Macroeconomic Sol_midterm-2014
Macroeconomic Analysis
ECON6022
Fall2014
Oct25,2014
INSTRUCTIONS:
?Timing and points:The exam lasts for150minutes.The maximal number of points to be attained for this exam is100points.
?Language:Please formulate your answers in English.
?Identi?cation:Please write your NAME and University NUMBER on the cover of the answer book that you use.
?Please write in an intelligible way,and write all your answers in the answer book.
1Capital Share[20Points]
In growth accounting,based on the production function Y=A·Kα·L1?α,the parameterαis determined by setting it equal to the average capital share of a country.
a.Explain the justi?cation for this procedure.[Hint:You may want to use some simple equations to
explain it.]
Solution:Since MP K=?Y
?K =α·A·Kα?1·L1?α,then MP K·K=α·Y.Thus,α=MP K·K
Y
.As
we know,MP K is the marginal product of capital.In competitive economy,it equals the rental rate of capital.Therefore,α=MP K·K
Y
represents the average capital share of a country.
b.Which particular assumptions are needed?[Hint:A short answer with one(or at most two)sentence(s)
is su?cient.]
Solution:Two assumptions are needed for this procedure:First,the production function takes the Cobb-Douglas form;Second,the economy is competitive,which ensures the rental rate of capital equals its marginal product.
2The Solow Model with Productivity Growth[40Points]
Consider a Solow model in which the production function is Y=F(K,AN)=Kα(AN)1?α,where Y is the aggregate output,K capital,N labor force,and A labor productivity.This form of technology is said to be
“labor-augmenting.”Assume that A is growing at a constant rate:?A/A=g.The capital accumulation equation is K t+1?K t=?K t=s·Y t?d·K t,where saving rate s,capital depreciation rate d and population growth rate n are constants.We de?ne a new variable,capital per e?ective labor,as follows,?k=K/(AN).
1.Derive the marginal product of capital(MP K)for the production function given above.Does the
function exhibit diminishing marginal product of capital?
Solution:The marginal product of capital is
?Y
?K
=αKα?1(AN)1?α.
Take second derivative with respect to K,we have
?2Y
?K
=α(α?1)Kα?2(AN)1?α<0.
So the production function exhibits diminishing marginal product.
2.Please show the following dynamics for?k:
??k t=s?kαt?(n+d+g)?k t.
Solution:?K t=sY t?dK t,plug in Y t=Kαt(A t N t)1?α,we have?K t=s·Kαt(A t N t)1?α?dK t.
Divide both sides by A t N t,we have
?K t A t N t =sKαt·(A t N t)?α?d
K t
A t N t
=s?kαt?d·?k t.(1)
Notice that
??k t ?k
t =
?K t
K t
?
?N t
N t
?
?A t
A t
??k t=?K t
K t
·?k t?
?N t
N t
·?k t?
?A t
A t
·?k t
?K t
A t N t
=??k t+n?k t+g?k t.(2) Equation(1)and(2)then show
??k t=s?kαt?(n+d+g)?k t.(3) 3.Solve for the steady state level of capital per e?ective labor,?k?.
Solution:Let??k t=0,we have?k?=(
s
n+g+d
)1/(1?α).
4.Similarly,de?ne?y=Y/(AN)to be the output per e?ective labor and y=Y/N the output per labor.
Solve for both the steady state level of output per e?ective labor?y?and output per labor,y?.
Solution:?y?=(
s
n+g+d
)α/(1?α)and y?t=A t(
s
n+g+d
)α/(1?α).
5.Suppose the economy stays in steady state before Period T.At Period T,the productivity growth rate
falls permanently from g to0.
Then take time derivatives of both sides,the growth rate of output per capita can be written as
y t y t =
?y t
?y t
+g
Before Period T,output per labor grows at a rate of g,because ?y t
?y t
=0.After Period T the capital per e?ective labor is increasing towards a new steady state,but the productivity growth
drops to zero,so output per labor will still increase at the convergence speed of?y t,which is
depicted as less than g in the graph.After the economy rests in the new steady state,the growth
rate of output per labor is zero.
c.Plot the trajectory of?y/y against time,before and after Period T.
Solution:The following?gure shows the trajectory of?y/y,which is the growth rate of output
per labor:
3Consumption and Savings with Borrowing Constraint[40Points] Consider a representative consumer who lives for three periods,he or she has an exogenous endowment stream given by y1,y2and y3and can borrow and lend at a given interest rate r,which is the interest rate. Assume that he/she starts out with no wealth(that is,b1=0).The discount factor isβ=1and utility function is increasing and concave in consumption.The consumer’s problem is:
max c
1,c2,c3
u(c1)+u(c2)+u(c3)
subject to
c1+b2=y1
c2+b3=y2+(1+r)·b2
c3=y3+(1+r)·b3
where b t is the?nancial wealth in Period t.
1.Write down the inter-temporal budget constraint.
Solution:The inter-temporal budget constraint for the consumer is
c1+
c2
1+r
+
c3
(1+r)2
=y1+
y2
1+r
+
y3
(1+r)2
2.Derive the Euler equations.
Solution:We can transform the consumer’s optimization problem as
max
c1,c2
u(c1)+u(c2)+u[(1+r)2(y1?c1)+(1+r)(y2?c2)+y3] First Order Conditions are:
FOC(c1):u (c1)+(?1)(1+r)2u (c3)=0
FOC(c2):u (c2)+(?1)(1+r)u (c3)=0
Rearrange the terms,we have the following Euler equations,
u (c1)=(1+r)2u (c3)(4)
u (c2)=(1+r)u (c3)(5) Or alternatively,
u (c1)=(1+r)u (c2)(6)
u (c2)=(1+r)u (c3)(7)
3.For simplicity,further assume that r=0.
a.Solve for the optimal consumption plan for the three periods,c?t,where t=1,2,3.
Solution:With r=0,we can get the household’s Euler equations:
c2 c1=1(8)
c3
c2
=1(9) The consumer’s inter-temporal budget constraint is now
c1+c2+c3=y1+y2+y3(10) Substituting equ(8)and equ(9)into(10),we get the optimal consumption plan for the consumer:
c1=c2=c3=y1+y2+y3
(11)
The result is consistent with the permanent income theory which says that consumers make consumption decision based on permanent income and would like to smooth the consumption over the lifetime.
b.Suppose y1=y3 c1≤y1.Find the optimal consumption plan for the three period,c?t,where t=1,2,3.(You don’t have to go through the math and it is su?cient to justify your answers in words.)Do the Euler equation hold in this case?Why or why not? Solution:Since y1 3 ,we know that in Period1,the consumer wants to borrow. However,the consumer is borrowing constrained,that’s c1≤y1,therefore he chooses c?1=y1. Since y2>y3,the consumer saves and therefore,c?2=c?3=y2+y3 2 c.Would your answer to(b)change,if the utility function is u(c)=c?Why or why not? Solution:If u(c)=c,the utility function is linear rather than concave.And the marginal utility of consumption is constant.Thus,the optimal consumption plan would be di?erent from what we get in sub-question(b).In this case,c1,c2and c3can be any combination as long as the consumer lifetime budget constraint c1+c2+c3=y1+y2+y3 and c1≤y1is satis?ed.The consumer does not care the consumption allocation along the life time since the marginal utility is constant in each period and does not change no matter how much he consumes in each period. —————end of paper—————