Hal R. Varian, chief economist at Google since 2007, was a 28 year old assistant professor at M.I.T. in 1975 when he taught my cohort the third in a sequence of four half-term courses that constituted MIT’s required core of graduate microeconomic theory. He assigned draft chapters from his graduate textbook Microeconomic Analysis (published in 1977). For this post I have transcribed the course outline, five problem sets and the final examination for the course.
Core microeconomic theory at MIT in 1974-75:
14.121 (linear models and production) was taught by Martin Weitzman,
14.122 (competitive and noncompetitive market structures) taught by Robert L. Bishop,
14.123 (theory of the consumer and resource allocation) was taught by Hal Varian,
14.124 (capital theory, uncertainty and welfare economics) was taught by Paul Samuelson.
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14.123—Microeconomic Theory III
Theory of the Consumer and Resource Allocation
Professor Hal R. Varian, E52-353, 3-2662
Spring, 1975
| Feb. 5 | advanced placement exam |
| Feb. 10 | utility; demand; expenditure |
| Feb. 12 | indirect utility; Slutsky equation |
| Feb. 17 | no class |
| Feb. 19 | no class |
| Feb. 24 | demand functions; duality |
| Feb. 26 | expected utility; properties |
| Feb. 28 | general equilibrium; existence |
| Mar. 3 | welfare theory |
| Mar. 5 | the core of an exchange economy |
| Mar. 10 | general equilibrium and production |
| Mar. 12 | dynamics and general equilibrium |
| Mar. 17 | malfunctions of the market mechanism |
| Mar. 19 | final exam |
Course text will be lecture notes available from me. Malinvaud and Arrow and Hahn are highly recommended secondary reading. There will be four or five problem sets and a problem session on Fridays, 9-10:30.
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14.123 Spring, 1975
Professor Hal R. Varian
Consumer Theory I
- Consider a consumer with a Cobb-Douglas utility function:
u(x,y) = a ln x + (1-a) ln y.
Calculate:- demand functions for x and y
- the indirect utility function
- the expenditure function
- the Hicksian demand functions.
- In a general equilibrium analysis, we cannot take income as an exogenous variable in the demand function since income, y = p.w, depends on the vector of relative prices. Derive the Slutsky equation for Dpm(p, p.w) in this case.
- At a general equilibrium price vector p*, we have aggregate supply equal aggregate demand:
Σ mi(p*, p*.w) = Σ wi. Show that if all agents have identical marginal propensities to consume each good (Dymi(p*, p*.w) = Dymj(p*, p*.w) for all i and j) then all aggregate demand curves must be downward sloping at equilibrium. More generally, show that Dp(Σ mi(p*, p*.wi)) is negative semi-definite. - Define eij = (-pj/xi) Dpjmi(p,y) be the cross price elasticity of good i with respect to price j, and ri = pi mi(p, y)/y, the income share of commodity i.
Show that r1e11 + r2e21 +r3e31 = r1.
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14.123 Spring, 1975
Professor Hal R. Varian
Consumer Theory II
- A consumer is found to have a utility function of the form
u = -1/x1 – 1/x2.- Starting from the utility function, compute the market demand functions for the consumer when he has income y and faces prices p1 and p2.
- Use the market demand functions to show that the indirect utility function is
u = -( √(p1) + √(p2))2/m. - Compute the expenditure function from the indirect utility function.
- Compute the consumer’s compensated and market demand curves from the expenditure function.
- Suppose at prices (p1, p2) = (5,10) and income y = $100, a rational consumer consumes the bundle (6,7). Assume that we have measured the following derivatives:
∂H1/∂p1 (p1, p2, ū) = -2
∂H1/∂p2 (p1, p2, ū) = +1
∂M1/∂y (p1, p2, y) = 2/7
where H1 and H2 are the Hicksian demand functions for goods 1 and 2 and M1 is the Marshallian demand function for good 1. Find an estimate of the consumption bundle of the consumer at (p1, p2) = 5,11).
- Suppose a consumer has an expenditure function of the form e(p, u) = u.g(p). Show that his utility function is homogenous of degree one. Suppose e(p,u) is of the form e(p,u) = h(u)g(p). How does the consumer’s behavior differ?
