Advanced Microeconomic Analsis Solutions to Midterm Exam Q1. (0 pts) An individual consumes two goods x 1 x and his utilit function is: u(x 1 x ) = [min(x 1 + x x 1 + x )] (a) Draw some indifference curves of this individual. The indifference curves will be the same as those of min(x 1 + x x 1 + x ). The first argument of the min is smaller when x 1 + x < x 1 + x x 1 < x Let s find the indifference curve for the utilit level u. If x 1 < x then u = x 1 + x x = u x 1 with a slope of. If x 1 > x then u = x 1 + x x = u/ x 1 / with a slope of 1/. Therefore the indifference curves look like this: 8 7 6 5 4 3 1 0 0 1 3 4 5 6 7 8 (b) Does the following propert hold for this utilit function? (You don t have to give an explanation). concave Consider the utilit level along the 45-degree line that is u(x x) as x increases. It is equal to 9x which is not concave. Therefore this utilit function is not concave. quasiconcave As we can see from the indifference curves the upper level sets are convex. Therefore u( ) is quasiconcave. homogeneous u(tx 1 tx ) = [min(tx 1 + tx tx 1 + tx )] 1
Therefore it is homogeneous of degree. homothetic = [t min(x 1 + x x 1 + x )] = t [min(x 1 + x x 1 + x )] A homothetic function can be formulated as an increasing transformation of a function that is homogeneous of degree 1. u(x 1 x ) = f(g(x 1 x )) where f(x) = x and g(x 1 x ) = min(x 1 + x x 1 + x ) is homogeneous of degree 1. Therefore it is homothetic. (c) Find the Marshallian demand for both goods. Suppose income is and prices of x 1 x are p 1. The optimal bundle will be the point(s) where the indifference curves are tangent to the budget line with slope p 1. > the upper left corner is optimal: x = (0 ) = all points on the upper half of the indifference curve are optimal: x = {t(0 ) + (1 t)( p 1 + p 1 + ) t [0 1]} If > p 1 > 1 the midpoint is optimal: x = ( p 1 + p 1 + ) = 1 all points on the lower half of the indifference curve are optimal: x = {t( p 1 0) + (1 t)( p 1 + p 1 + ) t [0 1]} < 1 the lower right corner is optimal: x = ( p 1 0) (d) Find the indirect utilit function. v(p 1 ) = ( ) If > p 1 > 1 v(p 1 ) = ) ( ) 1 v(p 1 ) = p 1 ( 3 p 1 + ) Q. (0 pts) Suppose an individual consumes two goods x 1 x and his indirect utilit function is v(p 1 ) = ( + p 1 + )p 1 1 p 1 (a) Find the Marshallian demand function for each of the two goods. (p 1 ) is demand for both goods positive? Using Ro s identit: At what levels of x 1 (p 1 ) = v/ p 1 v/ = (p 1 + + )p 3 1 p 1 p 1 1 p 1 p 1 1 p 1
= p 1 + p 1 x (p 1 ) = v/ v/ = + p 1 Both demands are positive when > p 1 and > p 1. (b) Find the individual s expenditure function. Using the relation v(p e(p u)) = u: e(p 1 u) = up 1 1 p 1 p 1 Q3. (0 pts) Suppose an individual lives for two periods; his goods are x 1 the amount consumed in period 1 and x the amount consumed in period. His utilit function over pairs (x 1 x ) is: u(x 1 x ) = ln(x 1 ) + ln(x ) At the beginning of period 1 he has an amount of wealth > 0 which can be allocated to three purposes: An amount 0 x 1 can be consumed in period 1. An amount 0 b can be stored; this will be available for consumption in period. An amount 0 z can be invested into a risk asset; this asset s returns will be available for consumption in period. Suppose that with probabilit p the net return is Rz where R > 1; with probabilit 1 p the net return is 0. Therefore the amount available for consumption in period will be b + Rz with probabilit p and b with probabilit 1 p. Find the choice of x 1 b and z that maximizes expected utilit E [u(x 1 x )] subject to the budget constraint x 1 + b + z. Since ln is a strictl increasing function the budget constraint will be satisfied with equalit in both periods. In period the individual will consume all available resources. The problem becomes: max E [u(x 1 x )] = max ln(x 1) + p ln(b + Rz) + (1 p) ln(b) s.t. x 1 + b + z = 0 x 1 bz x 1 bz L(x 1 b z λ) = ln(x 1 ) + p ln(b + Rz) + (1 p) ln(b) λ(x 1 + b + z ) b = x 1 = 1 x 1 λ = 0 p Rz + b + 1 p λ = 0 b z = pr Rz + b λ = 0 λ = x 1 + b + z = 0 3
This has a solution x 1 = b = R pr (R 1) z = pr 1 (R 1) Q4. (0 pts) Suppose there are a large number of identical firms in a perfectl competitive industr. Each firm has the long-run average cost curve: where q is the firm s output. AC(q) = q 1q + 50 (a) What condition must be satisfied in long-run equilibrium? In long-run equilibrium we assume that entr and exit of firms has driven profits to zero. Therefore an individual firm s profit must be: which is satisfied when p = AC(q). π(q) = pq c(q) = pq AC(q)q = 0 (b) What condition must be satisfied in a perfectl competitive industr? In a perfectl competitive industr we assume that firms are price-takers and the optimal quantit is achieved when: p = MC(q) = c (Q) = d dq q(q 1q + 50) = 3q 4q + 50 (c) Derive the long-run suppl curve for this industr. In the long run both conditions must be satisfied. Therefore we have p = AC(q) = MC(q) = q 1q + 50 = 3q 4q + 50 q = 6 p = 14 for an individual firm. Since there are man firms the suppl curve is horizontal at p = 14: firms will enter the market to produce as much as is demanded. Q5. (0 pts) An industr consists of man identical firms with cost function c(q) = q + 1. When there are J active firms each firm faces an identical inverse market demand p = 15q (J 1)q whenever the other J 1 firms produce the same output level q. 4
(a) In the short run suppose there is no entr or exit. What is the market price and quantit in the Cournot equilibrium with J firms? Consider firm i s profit as a function of its own quantit q i and q which is assumed to be the output of all of the other J 1 firms: π(q i q) = pq i q i 1 = ( 15q i (J 1)q)q i q i 1 = 16q i + ( (J 1)q)q i 1 The optimal choice of q i taking q as given satisfies the first-order condition: π q i = 3q i + (J 1)q = 0 qi (J 1)q (q) = 3 The Cournot-Nash equilibrium is when all firms individuall choose their optimal quantities taking other firms actions as given. Therefore each firm s quantit must satisf: q = (J 1)q 3 q = 3 + J 1 Total output is J 3+J 1. The market price is 170 31+J. (b) In the long run entr and exit is allowed. What will be the number of active firms? In the long run entr and exit drives profits down to zero. Equilibrium profits with J firms is: ( ) 16 + ( (J 1) 3 + J 1 1600 = (3 + J 1) J = 9 3 + J 1 ) 3 + J 1 1 = 0 5