Economics 401. Sample questions 1

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1 Economics 40 Sample questions. Do each of the following choice structures satisfy WARP? (a) X = {a, b, c},b = {B, B 2, B 3 }, B = {a, b}, B 2 = {b, c}, B 3 = {c, a}, C (B ) = {a}, C (B 2 ) = {b}, C (B 3 ) = {c}; (b) X = {a, b, c},b = {B, B 2 }, B = {a, b}, B 2 = {a, b, c}, C (B ) = {a}, C (B 2 ) = {c}; (c) X = {a, b, c, d},b = {B, B 2 }, B = {a, b, c}, B 2 = {a, b, d}, C (B ) = {a, c}, C (B 2 ) = {a, d}; WARP holds in some choice structure (B, C ( )) if: Assume B, B 2 B, {a, b} B B 2 ; a C (B ) and b C (B 2 ) a C (B 2 ). It follows from this definition that WARP has no leverage unless our observations give us at least two budget sets with two elements in common say a and b AND a is amongst the best elements in one budget set and b is amongst the best elements in the other budget set. These assumptions are not met in any of these cases above so WARP is trivially satisfied. 2. MWG,.B.3 Proof: We say that u ( ) represents on choice set X if the following statement is true for all x, y X, x y if and only if u (x) u (y) To complete the proof note that if f ( ) is a strictly increasing function defined on the real numbers then for all x, y X, x y if and only if f (u (x)) f (u (y)) MWG,.B.4

2 Proof: Denote ( ) for all x, y X, x y if and only if u (x) = u (y) ( ) for all x, y X, x y if and only if u (x) > u (y) Given that ( ) and ( ) are true we need to prove the definition in MWG.B.3 holds. Note that x y implies either x y in which case u (x) > u (y) or x y in which case u (x) = u (y); combining the two options then x y implies u (x) u (y). Conversely, u (x) u (y) implies either u (x) > u (y) in which case x y or u (x) = u (y) in which case x y. So combining the two options then u (x) u (y) implies x y. { } {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, 3. Let X = {a, b, c, d}, B = and (B, C ( )) be a {c, d}, {d, b, c}, {a, b, d}, {a, c, d} choice structure. Suppose C ( ) is such that d is the best choice whenever d is available and C ({a, b}) = {a}, C ({b, c}) = {b}, and C ({c, a}) = {c}. (a) Does C ( ) satisfy WARP? (b) Is there a rational preference relation which rationalizes C ( ) on B? (c) If {a, b, c} were another budget set could C ( ) satisfy WARP? Defend your answers to (a), (b) and (c) carefully. (a) WARP holds in some choice structure (B, C ( )) if: B, B B, {x, y} B, {x, y} B : x C (B), y C (B ) x C (B ). The only way to contradict WARP here is for some problem to arise with budget sets that don t contain d. But there is none of these with two or more elements in common, so the supposition in the definition of WARP is not satisfied and thus WARP must be true. Remember, anything is true of the empty set. (b) No, because a b c but c a, so transitivity doesn t hold. (c) Try C ({a, b, c}) = {a} ; but C ({c, a}) = {c} so WARP is violated. Try C ({a, b, c}) = {b} ; but C ({a, b}) = {a} so WARP is violated. Try C ({a, b, c}) = {c} ; but C ({b, c}) = {b} so WARP is violated. So the only way to satisfy WARP is to set C ({a, b, c}) = the empty set, which is not allowed. 4. Answer both parts. (a) What does it mean to say that a utility function, u( ), represents a preference relation on some choice set X? Prove that if u( ) represents preference relation, this preference relation must be complete and transitive. 2

3 (b) Now suppose X = R 2 + and (x, x 2) (x 2, x 2 2) when x > x 2, or x = x 2 and x 2 > x 2 2. Is this preference relation complete, transitive and continuous? Defend your answers. if (a) A utility function u : X R represents a preference relation on the choice set X x, y X, x y u(x) u(y). Completeness: x, y X, u(x) and u(y) are real numbers so either u(x) u(y) in which case x y or u(y) u(x) in which case y x. Transitivity: Suppose x, y, z X, and x y and y z. We need to show x z. Since u represents, u(x) u(y) and u(y) u(z). Since these are real numbers u(x) u(z) x z. (b) Lexicographic preferences are complete and transitive but not continuous. Completeness: Consider any two distinct points in R 2 + (x, x 2) (point a) and (x 2, x 2 2) (point b). If x > x 2, a b. If x 2 > x, b a. If x = x 2, since a and b are distinct, it must be that either x 2 > x 2 2 in which case a b or x 2 2 > x 2 in which case b a. Transitivity: Let a, b, c R 2 + where a = (x, x 2), b = (x 2, x 2 2) and c = (x 3, x 3 2). Suppose a b and b c. We want to prove a c. a b either x > x 2 (call this ) or x = x 2 and x 2 > x 2 2 (call this 2). b c either x 2 > x 3 (call this 3) or x 2 = x 3 and x 2 2 > x 3 2 (call this 4). If and 3 are true then x > x 3 in which case a c. If and 4 are true then x > x 3 in which case a c. If 2 and 3 are true then x > x 3 in which case a c. If 2 and 4 are true then x = x 3 and x 2 > x 3 2 in which case a c. Thus in every possible situation a c, which is what we wanted to prove. Continuity: Suppose and {x n } R 2 + and lim n x n = x {y n } R 2 + and lim n y n = y and x n y n, n. Then if were continuous we would be able to deduce that x y. Let x n = (/n, 0) and y n = (0, ). Then x = (0, 0), y = (0, ), x n y n n but y x. So lexicographic preferences are complete and transitive but they violate continuity. 5. You are given the following information about a consumer s purchases. Goods and 2 are the only goods consumed. Year Year 2 Quantity Price Quantity Price Good Good ? 80 3

