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ECO4145 Mathematical Economics II

Assignment 2

1. Consider a consumer who begins in period 0 with a stock of non-human wealth k0 . The consumer - known as the representative consumer in macroeconomics - is assumed to live forever. In each period t, t = 0, 1, ..., she has 1 unit of time that she aupplies in-elastically on the labor market at the wage rate ωt . The market rate of interest in period t is rt . The present

value of the stream of labor income (ωt)0, which is the stock of her human wealth is h0 = ω0 + + + ... + + ...

The sum of her human and non-human wealth is then given by k0 + h0 .


Let ct 0 be her consumption in period t, t = 0, 1, ... The present value of the stream of

consumption (ct)0 is

c0 + + + ... + + ...

Let u(c) denote her utility in a period, when her consumption in that period is c. Suppose that δ, 0 < δ < 1, is the factor the consumer uses to discount her future utilities. The problem faced by the consumer is

(1)   max(ct)0 Σ0 δt u(ct)

subject to the intertemporal budget constraint

(2)   k0 + h0 = c0 + + + ... + + ...


Solve the representative problem. In answering this question assume the following func- tional form for the single-period utility function: u(c) = Log[c].

The present value of the consumption in period t is

(3) t = , t = 0, 1, ...

Using (3), we can rewrite the intertemporal budget constraint (2) as follows:

(4)   k0 + h0 = 0 + 1 + 2 + 3 + ... + t + ... Now equation (3) can also be written as

(5)   ct = t(1 + r0)(1 + r1)×(1 + r2) ... (1 + rt-1). The social welfare in period t is thus given by

(6)   Log[ct] = Logt(1 + r0)(1 + r1)×(1 + r2) ... (1 + rt-1)

= Log[t] + Log[(1 + r0)×(1 + r1)×(1 + r2) ... ×(1 + rt-1)].

Using (6), we can rewrite the objective function (1) as


(7) Σt0 δt u[ct] = Σt0 δt Log[ct]

= Σt0 δt Log[t] + Log[(1 + r0)×(1 + r1)×(1 + r2) ... ×(1 + rt-1)]

= Σt0 δt Log[t] + δt Log[(1 + r0)×(1 + r1)×(1 + r2) ... ×(1 + rt-1)]

= Σt0 δt Log[t] + Σt0 δt Log[(1 + r0)×(1 + r1)×(1 + r2) ... ×(1 + rt-1)]

= Σt0 δt Log[t] + κ ,

where we have let

(8) κ = Σt0 δt Log[(1 + r0)×(1 + r1)×(1 + r2) ... ×(1 + rt-1)] = constant.


Thus, maximizing (7) is the same as maximizing Σt0 δt Log[t], and the problem of the representative consumer is reduced to

(9)   max(ct)0 Σt0 δt u[t]

subject to t 0, t = 0, 1, ...,

and the intertemporal budget constraint

(10) k0 + h0 = 0 + 1 + 2 + 3 + ... + t + ....

The maximization problem constituted by (9) and (10) is nothing other than the cake-eating problem with k0 + h0 as the size ofthe cake. The solution is given by

(11) t= δt (1 - δ )(k0 + h0), t = 0, 1, 2, ... or

(12) ct = t(1 + r0)(1 + r1)×(1 + r2) ... (1 + rt-1), t = 0, 1, 2 ...

The solution of the cake-eating problem with logarithmic utility function

Suppose that we begin at time 0 with a cake ofsize x and must find a consumption program to eat this cake in infinite time. Let v[x] be the discounted social welfare obtained by optimally eating this cake in infinite time. Invoking the principle of optimality, we can assert that v[x] satisfies the following Bell- man equation:

(11) v[x] = max0≤ c≤ x(Log[c] + δ v[x - c]).

The following first-order condition characterizes the value of c , the optimal consumption in period 0, that maximizes the expression on the right-hand side of (1):

(12) - δ v'[x - c] = 0.

Let c[x] be the solution of (2). As defined, c[x] is the optimal consumption in any period when x is the remaining cake. It follows directly from the definition ofc[x] that

(13) - δ v'[x - c[x]] = 0.

Also, substituting c[x] for c in (11), we obtain

(14) v[x] = Log[c[x]] + δ v[x - c[x]].

Dierentiating (14) with respect to x, we obtain

(15) v'[x] = c'[x] + δv'[x - c[x]](1 - c'[x])

= c'[x] - δv'[x - c[x]] + δv'[x - c[x]]

= δv'[x - c(x)].

Observe that the last equality in (15) has been obtained with the help of (3).

