, αβ, > 0 is strictly quasi-concave on

Transcription

1 John Riley 8 Setember 9 Econ Diagnostic Test Time allowed: 9 minutes. Attemt all three questions. Note that the last two arts of questions and 3 are marked with an asterisk (). These do not carry many oints so should be viewed as bonus questions to be considered at the end ONLY if you have time.. Consumer demand / Ale has utility function U( ) = 8 +. The rice vector is = (,) and income is 9. (a) Solve for Ale s otimal consumtion vector if (i) = 4 (ii) =. (b) Solve for Ale s demand function ( ) for all strictly ositive.. Concave and quasi-concave functions (a) Suose that g : and : n + f X + +. If g is strictly increasing and strictly concave and h ( ) = g( f( )) is strictly concave show that f is strictly quasi-concave on X. (b) Use this result to show that f( ) α β =, αβ, > is strictly quasi-concave on = { ++, >, > }. (c) Either rove the following or elain why the statement is false. If f is strictly quasi-concave and df d ( ) = then f has a local maimum at. (d) If f is concave on n show that for any f f f + ( ) ( ) ( ) ( ) n,, (e) Returning to art (b), is f strictly quasi concave on (f) Is f quasi-concave on +? +? 3. Choice over time

2 John Riley 8 Setember 9 t Bev has lifetime a utility function Uc ( ) = ln c+ δ ln c δ ln c T where δ (,). Her initial financial caital is K > and she has an income stream { y } T t t + where yt, t =,..., T. She can borrow or lend at the interest rate r >. (a) Let W be the resent value of initial caital lus the income stream. Show that the constraints on the Bev s otimal choice can be reresented by a single life-time budget constraint c c3 ct c W T. + r ( + r) ( + r) (b) Elain why the constraint qualification holds for any { c } T t satisfying this constraint with equality. (c) Write down the first order (Kuhn-Tucker) conditions for the three eriod case and use these to solve for the first eriod consumtion as a function of total wealth W and arameters. (d) Are the necessary conditions also sufficient for a maimum? Elain. c (e) What is the marginal roensity to consume out of current income, that is y? (f) Either formally or informally, etend the argument to the T eriod case. c (g) How would y change if Bev lans her consumtion so as to leave a bequest of K T +? Answers:

3 John Riley 8 Setember 9. Consumer demand (a) The constraint qualification is satisfied because (i) the gradient of h ( ) = is not zero and (ii) since the constraint in linear, the linearized feasible set is the budget set and this has an on-emty interior since I = 8. Form the Lagrangian. FOC L= (9 + ). / L = / 4, with equality if > L =, with equality if > Suose that >> then = and so 4 =. Then / 6 =. (i) = 4 then (ii) = then = and = 6 and = 9 = 5 = 9 6= 7 Clearly the latter is wrong. Then try FOC = so = I= 9. Then 9 =. L = / 4 = since > L =, since = Substituting into the FOC for L 4 = = 3 Then the FOC for is also satisfied.

4 John Riley 8 Setember 9 (b) We first see when >. If so 6 6 = and so = = 9. 9 Thus = 6 if and only if 9/6. Otherwise. Concave and quasi-concave functions (a) h ( ) is strictly concave. Thus for any That is, h h h ( ) > ( ) ( ) + ( ) g f g f g f ( ( )) > ( ) ( ( )) + ( ( )) = and 9 =., and conve combination, < < Suose that f ( ) f( ). Since g is increasing, it follows that g f ( ( )) g( f( )) and so g f ( ( )) g( f( )) >. Since g is strictly increasing it follows that f ( ) f( ) >. (b) Choose g( y) = lny. Then h ( ) = ln = αln + βln. α β Note that h is the sum of two strictly concave functions (strictly negative second derivatives). Then h is strictly concave. (c) A counter eamle is deicted below. The sloe is zero at the oint of inflection. Each of the uer contour sets is an interval and hence f is quasi-concave. Also conve combinations of oints in an uer contour set must lie in the interior so the uer contour sets are strictly quasiconcave. (d) See Aendi (e) Along the aes f =. Thus the contour set f( ) = is as deicted below.

5 John Riley 8 Setember 9 It is not the case that f is strictly higher for conve combinations so the uer contour set is conve rather than strictly conve. (f) From the above argument the function is quasi-concave. f ( ) 3. Choice over time (a) K = t ( + r)( K + t y + t ct) Then

6 John Riley 8 Setember 9 K K y c ( + r) ( + r) ( + r) ( + r) Summing over t Rearranging, t+ t t t t t t t KT + K PV{ yt} PV{ ct} T ( + r) K K PV{ c } K + PV{ y } = W ( + r ) ( + r ) T+ T+ t t T T With no bequest motive the last term is zero. (b) The gradient vector is non-zero and since W > the (linear) feasible set has a non-emty interior. (c) c c 3 L= ln c+ δ ln c + δ ln c3+ ( W c ). + r ( + r) Look for a non corner solution c >>. FOC L = =, c c L δ = = c c + r L δ = = c3 c3 ( + r). From the first two conditions c = ( + r) δc. From the first and third condition, c = ( + r) δ c. 3 Substitute into the budget constraint c c3 c+ + = c + δc+ δ c = W. + r ( + r) W Hence c = = ( K + y+...). + δ + δ + δ + δ

7 John Riley 8 Setember 9 (d) Each term in the utility function is concave so U is concave (sum of concave functions) and hence quasi-concave. The gradient vector is strictly ositive for all c >>. The constraint function h ( ) is linear and hence quasi-concave. Thus the necessary conditions are sufficient. c (e) From art (c) = y + δ + δ. KT+ y KT+ (g) The budget constraint is PV{ c } W = y ( + r) + r ( + r) Thus the derivative is unaffected. t T T