4 For what values of good 2 consumed in year 2 would you conclude: (a) that the consumer s consumption bundle in year is revealed preferred to that in year 2? (b) that the consumer s consumption bundle in year 2 is revealed preferred to that in year? (c) that her/his behaviour contradicts the weak axiom? (d) that good is an inferior good somewhere for this consumer (assume WARP holds)? (e) that good 2 is an inferior good somewhere for this consumer (assume WARP holds)? (a) Denote the consumption of good 2 in year 2 by x 2 2. If the consumer s consumption bundle in year is revealed preferred to that in year 2 then the year 2 bundle must be affordable with year prices and income. So (00) (00) + (00) (00) (00) (20) + (00) x 2 2 or x (b) If the consumer s consumption bundle in year 2 is revealed preferred to that in year then the year bundle must be affordable with year 2 prices and income. So (00) (20) + 80x 2 2 (00) (00) + (80) (00) or x (c) This requires that the inequalities in (a) and (b) both hold, so 75 x (d) x 2 2 < 75. (e) 80 < x Consider a price-taking consumer in a two-good world. Let w denote money wealth, (, ) prices, x j (,, w) Marshallian demands, h j (,, u) Hicksian demands, V (,, w) the indirect utility function and e (,, u) the expenditure function. Fill in the following tables, and verify the Slutsky equation for the effect of changing the price of good on the demand for good 2, for someone whose preferences are represented by: u = x x 2. 4

5 x (,, w) x 2 (,, w) V (,, w) Range h (,, u) h 2 (,, u) e (,, u) Range x (,, w) x 2 (,, w) V (,, w) Range w w w , > 0, w 0 h (,, u) h 2 (,, u) e (,, u) Range p /2 /2 2 u /2 /2 p /2 2 u /2 2/2 /2 2 u /2, > 0, u 0 Slutsky equation: from the duality between the UMP and the EMP we know x 2 (,, e (,, u)) = h 2 (,, u). Differentiate this with respect to and use the envelope theorem to obtain x 2 x 2 + x w = h 2 LHS = 0 + w 2 2 But w = e (,, u) = 2/2 /2 2 u /2, so LHS = 2 p /2 p /2 2 u /2 = RHS 7. Consider a price-taking consumer in a two-good world. Let w denote money wealth, (, ) prices, x j (,, w) Marshallian demands, h j (,, u) Hicksian demands, V (,, w) the indirect utility function and e (,, u) the expenditure function. Fill in the following tables for someone whose preferences are represented by: u = min (x, 2x 2 ). x (,, w) x 2 (,, w) V (,, w) Range h (,, u) h 2 (,, u) e (,, u) Range 5

6 x (,, w) x 2 (,, w) V (,, w) Range 2w 2 +, > 0, w 0 2w 2 + w 2 + h (,, u) h 2 (,, u) e (,, u) Range u u/2 u ( + /2), > 0, u 0 The Slutsky substitution matrix is S = = [ h h h 2 h 2 [ ]. ] 8. Susan is a price taker and lives in a three-good world. Denote her money wealth by w, prices by (,, ) and her consumption bundle by (x, x 2, x 3 ). Find her Marshallian demands if her preferences can be represented by u = ln x + 2 ln x 2 + x 3. x (,, w) x 2 (,, w) x 3 (,, w) Range 2 w 3 w 3 w 2 w w < 3 9. Suppose a consumer s preferences can be represented by u = 2x /2 + 2x /2 2 + x 3. Find Marshallian demands and the indirect utility function. Find Hicksian demands and the expenditure function. Be sure to specify the range over which each function is valid. Verify the Slutsky equation for the effects of a change in the price of good 2 on the demand for good 3. The Marshallian demands are: 6

7 x x 2 x 3 V (,, w) Range 3 w ( + ) 3 2 w w ( + ) 0 2 w + + w > p2 3 + p2 3 p ( ) 2 /2 w(p + ) w p2 3 + p2 3 The Hicksian demands are: ( h h 2 h ( 3 ) e(,, u) Range ( ) 3 3 u 2p u p2 3 p2 3 u > 2 + p ) 2 2 ( 2 ( ) u p 2( + )) 0 u 2 4( + u 2p ) 3 + u 2( + ) The Slutsky equation we want is Verifying we have h 3 = x 3 x 3 + x 2 w Range LHS RHS Range u > 2 ( + p2 ) p w > p2 3 + p2 3 u 2 ( + p2 ) 0 0 w > p2 3 + p Consider the utility function Assume x, x 2 > 0. u = x a x b 2, a > 0, b > 0. (a) Under what conditions is this function strictly concave? (b) Under what conditions is this function strictly quasi-concave? (c) Use your answers to (a) and (b) to discuss the relationship between concavity and quasi-concavity. The function f (x, x 2 ) = x a x b 2, a > 0, b > 0 is strictly concave only if f < 0 and f f 22 f2 2 > 0. And the function is strictly quasi-concave only if f 2 f 22 2f f 2 f 2 +f2 2 f < 0. We have 7

