20. Chamley's Model (1986, Econometrica)

The model considers the optimal taxation of capital income in general equilibrium. It adopts the neoclassical growth model but allows the utility function to be broader than the time additively separable case. The conclusion is that in steady state, the optimal tax rates on capital should be zero.

We will be analyzing the continuous time version of this model.

20.1 Set-up

• There is only one good, which can be used for private consumption \(c(t)\), government expenditure \(g(t)\) and net capital investment \(\dot k(t)\) at any time \(t\).
• The production needs two inputs: labor \(u(t)\) and capital \(k(t)\). The production function is

$$ Y(t)=F\big(k(t),u(t)\big) $$

note that the production function is net-of-depreciation, which means that no further depreciation of capital need to be considered.

• The representative household:
  • has 1 unit of time at each point \(t\), which is divided between working \(u(t)\) and leisure \(x(t)=1-u(t)\);
  • has infinite length of life;
  • has utility as a function of consumption \(c(t)\), government expenditure \(g(t)\) and leisure \(1-u(t)\), i.e. \(U\big(c(t),\,1-u(t),\,g(t)\big)\);
    • the utility function here is assumed to be additively time separable, but later we will discuss a broader family of utility functions;
  • has utility discount rate \(\rho\).
• The Resources Constraint (RC) of the economy, or the law of motion of capital, is

$$ \dot k(t)=F\big(k(t),u(t)\big)-c(t)-g(t) \tag{20.1} $$

where \(\dot k(t)\equiv\frac{dk(t)}{dt}\).

• Government's choice variables:
  • \(\{g(t):t\ge0\}\), government expenditures, which is endogenously chosen as it enters the household's problem;
  • \(\{\tau_k(t),\tau_u(t):t\ge0\}\), flat tax rates, where \(\tau_k(t)\) is the tax rate on capital income, and \(\tau_u(t)\) is the tax rate on labor income;
    • we will impose the restriction \(\tau_k(t)\le 1\) \(\forall t\).
• Return to factors:
  • pre-tax capital income: \(r(t)=F_k(k(t),u(t))\)
  • pre-tax labor income: \(w(t)=F_u(k(t),u(t))\)
  • after-tax capital income: \(\bar r(t)=(1-\tau_k(t))F_k(k(t),u(t))\)
  • after-tax labor income: \(\bar w(t)=(1-\tau_u(t))F_u(k(t),u(t))\)
• Discount factor: the effective interest rate to household is \(\bar r(t)\), which means that when making decisions on saving, the rate of return on capital will be \(\bar r(t)\); let \(R(t)\equiv\int_0^t \bar r(s)\,ds\), then the present discounted value of 1 unit of income at \(t\) is \(e^{-R(t)}\).

20.2 Strategies for solving the model

• First, we will be looking at representative household's problem to obtain the f.o.c.
• Second, we will use the household problem's f.o.c. and household's budget constraint to obtain the implementability constraint (IC).
• Then, we will solve the government's problem with IC and RC by Hamiltonian method to obtain part of the conditions for government problem's.
• Finally, from what we derived we can see the optimal zero capital tax rate implication.

20.3 Representative household's problem

20.3.1 Utility maximization problem

$$ \max_{\{u(t),c(t),x(t)\ge0\}}\int_0^{\infty}e^{-\rho t}\,U\big(c(t),\,1-u(t),\,g(t)\big)\,dt $$

$$ \text{s.t.}\quad \int_0^{\infty}e^{-R(t)}\big[c(t)-\bar w(t)u(t)\big]\,dt\le A_0 \tag{20.2} $$

where \(A_0\) is household's initial capital wealth. Note that the capital income doesn't directly show up but it's embedded into the discounting rate \(R(t)\equiv\int_0^t\bar r(s)\,ds\).