Using (15), we can rewrite (13) as follows:

(16) - v'[x] = 0.

Now suppose that we begin in period 0 with a cake ofsize x0. Next, let (xt(*))0 and (ct(*))0 be two infinite sequences defined recursively as follows.

(17) x0(*) = x0, c0(*) = c[x0(*)],

x1(*) = x0(*) - c0(*), c1(*) = c[x1(*)],

...,

xt(*) = xt-(*)1 - ct(*)-1, ct(*) = c[xt(*)],

...

As defined, (ct(*))0  is the optimal program for eating the cake of size x0  in infinite time, and xt(*), t = 0, 1, ..., is the remaining cake at the beginning ofperiod t. Next let

(18) λt = v'[xt(*)], t = 0, 1, ...,

be the current shadow price of cake in period t along the optimal trajectory. Applying (15) in period t, with x = xt(*), we obtain

(19) v'[xt(*)] = δv'[xt(*) - c[xt(*)]] = δv'[xt(*) - ct(*)]     = δv'[x1].

Using the definition ofλt, as given by (18), we can rewrite (19) as

(20) λt = δ λt+1, t = 0, 1, 2, ...  It follows directly from (20) that

(21) λ1 = ,

λ2 = = ,

...,

λt = = ,

...

Applying (16) in period t, and using u[c] = Log[c],we obtain

(22) - λt = 0, t = 0, 1, ...    Using (22) and (21), we obtain

(23) ct(*) = = , t = 0, 1, ...


Summing (23) over t = 0, 1, ..., we obtain the following stock constraint:

(24) x0 = ∑0 ct(*)      = 0

= 1 1

It follows directly from (24) that

(25) = (1 - δ ) x0.

Hence

(26) ct(*) = δt (1 - δ ) x0, t 0.


(27) v[x0] = ∑0 δ t Logδ  (1t - δ ) x0

= ∑0 δ t Logδ + Log[(1t - δ ) x0]

= ∑0 δt t Log[δ ] + ∑0 δt Log[1 - δ ] + 0 δt Log[x0]

= Log[δ ]Σt0 tδt + + .

Now let

s = Σt0 t δ t = 0 + δ + 2δ 2 + 3δ 3 + 4δ 4 + ... tδt + ...

We have

δs = δ 2 + 2δ 3 + 3δ 4 + ... + (t - 1) δ t + ...,

and

δs + δ 2 + δ 3 + δ 4 + ... + δ t + ... = 2δ 2 + 3δ 3 + 4δ 4 + ... + tδt + ... = s - δ .

Thus

δs + Σt2 δ t = s - δ ,

or

s(1 - δ ) = Σt2 δ t + δ = Σt1 δ t = ,

and

(28) s = .


Using (28) in (27), we obtain

(29) v[x0] = + + .

Computation of the Bellman function by Mathematica

In[]:= c [t_ ] := δ t (1 - δ) x [0]

In[]:= f [t_ ] := Log[c [t ]] // PowerExpand

对数 幂展开

In[]:= f [t ]

Out[]= Log[1 - δ] + t Log[δ] + Log[x [0]]

In[]:= v = f [t ] δ t

t =0

Out[]=

In[]:= v = Collect[v, Log[x [0]]]

并同类项 对数

Log[1 - δ] - δ Log[1 - δ] + δ Log[δ]      (1 - δ) Log[x [0]]

Out[]= +

(- 1 + δ)2                                                           (- 1 + δ)2

In[]:= v = Collect[v, Log[1 - δ]]

并同类项 对数

1 δ

Out[]=

(- 1 + δ)2         (- 1 + δ)2




In[ ]:= v1

Out[ ]= - Log[1 - δ]

In[ ]:= v1= v1// FullSimplify

完全简化

Out[ ]=

In[ ]:= v2

Out[ ]=


In[ ]:= v3

Out[ ]=

In[ ]:= v3= v3// FullSimplify

完全简化

Out[ ]=

In[ ]:= v = v1+ v2+ v3

Out[ ]= + +

In[ ]:= ClearAll[c, f, v ]

清除全部

2. In this exercise, we analyze a more complex q model. Suppose that a firm faces a market interest rate r and has the following production function in year t: Yt= At F[Kt], where At  is a productivity parameter; Kt is its capital stock; and Yt is the output -- all in year t, t = 0, 1,... Here we treat the labor input as fixed because we want to concentrate on the investment decisions of the firm. The objec- tive of the firm is to maximize the present value of profits. The exercise deals with the real-life problem that capital cannot be installed, or dismantled and moved into a different line of work, without incurring frictional costs. And these costs will be typically higher the more dramatic is the capital-stock change: management becomes spread more thinly, and there is greater disruption to current production, and so forth.


