Optimal Growth: Discrete Time Analysis

Abstract

We consider in this chapter the discrete time version of some of the issues discussed in the previous chapter. We introduce a government in the economy and define and characterize the competitive equilibrium. The intertemporal government budget constraint, the relationship between the competitive equilibrium allocation and that of the benevolent planner mechanism, and the Ricardian doctrine, can be all analyzed in discrete time in a similar fashion as we have done in the continuous time version of the model. Dealing with all the details of the discrete time version of the Cass–Koopmans economy is very instructive in order to be able to formulate alternative, more complex growth models, as well as to perform policy analysis, as we do toward the end of the chapter.

Appendices

4.3 Appendices

4.1.1 4.3.1 Appendix 1: A Reformulation of the Stability Condition for the Deterministic Version of the Model

Condition (4.42) is the approximate linear representation of the stable manifold for this problem, which we characterized graphically in previous sections, and it is called the stability condition of the system.¹⁰ This condition imposes orthogonality between the row of Γ⁻¹ associated to the unstable eigenvalue, and the vector of initial deviations with respect to steady-state (k ₀ − k _ss, c ₀ − c _ss). In fact, the row of Γ⁻¹ associated to the unstable eigenvalue of D, μ ₁, is: $(u_{1},v_{1})=\frac { d_{12}}{\mu _{2}-\mu _{1}}(\frac {\mu _{2}-d_{11}}{d_{12}},-1)$, where we have skipped the proportionality constant when imposing the orthogonality condition: (u ₁, v ₁) ⋅ (k ₀ − k _ss, c ₀ − c _ss)^′ = 0.

That the stability condition satisfying the transversality condition can be written this way is not casual. The linear approximation to the deterministic, dynamic system (4.36) can be written,

$$\displaystyle \begin{aligned} B_{0}\tilde{x}_{t+1}=B_{1}\tilde{x}_{t}, \end{aligned}$$

with $\tilde {x}_{t}$ the vector of deviations around steady-state. In the tax reform analysis in the previous section, $\tilde {x}_{t}=(k_{t}-k_{{\textit {ss}}}$, c _t − c _ss)^′.

Provided B ₀ is invertible, we have,

$$\displaystyle \begin{aligned} \tilde{x}_{t+1}=B_{0}^{-1}B_{1}\tilde{x}_{t}=D\tilde{x}_{t}, \end{aligned}$$

and using the spectral decomposition of D:

$$\displaystyle \begin{aligned} \tilde{x}_{t+1}=D\tilde{x}_{t}=\Gamma \Lambda \Gamma ^{-1}\tilde{x}_{t}\text{ ,} {} \end{aligned} $$

(4.45)

or,

$$\displaystyle \begin{aligned} \tilde{z}_{t+1}=\Lambda \tilde{z}_{t}\text{ ,} \end{aligned}$$

after premultiplying in (4.45) by Γ⁻¹ and defining $\tilde {z}_{t}=\Gamma ^{-1}\tilde {x}_{t}$. Each element in $\tilde {z}_{t}$ is a linear combination of deviations from steady-state for all variables in $\tilde {x}_{t}$. By repeated substitutions, taking into account the diagonal structure of Λ, we get,

$$\displaystyle \begin{aligned} \tilde{z}_{t}=\Lambda ^{t}\tilde{z}_{0}\text{ ,} \end{aligned}$$

a system of linear equations which will be stable, in the sense of satisfying the transversality conditions, only if the elements in the diagonal of Λ, i.e., the eigenvalues of D are less than $1/ \sqrt {\beta }$.

In the tax reform analysis, the system is 2 × 2, and each element in $\tilde {z} _{t}$ is a linear combination of deviations from steady-state for both variables, k _t − k _ss, c _t − c _ss. We have already shown that, in that system, the μ ₁-eigenvalue has absolute value greater than 1 so the system becomes explosive.

The only way to avoid the explosive path is by fixing $\tilde {z}_{1t}=0\ \forall t$, which amounts to setting to zero each period the inner product of the first row in Γ⁻¹ (which is the left- eigenvector associated to the explosive eigenvalue) times the vector of variables in deviations from steady state. Needless to say, had we assumed that μ ₁ was the stable eigenvalue, with μ ₂ being unstable, we would have concluded the need to set to zero the inner product of the second row of Γ⁻¹ and the vector of deviations from steady-state.

Some observations are worthwhile at this point:

• There are infinite linear trajectories passing through the optimal steady state, all having the form: c _t − c _ss = b(k _t − k _ss) for a certain range of slope values, b. If we choose c _t each period to satisfy any one of these conditions, given the stock of capital k _t chosen at the end of the previous period, the economy will converge to the optimal steady state. The solution procedure described in the previous section can be seen as selecting, among all those linear trajectories, the one approximating better the true model, at least in a neighborhood of the optimal steady-state, since it is not possible to characterize that stable manifold analytically. In fact, in the numerical exercises we present in EXCEL files, we check the amount by which the Keynes–Ramsey condition, not used in the generation of the solution, is not fulfilled by the numerical solution. The stable root takes care that the law of motion for the stock of capital takes k _t in the right direction, towards k _ss, so we obtain stable time series for the stock of capital as well as for consumption. However, these time series would satisfy just part of the model, the budget constraint, but not the Keynes–Ramsey condition.