20.3.2 First-order conditions

We can form the Lagrangian as

$$ \mathcal{L}=\int_0^{\infty}e^{-\rho t}U\big(c(t),1-u(t),g(t)\big)\,dt+\lambda\left[A_0-\int_0^{\infty}e^{-R(t)}\big[c(t)-\bar w(t)u(t)\big]\,dt\right] $$

The f.o.c. for \(c(t)\) is

$$ e^{-\rho t}U_c\big(c(t),1-u(t),g(t)\big)=\lambda e^{-R(t)} \tag{20.3} $$

The f.o.c. for \(u(t)\) is

$$ e^{-\rho t}U_x\big(c(t),1-u(t),g(t)\big)=\lambda e^{-R(t)}\bar w(t) \tag{20.4} $$

Use the f.o.c. for \(c(t)\) in (20.3) evaluated at steady state:

$$ \begin{aligned} \log\big(e^{-\rho t}U_c(c(t),1-u(t),g(t))\big)&=\log\big(\lambda e^{-R(t)}\big)\\ \Rightarrow\ -\rho t+\log\big(U_c(c(t),1-u(t),g(t))\big)&=\log(\lambda)-R(t)\\ \Rightarrow\ \log\big(U_c(c(t),1-u(t),g(t))\big)&=\log(\lambda)+\rho t-R(t)\\ \Rightarrow\ \frac{d\log\big(U_c(c(t),1-u(t),g(t))\big)}{dt}&=\rho-\bar r(t) \end{aligned} \tag{20.5} $$

When evaluating (20.5) at the steady state, we have that

$$ 0=\frac{d\log\big(U_c(c^{ss},1-u^{ss},g^{ss})\big)}{dt}=\rho-\bar r^{ss}\ \Rightarrow\ \rho=\bar r^{ss}=(1-\tau_k^{ss})F_k(k^{ss},u^{ss}) \tag{20.6} $$

20.4 Implementability constraint

By household's f.o.c. (20.3) for \(c(t)\):

$$ e^{-\rho t}U_c\big(c(t),1-u(t),g(t)\big)=\lambda e^{-R(t)}\ \Rightarrow\ e^{-R(t)}=\frac{e^{-\rho t}U_c(c(t),1-u(t),g(t))}{\lambda} \tag{20.7} $$

By household's f.o.c. (20.4) for \(u(t)\):

$$ e^{-\rho t}U_x\big(c(t),1-u(t),g(t)\big)=\lambda e^{-R(t)}\bar w(t)\ \Rightarrow\ \bar w(t)=\frac{e^{-\rho t}U_x(c(t),1-u(t),g(t))}{\lambda e^{-R(t)}} \tag{20.8} $$

For convenience, denote \(U_c(c(t),1-u(t),g(t))\) by \(U_c(t)\) and denote \(U_x(c(t),1-u(t),g(t))\) by \(U_x(t)\). Plug (20.7) and (20.8) into the household's budget constraint (20.2):

$$ \begin{aligned} &\int_0^{\infty}e^{-R(t)}\big[c(t)-\bar w(t)u(t)\big]\,dt-A_0\le 0\\ \Rightarrow\ &\int_0^{\infty}\frac{e^{-\rho t}U_c(t)}{\lambda}\left[c(t)-\frac{e^{-\rho t}U_x(t)}{\lambda e^{-R(t)}}u(t)\right]dt-A_0\le 0\\ \Rightarrow\ &\int_0^{\infty}\frac{e^{-\rho t}U_c(t)}{\lambda}\left[c(t)-\frac{e^{-\rho t}U_x(t)}{\lambda\cdot\frac{e^{-\rho t}U_c(t)}{\lambda}}u(t)\right]dt-A_0\le 0\\ \Rightarrow\ &\int_0^{\infty}e^{-\rho t}U_c(t)\left[c(t)-\frac{U_x(t)}{U_c(t)}u(t)\right]dt-\lambda A_0\le 0\\ \Rightarrow\ &\int_0^{\infty}e^{-\rho t}\big[U_c(t)c(t)-U_x(t)u(t)\big]\,dt-\lambda A_0\le 0\\ \Rightarrow\ &\int_0^{\infty}e^{-\rho t}\big[U_c(t)c(t)-U_x(t)u(t)\big]\,dt-U_c(0)A_0\le 0 \end{aligned} $$

