involving the parameters w₂⁰ and w₁.

f.

Assuming that x₂ can be held fixed, explain the meaning of the following identity:

125)

The minimum average cost function is given by AC^*(w₁, w₂, y⁰). Find the derivative of this with respect to y⁰ and simplify to get an expression for marginal cost. Interpret this equation.

126)

Suppose that x₁ is an inferior factor.

a.

What must be the sign of ¶l^c /¶w₁? And what must be the sign of ¶AC^* /¶w₁?

b.

Given the sign of the two derivatives, draw a marginal cost curve, and then carefully graph how the marginal cost curve changes when w₁ changes.

127)

Consider a general minimization problem of the type:

a.

Let f(a, k) be the minimum value of y for a given level of (a, k). On a graph with y on the vertical axis and a on the horizontal axis, plot the functions f(a, k) and h(x₁⁰, x₂⁰) +ax₁⁰, where the `0' indicates fixed values. Graphically show the envelope theorem, and that phi_aa < 0 at some appropriate a⁰.

b.

Prove (ie. do not just assert) the following:

       f_a = x₁^*(a,k).

       f_k = l^*(a, k).

c.

Show that

128)

Consider an individual endowed with nonwage income Y⁰ and 24 hours of time per day, who chooses some mix of leisure (L) and income (Y). Income can be augmented at market wage rate w. The individual maximizes U(Y,L) subject to the budget constraint Y = w(24-L) +Y⁰ yielding the Marshallian demand functions Y = Y^M(w, Y⁰) and L = L^M(w, Y⁰).

a.

Explain the meaning of the Marshallian demand functions, show explicitly how they are derived, state any homogeneity that exists, and identify the indirect objective function. Explain the derivation of the Hicksian demand curves. What are they functions of, state any homogeneity that exists, and explain the meaning of the associated indirect objective function.

b.

Carefully explain the meaning of the following identity:

c.

Why is the Marshallian demand for leisure not necessarily downward sloping when leisure is a normal good.

129)

Suppose Ticket Master is underpricing the tickets to a Spice Girls concert, and as a result a queue is created. Suppose also that Ticket Master has restricted each person to purchase at most 2 tickets.

a.

Describe using the Barzel graph, how the equilibrium price in terms of time is achieved. (Minor Hint: Be careful to note that the price of the ticket is not zero.)

b.

Suppose that Ticket Master decides to increase the number of tickets that each person can purchase to 4, but that the Spice Girls are still only going to do one concert. What happens to the price per ticket in terms of time, and the total time spent waiting in line per person?

c.

In general, is it true that the individuals in a line are the individuals with the lowest time cost?

130)

What is the difference, in a many-commodity model, between diminishing marginal rate of substitution between any pair of commodities, and quasi-concavity of the utility function? Which is the more restrictive concept?

131)

Show that if l = l^M(p₁) then ¶x₁/¶M = 1/p₁. [Hint: Do not use any matrix algebra!]

132)

In price/quantity space, show me a graph comparing EV, CV, and CS when x₁ is inferior.

133)

Mormons are big on education (BYU is the largest university in the US), Jehovah's Witnesses think the world is ending tomorrow anyway, so why bother going to college. Both religious groups are growing. Which group do you think gives more in terms of money to their church, and which one do you think gives more in terms of time? Briefly explain why.

134)

It is often said that accountants and dentists make so much money because their wage reflects the fact that they have a boring job. Suppose the wage difference between an economist and accountant was $20,000. Under what conditions would this be an accurate measure of the ``value of boredom''?

135)

This winter I am going to New York for the American Economic Association meetings. When I'm there, I'll probably eat at McDonalds. When I eat at McDonalds at home I always have a Quarter Pounder and a medium fries. When I eat there in New York, am I more likely to have a ``higher quality'' meal? For example, a crispy chicken sandwich and a large fries?

136)

Suppose a crazy terrorist builds and explodes several neutron bombs in the USA killing half of the people, but not damaging any of the physical capital. What would you predict would happen to the real interest rate? Briefly explain using a graph.

137)

The Prime Minister recently defended his low interest policy over a tax cut. He claimed that a lower interest rate meant a savings of perhaps $400 per month on the average mortgage, while a tax cut may only mean an increase in income by about $600 per year. He concluded that a low interest rate policy is better for Canadians than a tax cut. Supposing his numbers are correct, what little detail is he glossing over? Use a graph in your answer.

138)

``A certain former Soviet Union tank commander is planning to hold XYZ stock for at most 5 years. As a result, he should not be concerned about XYZ dividends after that date.'' T/F/U Explain.

139)

Consider the following model:

a.

Derive any refutable hypotheses from this model.

b.

Prove the following reciprocity result:

c.

Suppose that x_n is held fixed at x_n⁰, its previously maximizing value. Let a¢ = (a₁ ...,a_n-1), and therefore, define the short run choice function as x^s_i(a¢,b,x_n⁰). Using the relevant identity between x^*_i(a, b) and x^s_i prove the Le Chatelier result

140)

Suppose that the utility function is given by U(x₁, x₂, p₁). That is, utility depends on the price of good 1, as well as the quantities. Derive the Slutsky equationfor changes in x₁ with respect to p₁, and discuss how this differs, if at all, from the usual Slutsky equation.

141)

Neil's von Neumann-Morgenstern utility function is uⁿ(w) = 7-3w^-3, while Robert's is u^r(w) = 3-8e^-w

a.

Find the measure of absolute risk aversion for each.

b.

Is one globally more risk averse than the other? If the answer is yes, which one? If the answer is no, explain why.

142)

Suppose that initially an individual owns $4 and a lottery ticket. The lottery ticket will be worth $12 with probability 1/2 and worth $0 with probability 1/2. The individual's VM utility function is u(w) = w^1/2 where w is wealth. What is the lowest price at which the individual would be willing to sell the lottery ticket?

143)

If U(c₁,c₂) = pv(c₁) + (1-p) v(c₂) is homothetic, then the individual has constant absolute risk aversion. T/F/U. Explain.

144)

``If the quality and quantity attributes of a good are complements or independent, then the Alchian and Allen ``Shipping the good apples out'' proposition, holds.'' T/F/U. Explain.