Macroeconomics

1. (30 marks) Consider the life-cycle model discussed in Lecture 3-5. Now suppose that

capital depreciates after production, and the capital depreciate rate is 0 < δ < 1.1

(a) (10 marks) Formulate the individuals’ utility maximisation problem and solve it

to find the optimal consumption plan and saving of a generation t individual.

(b) (5 marks) Formulate the firms’ profit maximisation problem, and write down the

equations that characterise the solution.

(c) (5 marks) Write down the market clearing condition for the labour and capital

markets.

(d) (5 marks) Combine the equations obtained in (a)-(c) to find the transition equation

that describes how kt ≡

Kt

N

evolves over time.

(e) (5 marks) Show that national saving equals national investment.

2. (30 marks) Assume the following parameter values: N = 100, A = 10, K1 = 150,

α = 0.7, β = 0.3, δ = 0.6.2

(a) (15 marks) Write down the equations you will use to compute the transition paths

of kt

, rt

, wt

, cyt, cot, Kt

, Yt

, Ct ≡ Ncyt + Ncot, St

, It

, yt ≡ Yt/N, and st ≡ St/Yt

.

Then calculate their values (in Excel or other software) for periods 1 through 10,

and report kt

, rt

, wt

, cyt, cot, Yt

, Ct

, It

, yt

, and st

in a table.

1

In Lecture 3-5, we assumed capital does not depreciate, i.e., δ = 0, so that old individuals can get back

the full assets they invest in firms plus interests. However with capital depreciation, the assets they invest

in firms will lose some value after production. This is the only difference from the lectures.

2These values are more realistic than those used in Lecture 5, especially when a period in our model refers

to 30 years. In particular, a higher value of α reflects that individuals value present consumption more than

future consumption, and δ = 0.6 implies an annual depreciation rate of roughly 3 percent.

1

(b) (7 marks) Plot the time paths of kt

, yt

, rt

, wt

, cyt, cot, and st (put wt and rt on

one figure, and cyt and cot on one figure).

(c) (8 marks) Calculate the steady state values of kt

, Yt

, yt

, wt

, rt

, cyt, cot, and st

.

Compare them to the steady state values in Table 5.1 of Lecture 5 and explain

any differences and similarities.

3. (25 marks) Suppose in period 1 the economy is in the steady state calculated in problem