8 u = au x u 2 = bu x 2 a(a )u u = x 2 u 2 = abu x x 2 b(b )u u 22 = x 2 2 Thus u u 22 u 2 2 = ( ab(a )(b ) a 2 b 2) u 2 = ab( a b) u2 x 2 x 2 2 x 2 x 2 2 and = u 2 u 22 2u u 2 u 2 + u 2 2u ( ) 2 ( ) ( ) ( ) 2 au b(b )u au bu abu bu a(a )u 2 + x x 2 x x 2 2 x x 2 = abu3 (a(b ) 2ab + b(a )) x 2 x 2 2 = abu3 (a + b) x 2 x 2 2 So strict concavity holds only if a+b < but strict quasi-concavity holds for any a, b > 0. Thus strict quasiconcavity is a weaker condition than strict concavity.. Suppose a consumer s preferences can be represented by u = x + 2x /2 2. Find Marshallian demands and the indirect utility function. Find Hicksian demands and the expenditure function. Be sure to specify the range over which each function is valid. Verify the Slutsky equation for the effects of a change in the price of good 2 on the demand for good. The Marshallian demands are: 8 x 2 x 2

9 The Hicksian demands are: x x 2 V (,, w) Range w/ / / 2 w/ + / w / 0 w/ 2(w/ ) /2 w < / The Slutsky equation we want is h h 2 e(,, u) Range u 2 / / 2 u / u 2 / 0 u 2 /4 u 2 /4 u < 2 / Verifying we have h = x x + x 2 w Range LHS RHS Range u 2 / 2 / 2 / 2 + ( / 2)(/ ) w / u < 2 / 0 0 w < / 2. Answer both parts of this question. (a) Prove that the expenditure function, e(,, u) is concave in prices (, ). (b) Consider a three-good demand system where x = a + b / + c / + d / 3 x 2 = e + f / + g / + h / 3. where the demand function for the third good follows from the person s budget constraint. If this system is derived from a well-behaved utility-maximizing problem what are the restrictions on a, b, c, d, e, f, g and h? (a) Consider two sets of prices (, 2) and (, 2) and the convex combination of them (p t, p t 2) = t(, 2) + ( t)(, 2) for 0 t. Then e(,, u) is concave in prices (, ) if e(p t, p t 2, u) te(, 2, u) + ( t)e(, 2, u). Let (h, h 2) be the expenditure-minimizing bundle for prices (p t, p t 2) and utility level u. Then 9

10 e(p t, p t 2, u) = p t h + p t 2h 2 = (t + ( t) )h + (t 2 + ( t) 2)h 2 = t( h + 2h 2) + ( t)( h + 2h 2) te(, 2, u) + ( t)e(, 2, u). (b) If this system is derived from a well-behaved utility-maximizing problem then it must satisfy the budget constraint, the demands must be homogeneous of degree zero in prices and money income and the Slutsky substitution matrix must be symmetric and nsd. The demand for the third good is defined so that the budget constraint holds. Inspection of the equations for x and x 2 shows that scaling prices and income by the same positive number has no effect on the demands, so homogeneity is satisfied. Inspection of the demands for the first two goods also shows that they are independent of income so not only are they Marshallian demands but they are also Hicksian demands. Thus S = = h h S 3 h 2 h 2 S 23 S 3 S 32 S 33 b/ + d / 3 c/ + d / 3 S 3 f/ + h / 3 g/ + h / 3 S 23 S 3 S 32 S 33 Note that the budget constraint and homogeneity imply the rows and columns of S are linearly dependent so we need check only the upper 2 X 2 submatrix of S. For symmetry we must have c/ + d / 3 = f/ + h / 3,,, > 0. This means that d = h = 0 and c = f. Then S 0 means b 0 and S 22 0 means g 0. S S 22 S implies bg c 2 0. Finally, if we want the demands to be positive for all prices then a, e Consider a life-cycle consumer whose lifetime is T years and whose utility function is ln(c ) + + ρ ln(c 2) ( + ρ) ln(c T T ), where c i, i =,..., T, is real consumption in period i and ρ > 0 is the utility discount rate. Suppose the real interest rate is r. Under what conditions will her consumption-age profile (annual consumption graphed against age) be upward-sloping? Justify your answer. In an optimal consumption plan 0

11 MRS ctc t+ = + r so (+ρ) t c t (+ρ) t c t+ = + r, t =,..., T. Rewriting we have c t+ c t = + r + ρ. Thus if r > ρ, c t+ > c t and the consumption-age profile slopes upward throughout the person s life cycle.