1    The Basic Solow Growth Model

Consider the basic Solow growth model presented in Lecture Notes 1 (Sections 1-4).  We have shown that for any initial level of the capital-to-labor ratio k0  ≥ 0, the dynamics of the economy are given by:

kt+1 = G(kt),        At = 0, 1, . . . ,

where G : R+ 一 R+is the function defined by:

G(k) = sf(k) + (1 - δ - n)k.

1.  Prove that for any kt  < k* ,

kt+1  e (kt, k* ).

You may use Lemma 1 in the lecture notes in your proof.

2.  Use the implicit function theorem to show that the steady state level of capital k* is strictly increasing in s. Provide intuition for this comparative static.

3.  Let s  e  (0, 1) and δ + n  <  0.  Prove that only one steady state of this economy exists: the trivial, zero steady state at k*  = 0.  That is, prove that a non-zero steady state in this economy does not exist.

2    Unproductive Government Spending

Consider the Solow model presented in Lecture Notes 1.  Suppose we introduce a government into the economy which taxes income every period in order to finance government spending?t. For this problem we assume that government spending is unproductive: it does not contribute to production nor capital accumulation. The resource constraint of the economy is given by:

Ct+ ?t+ It = Yt

Let gt 三 ?t/Lt denote government spending per capita. The resource constraint of the economy (in intensive form) becomes:

ct+ gt+ it = yt.

We assume that government spending is financed with a proportional income tax τ e (0, 1):

gt = τyt

which implies that after-tax disposable income is given by (1 — τ)yt. We continue to make Solow’s assumption that consumption is a constant fraction (1 — s) of after-tax disposable income.

1.  In this  economy,  state  and prove the  analog of Proposition 3 in Lecture Notes  1.   In

particular, provide the policy function for capital accumulation, G(k),

kt+1 = G(kt)

and the function γ(k) that describes the growth rate of capital,

 = γ(kt),

for this economy.

2.  Define the function ϕ : R+ 一 R+as the output-to-capital ratio:

ϕ(k) 三  > 0,        Ak > 0.

Using your answer to part (1) of this problem, prove that a non-zero steady state exists and is characterized by:

ϕ(k* ) =  .

3.  How does the steady state level of capital k* vary with τ? Provide intuition.

4.  Suppose  we  are  in  the  steady  state  described  in  part  (2)  of this  problem  for  some τ,s,δ,n and f.  Starting from this steady state, suppose there is a permanent increase in government consumption that is financed by a permanent increase in the tax rate from τ to τ′with τ′  > τ . Use a figure to describe the transitional dynamics to the new steady state.

3    Productive Government Spending

Consider the model with government spending described in the previous problem. We continue to assume that government spending is financed with a proportional tax τ ∈ (0, 1) on income. However, for this problem we now assume that government spending is productive.

yt = f(kt, gt) = kt(α)gt(β)                                                           ( 1)

where

α ∈ (0, 1),        β ∈ (0, 1),        and         α + β < 1.

Government spending can thus be interpreted as infrastructure or other productive services and β represents the elasticity of output with respect to government spending.  The resource constraint is given by

ct+ gt+ it = yt = f(kt, gt).

Finally, we continue to make Solow’s assumption that consumption is a constant fraction (1−s) of after-tax disposable income.

1.  Given that government spending is financed with a proportional income tax, show that output can be written as a function of kt  and τ alone.   In particular, show that output satisfies:

yt = kτβ(ˆ)

for some positive scalarsˆ(α) andβ(ˆ) .

2.  Isˆ(α) strictly greater than, less than, or equal to α?  Provide an economic interpretation of this result.

3.  Solve for the steady-state level of capital k* in closed form.  Note that this is possible now that we have specified a particular production function in (1).

4.  Show  that  k*   is  non-monotonic  in  the  tax  rate  τ .    Provide  economic  intuition  and contrast your answer here with the previous problem in which government spending was completely unproductive.

5.  Solve for the tax rate τ that maximizes steady state capital k* . Call this tax rate the steady- state maximizing tax, τ** . How does τ** depend on β? Provide intuition.