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Exercises for Microeconomic theory I -individual choice

Lars-Gunnar Svensson

071029

1 Production theory

1. A company produces one good (y) and use two factors of production(x1, x2). The company can choose between two alternative technologiesrepresented by two concave production functions y = f1(x1, x2) andy = f2(x1, x2), with the corresponding cost functions c1(w1, w2, y) =(2w1 + w2)y and c2(w1, w2, y) = (w1 + 2w2)y.

(a) Calculate the cost function corresponding to the aggregate tech-

nology of the company.

(b) Examine if the aggregate input set is convex.

(c) Examine if the production x1 = 1, x2 = 2, y = 8 is efficient.

2. Let B(y), y R+, be a regular, convex and monotonic input set andlet c(w, y) = (2y + y2)

w1w2 be the corresponding cost function.

Show that the corresponding technology (production function) revealsdecreasing returns to scale.

3. Consider a production sector with two goods y R2 and correspondingprices p R

2

+. The technology Y is closed, convex and monotonic (freedisposal). The profit function is given by

(p) =

0 ifp1 2p22p2 p1 if p2 p1 2p2 ifp1 < p2

(a) Calculate supply and demand for prices p1 > p2.

(b) Show that the technology Y, defined by the profit function ,reveals non-increasing returns to scale. Describe the technology.

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2 Individual choice

1. Let A = {a,b,c} be a set with three alternatives and consider thefollowing four preference relations < on A (each column represents onepreference relation).

1 2 3 4a < b a < b a < b a < ba < c a < c a < c a < cb < a b 6< a b 6< a b < ab < c b 6< c b < c b < c

c 0 bean indirect utility function. Calculate the corresponding direct utilityfunction assuming convex preferences.

2. Let u = f(x1, x2) be a utility function, x1

0, x2

0. the utility

function is continuous, monotonic and concave and the correspondingexpenditure function is:

e(p, u) = (2p1 + p2)u if 0 p1 p2,e(p, u) = (p1 + 2p2)u if 0 p2 p1.

Calculate the two indifference curves

x R2+; f(x) = 1

and

x R2+; f(x) = 2

and give a graphic illustration. Also give a mathematical expressionfor the utility function f(x).

3. Let u(x), xR2

+

, be an ordinal utility function that represents mono-tonic, convex and continuous preferences. The corresponding indirectutility function is

v(p, m) = p1 + 2p2

2

m,

where p > 0 is the price vector and m > 0 is the income.

Calculate the corresponding demand functions and the utility functionu(x).

4. An individual has a demand function x(p, w), where p R2

++ is theprice vector and w R+ is the wealth of the individual. For k = 1, 2, 3the following observations are being made, xk = x(pk, 10) :

p1 = (2, 2), p2 = (1, 3), p3 = (4, 1),x1 = (2, 3), x2 = (4, 2), x3 = (2, 2).

Decide whether these observations are consistent with the Weak Ax-iom of Revealed Preference.

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