- Suppose a consumer has a differentiable expected utility function for income with Dyu(y) strictly positive. Show that he will always take a small enough bet as long as it has positive expected value.
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14.123 Spring, 1975
Professor Hal R. Varian
General Equilibrium III
- Show that Walras law holds for a production economy with fully distributed profits.
- Prove the theorem that a general equilibrium is pareto efficient for an economy with production.
- Suppose we have a productive economy with two agents. The producer has a production function x = q1/2 where x is output and q is labor.
The consumer has a utility function u(x,q) = x1/2(1-q)1/2. Calculate the general equilibrium real wage and equilibrium level of profits.
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14.123 Spring, 1975
Professor Hal R. Varian
General Equilibrium Theory and Welfare Economics I
- Show that any solution to
max Σ ai ui(xi), ai>0
s.t. Σ xi ≤ w
is necessarily pareto efficient.
- Suppose we have two agents with indirect utility functions
v1(p1, p2, y) = ln y –a ln p1 – (1-a) ln p2
v2(p1, p2, y) = ln y –b ln p1 – (1-b) ln p2
and initial endowments
w1 = (1,1)
w2 = (1,1)
Calculate the market clearing price.
- We have two agents with utility functions
u1(x1, y1) = a ln x1 +(1-a) ln y1
u2(x2, y2) = b ln x2 + (1-b) ln y2
and initial endowments
w1 = (1,0)
w2 = (0,1)
Calculate the market equilibrium prices in terms of the parameters a and b.
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14.123 Spring, 1975
Professor Hal R. Varian
General Equilibrium Theory and Welfare Economics II
- Two agents with strictly convex preferences have equal initial endowments
w1 = w2. They trade to an arbitrary allocation in the Core (w1,w2), (x1,x2). Prove that this allocation is necessarily fair:- Draw an Edgeworth box and give a geometric argument;
- give an algebraic argument in the general case (there is a one-line proof.)
- Show in a three person economy there are allocations in the equal division core that are not fair.
- Suppose we have n agents with identical, strictly convex preferences and we have some initial bundle of k goods to be divided among them. Let x be a fair allocation; show that x must give the same bundle to each agent. (Recall that a fair allocation is one that is strongly pareto efficient and such that no agent prefers any other agent’s bundle to his own.)
- Show that under appropriate assumptions of convexity, every pareto efficient allocation is necessarily a solution to a problem of maximizing a weighted sum of utilities. What is the economic interpretation of the weights?
- Suppose we are at a market allocation that is considered good. Since it is a market equilibrium it is pareto efficient and therefore maximizes a certain weighted sum of utilities Σ ai* ui(x). Accordingly, we will use Σ ai* ui(x) to evaluate small projects. Suppose we are considering a small project that will change x = (x1,…, xn) to x´= (x1´,…, xn´). Show that it should be undertaken if and only if it increases national income; that is, iff Σ p.(xi´-xi) >0.
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14.123 FINAL EXAMINATION
March 19, 1975
Professor H. Varian
Answer any 2 out of 4. All questions have equal weight. Good luck!
- A consumer has a utility function of the form u(x1, x2) = ln x1 + x2. He faces prices p1 and p2 and has income y. Calculate:
- his Marshallian demand functions for each good
- his indirect utility function
- his Hicksian demand functions
- his expenditure function.
- There are two consumers A and B with the following utility functions and endowments:
UA(XA1, XA2) = a ln XA1 + (1-a) ln XA2 , WA = (0,1)
UB(XB1, XB2) = min (XB1, XB2) , WB =(1,0)
Calculate the market clearing prices and the equilibrium allocation.
- We have n agents with identical strictly concave utility functions, u1(x1),…,un(xn). There is some initial bundle of goods w. Show that equal division is a pareto efficient allocation.
- A consumer has a differentiable expected utility function u(y) with u´(y) > 0. (There are no conditions on u´´(y)). His initial level of wealth is w and he is contemplating a bet which gives him $e with probability p > ½ and he loses $e with probability 1-p. (Notice the bet has positive expected value.) Show that he will always take the bet if e is small enough. (Hint: try Taylor series.)
Source: Personal copies.
Image Source: Detail from 1976 departmental group photo.