Suppose that if It is the investment or disinvestment made in year t, then the adjustment cost is χ , where χ > 0 is a constant. The profit in year t is thus given by At F[Kt] - It - χ , and the capital stock in the following year is Kt+1 = Kt+ It . The problem that the firm must solve is to find an investment program (It)0 to maximize the following discounted profit:


(1)   ∑0 At F[Kt] - It -


subject to the capital accumulation constraint


(2)   Kt+1 = Kt+ It, K0 is given.

For each t = 0, 1,..., let


(3)   vt[Kt] = max(Is)t t s-t As F[Ks] - Is - χ

subject to


(4)   Ks+1 = Ks+ Is, Kt is given.


As defined, v[Kt] gives the optimal discounted profits -- discounted back to year t -- given that the firm begins in year t with Kt as its capital stock. The function vt[Kt] is known as the optimal-value function or the Bellman function of the firm.

Let (It(*))0 be the solution of the problem of maximizing (1) subject to (2), and Kt(*) be the capital stock

in year t under the optimal investment program, i.e.,

(5)   Kt(*)+1 = Kt* + It(*), Kt  = K*0 for t = 0.


(a) Let qt= vt[Kt*] be the shadow price of capital (or the Tobin's q) in period t along the optimal trajectory. Find the difference equations that link (qt+1, Kt(*)+1) to (qt* , Kt* ). Assume that the productiv- ity parameter is constant over time, say At = A = constant . Draw a phase diagram with K on the horizontal axis and q on the vertical axis, then explain how the system converges to the steady state.

(b) Suppose that the system is in steady state. Suddenly, there is an unanticipated permanent increase in A. Show the new steady state and the transition to the new steady state.

To solve the above problem constituted by (3) and (4), we use the method of dynamic programming. Let It be an arbitrary level ofinvestment made in period t. Under this action, the profit made in year t is At F[Kt] - It - χ . The capital stock at the beginning of period t + 1 is then Kt+1 = Kt + It, and if the firm continues optimally from period t + 1 on after that, then the discounted value of the stream of profits  -  discounted  to  period  t + 1  -  under  the  optimal  investment  program  is  given  by vt+1[Kt+1] = vt+1[Kt + It]. Thus, if the firm makes an arbitrary investment It in period t, and then contin- ues optimally from time t + 1 until kingdom comes, then the discounted profit - discounted to period t - under such a strategy is given by

(5)   At F[Kt] - It - χ + vt+1[Kt + It].

The optimal investment program starting from year t is thus obtained by choosing It to maximize (5), and the discounted profit - discouted to period t - is equal to vt[Kt]. We have just established  the following equation known as the Bellman equation for period t:

(6)   vt[Kt] = maxIt At F[Kt] - It - χ + vt+1[Kt+ It] .




time


- It -   χ



v +1[K +1] = v +1[K +I ]



vt [Kt] = maxIt At F[Kt] - It - χ + vt+1[Kt +It]

Analysis of the model

The following first-order condition characterizes the solution of the maximization problem on the right-hand side of the Bellman equation (6).

-1 - χ + vt+1[Kt + It] = 0,

from which we obtain

(7)   It = Kt.

Applying the envelope theorem to (6), we obtain

(8)   vt[Kt] = At F'[Kt] + χ + vt+1[Kt + It].

Now let qt= vt[Kt], qt+1 = vt+1[Kt+1], ... We can then rewrite (7) and (8), respectively, as follows:

(9)   It= Kt+1 - Kt = Kt,

(10) qt= At F'[Kt] + χ + qt+1.

For s ≥ t, using (9), we obtain the following difference equation which governs the dynamics of capital accumulation:

(11) Ks+1 - Ks = t.

Also, the dynamics of the current shadow price ofcapital, namely (10), can be rewritten as follows:

(12) qs= As F'[Ks] + χ + qs+1, s t.

In economics the symbol I(uppercase) is used to deoite investment. However, in Mathematica I repre- sents the imaginary number whose square is equal to -1. Thus, when we tell Mathematica to com- pute, we cannot use I to represent investment; a different symbol must be used to represent invest- ment in the Mathematica computations.  Thus, in the Mathematica computations we shall use i(lower - case) to represent investment.




In[]:= I

虚数单位

Out[]=

In[]:= I2

Out[]= - 1

In[]:= i

Out[]= i

In[]:= i2

Out[]= i2