where the last equality holds because \(\lambda\) is the marginal utility increase (shadow value) from one margin increase in current capital, which is the definition of \(U_c(0)\). So, that last line gives us the implementability constraint:

$$ \int_0^{\infty}e^{-\rho t}\big[U_c(t)c(t)-U_x(t)u(t)\big]\,dt-U_c(0)A_0\le 0 \tag{20.9} $$

which incorporates both the f.o.c. and the budget constraint of the household.

20.5 Government's problem

Now, with the implementability constraint well-defined, the government is no longer choosing from \(\{g(t),\tau_k(t),\tau_u(t):t\ge0\}\). Instead, it is choosing from the allocations \(\{c(t),k(t),g(t):t\ge0\}\) that will pin down \(\{\tau_k(t),\tau_u(t):t\ge0\}\).

20.5.1 The Hamiltonian

As discussed in section 4.2.5, we can construct the instantaneous return function as (where \(v\) is the IC multiplier and the bracket is the IC integrand component):

$$ W(c,u,g)=U(c,1-u,g)+v\big(U_c\,c-U_x\,u\big) $$

where this function is evaluated at steady state. And the Hamiltonian of government's problem can be constructed as (the bracket \(=\dot k\), i.e. RC):

$$ H(k,c,u,g)=W(c,u,g)+\lambda\big(F(k,u)-c-g\big) $$

where \(\lambda\) is the co-state variable.

20.5.2 Part of the conditions for the solution at steady state

To reach the conclusion of optimal zero capital tax rates, we don't even need to solve the whole government problem. We only need the following part of the conditions.

• For control variable \(c\):

$$ \frac{\partial H}{\partial c}=W_c-\lambda=0 \tag{20.10} $$

• For co-state variable \(\lambda\):

$$ \dot\lambda=\rho\lambda-H_k\ \Rightarrow\ \frac{\dot\lambda}{\lambda}=\rho-\frac{H_k}{\lambda}\ \Rightarrow\ \frac{\dot\lambda}{\lambda}=\rho-F_k(k,u) \tag{20.11} $$

Consider (20.10):

$$ \begin{aligned} W_c-\lambda&=0\ \Rightarrow\ W_c=\lambda\ \Rightarrow\ \log W_c=\log\lambda\\ \Rightarrow\ \frac{d\log W_c}{dt}&=\frac{d\log\lambda}{dt}\ \Rightarrow\ 0=\frac{d\log W_c}{dt}=\frac{\dot\lambda}{\lambda} \end{aligned} \tag{20.12} $$

where the last line is true because everything is evaluated at the steady state. So, by (20.11) and (20.12) and explicitly write out that everything is evaluated at the steady state,

$$ \rho=F_k(k^{ss},u^{ss}) \tag{20.13} $$

Compare (20.6) and (20.13) and we can conclude that

$$ F_k(k^{ss},u^{ss})=(1-\tau_k^{ss})F_k(k^{ss},u^{ss})\ \Rightarrow\ \tau_k^{ss}=0 $$

20.6 Extension of utility function

If we use the discrete setting for this model, then we will be able to use the following utility function, which is known as the Koopmans-Diamond-Williamson (KDW) utility function:

$$ W(c_0,c_1,c_2,\ldots)=V\big(c_0,\,W(c_1,c_2,c_3,\ldots)\big) $$

for some function \(V(c,W)\), where \(c_0,c_1,c_2,\ldots\) is a consumption sequence. Note that the utility from future does not depend on today, and the utility from today does not depend on the past, which means that there is no habit formation. But the utility of today does depend on the future's utility, so this utility function is not additively separable. The rest of details about this discussion is in Chamley's paper (1986, Econometrica).