G30D, Revision Lecture

OutlineTopic 1: Consumer Theory

Topic 2: Choice Under UncertaintyTopic 3: Production TheoryTopic 4: General Equilirium

G30D, Revision Lecture

Martin K. Jensen (U. B’ham)

April 2013

Martin K. Jensen (U. B’ham) G30D, Revision Lecture



1 Topic 1: Consumer TheoryBasics: Preference Relations and Utility FunctionsThe Consumption DecisionComparative Statics

2 Topic 2: Choice Under Uncertainty

3 Topic 3: Production Theory

4 Topic 4: General EquiliriumBasics, WE, WEAThe Welfare Theorems




Basics: Preference Relations and Utility FunctionsThe Consumption DecisionComparative Statics

We started out with consumers described by a consumptionset X and a preference relation �.

Five key assumptions on consumers which you must know forthe exam (completeness, transitivity, strong monotonicity,continuity, strict convexity).

It is important that you are able to draw an indifferencediagram from a utility function and argue graphically thatpreferences are e.g. convex.

From preference relations we then made the connection toutility functions through our representation theorem:





























Theorem

(Utility Representation Theorem) Let a preference relation �on X = Rn

+ satisfy assumptions 1-5 of week 1. Then there exists autility representation u : X → R which is a continuous, stronglymonotone, and strictly quasi-concave.

You must know these conditions on utility functions and be able toprove monotonicity (through differentiation). As a minimum youmust also be able to argue from an indifference diagram that theunderlying utility function is strictly quasi-concave.





Next we turned to the objective of the consumer in acompetitive economy. This is to choose the consumptionvector within her feasible set (which you must be able todefine!) which yields the highest level of satisfaction. Thehighest level of satisfaction translates into “on the highestindifference curve” in an indifference diagram.

Assume that the consumer has utility function u : X → R.

The consumer’s decision problem is then:

max u(x1, . . . , xn)

s.t.

{ ∑i pixi ≤ M

xi ≥ 0 for i = 1, . . . , n(1)








max u(x1, . . . , xn)

s.t.

{ ∑i pixi ≤ M

xi ≥ 0 for i = 1, . . . , n(1)








max u(x1, . . . , xn)

s.t.

{ ∑i pixi ≤ M

xi ≥ 0 for i = 1, . . . , n(1)





A key result here was the Existence and uniqueness result(under the three standard assumptions).

Writing the solution to the consumer’s decision problem givenprices p and income W is denoted byx(p,M) = (x1(p,M), . . . , xn(p,M)), this is the demandfunction. You MUST be able to find a demand function(using for example Lagrange’s method).

We looked at the Cobb-Douglas and CES utility functions insome detail and calculated the MRS (you must know thisdefinition and be able to illustrate it in an indifferencediagram).

Another thing to remember is that the fact that the MRSdecreases along indifference curves is referred to as theprinciple of the diminishing marginal rate of substitution(and that it holds because u is strictly quasi-concave).





























Our final topic in consumption theory under certainty wascomparative statics.

Here you need to be able to draw an ICC and a PCC.

You must understand the concept of a normal good and aninferior good (and how the ICC looks in either case —increasing/decreasing respectively).

Likewise, you must know what a normal and a Giffen goodare, and in all cases be able to show from a demand functionwhat kind of good you are looking at (differentiation isnormally the easiest way).

Finally, income and substitution effects (be able to illustratethese and explain in detail how we get a Giffen good).








































Our second topic is special as far as the exam is concernedsince it is not really a “calculations kind of topic”.

For the exam, you “just” need to understand what we did atthe lecture so that you can answer basic questions about thematerial. Most importantly: you need to know what a lotteryis, what a set of alternatives is, and,

Know what a utility function on lotteries is.

Know what a von Neumann-Morgenstern utility function is,and finally,

Understand the concept of risk-aversion and be able to relatethis to the Bernoulli utility function (for example, theconsumer is risk-averse if and only if u is strictly concave).

Finally, do study the examples we looked at (for example, thebuying a used car example).

















































Next we turned to production theory starting with productionfunction f : R2

+ → R+.

The Key definitions are: Input requirement set and isoquant.You MUST be able to illustrate in a diagram (draw anisoquant diagram).

The main assumptions were: Strictconcavity/concavity/quasi-concavity, continuity, possibility ofinaction. You must be able to illustrate (strict)quasi-concavity!

Then on to Marginal Products and the MRTS (you MUST beable to illustrate the MRTS!).

And then returns to scale (know what they are and how toprove something exhibits for example DRS). Also know what ahomogenous of degree 1 function is.





+ → R+.









+ → R+.









+ → R+.









+ → R+.








Given prices and a target level of income, we have the COSTMINIMIZATION PROBLEM. Be able to illustrate this(isoquants and isocost lines), and of course solve this in orderto find the CONDITIONAL demand function.

Given input and output prices, we have the PROFITMAXIMIZATION PROBLEM (PMP). Be able to solve this inorder to find the (FACTOR) DEMAND FUNCTION.Remember that this problem has no solution under IRS! If theproduction function is strict concave the solution is unique (sothis is when we get a demand FUNCTION).

That’s it for production theory.
















Basics, WE, WEAThe Welfare Theorems

The basics here to know and understand are (1) Net supplyfunctions and production sets, (2) profit functions, (3)consumers’ income [based on the “three sources of income”].

We looked at Walrasian Equilibrium (WE) and WalrasianEquilibrium Allocation (WEA) in two settings: Privateownership economies (no taxes and transfers+consumers’ ownthe firms), and economies with taxes and transfers.

A key result is the existence result which tells us when a WEexists — you should know this result (including itsassumptions).



















After defining WE/WEA and speaking of existence we turnedto the welfare theorems.

You must know by heart the definition of (1) a feasibleallocation, (2) a Pareto optimal allocation.

The first welfare theorem [requires just “greed”]. We had twoversions: With and without lump-sum transfers.

The second welfare theorem [requires the whole battery ofassumptions used also in the existence theorem].

You must finally also be able to discuss the second welfaretheorem (especially its limitations). See the slideshow GET-IIfor more details.






































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G30D, Revision Lecture