• The discussion in this section generalizes to more general models, as we will have a chance to see in subsequent chapters. In general, we will have a vector $\tilde {x}_{t}$ of q variables in deviations with respect to steady-state, r of which will be control or decision variables, the remaining q − r being state variables. In the Cass–Koopmans model, r = 1, q = 2, with consumption as the single control or decision variable, and the stock of capital as the state variable. For the model to have a single stable solution, it is necessary to have as many stability conditions as control variables, r, so that the matrix of coefficients D in the first order vector autoregression (4.45) will need to have q − r stable eigenvalues, i.e., less than $1/\sqrt {\beta }$ in absolute value, and r unstable eigenvalues, and the rows of matrix Γ⁻¹ associated to the unstable eigenvalues will provide us with stability conditions, just like we have done in the Cass–Koopmans model.

• As we will see in the next chapter, in stochastic models, stability conditions generate a set of relationships between stochastic shocks to the model and expectation errors, which can be interpreted as approximating the way rational expectations errors depend on the innovations to the exogenous stochastic processes. Additionally, these relationships allow the researcher to generate time series for the expectations errors from the time series for the exogenous processes once the model has been solved. In turn, these time series can be used for implementing rationality tests on the expectations errors: zero mean, no serial correlation, and lack of correlation with any variable contained in the information set available to the agent when forming their expectations. If the numerical solution is a good approximation to the true solution, the expectations data should not fail these rationality tests. Unfortunately, this type of validation of the numerical solution is not very often implemented in practice.

4.1.2 4.3.2 Appendix 2: The Intertemporal Government Budget Constraint

We present in this section an intertemporal analysis of the government budget constraint, following the lines of the presentation we made on the continuous time version of the model. The reader will recognize that the qualitative results we reach and the expressions we obtain are similar to those we obtained in Sect. 3.4. Familiarization with the analytic details of this presentation is needed to discuss more general questions relating to government financing in discrete time models. For simplicity, we assume zero population growth (n = 0).

4.1.2.1 4.3.2.1 Government Budget Constraint

We can rewrite (4.11) as:

$$\displaystyle \begin{aligned} b_{t+1}=g_{t}+(1+r_{t})b_{t}-\tau _{t}, {} \end{aligned} $$

(4.46)

and for the following period:

$$\displaystyle \begin{aligned} b_{t+2}=g_{t+1}+(1+r_{t+1})b_{t+1}-\tau _{t+1}, \end{aligned}$$

and the two expressions together lead to:

$$\displaystyle \begin{aligned} b_{t+2}=g_{t+1}+(1+r_{t+1})\left[ g_{t}+(1+r_{t})b_{t}-\tau _{t}\right] -\tau _{t+1}, \end{aligned}$$

so that the present value of the level of debt outstanding at period t + 2, is:

$$\displaystyle \begin{aligned} \frac{b_{t+2}}{(1+r_{t+1})(1+r_{t})}=\frac{g_{t+1}-\tau _{t+1}}{ (1+r_{t+1})(1+r_{t})}+\frac{g_{t}-\tau _{t}}{(1+r_{t})}+b_{t}. \end{aligned}$$

Repeating the process for the stock of debt outstanding at time t + 3, we get:

$$\displaystyle \begin{aligned} \begin{array}{rcl} b_{t+3} & =&\displaystyle g_{t+2}+(1+r_{t+2})b_{t+2}-\tau _{t+2} \\ & =&\displaystyle g_{t+2}+(1+r_{t+2})\left[ g_{t+1}+(1+r_{t+1})\left[ g_{t}+(1+r_{t})b_{t}- \tau _{t}.\right] -\tau _{t+1}\right] \\ & &\displaystyle -\tau _{t+2}, \end{array} \end{aligned} $$

so that the present value of debt outstanding at time t + 3 is:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{b_{t+3}}{(1+r_{t+2})(1+r_{t+1})(1+r_{t})} & =&\displaystyle \frac{g_{t+2}-\tau _{t+2} }{(1+r_{t+2})(1+r_{t+1})(1+r_{t})} \\ & &\displaystyle +\frac{g_{t+1}-\tau _{t+1}}{(1+r_{t+1})(1+r_{t})}+\frac{g_{t}-\tau _{t}}{ (1+r_{t})}+b_{t}. \end{array} \end{aligned} $$

Repeating the process T times, we obtain that the present value of debt outstanding at time t + T + 1:

$$\displaystyle \begin{aligned} \frac{b_{t+T+1}}{\prod_{s=0}^{T}(1+r_{t+s})}=\sum_{j=0}^{T}\frac{ g_{t+j}-\tau _{t+j}}{\prod_{s=0}^{j}(1+r_{t+s})}+b_{t}. \end{aligned}$$

Taking limits on T:

$$\displaystyle \begin{aligned} \lim_{T\rightarrow \infty }\frac{b_{t+T+1}}{\prod_{s=0}^{T}(1+r_{t+s})} =\sum_{j=0}^{\infty }\frac{g_{t+j}-\tau _{t+j}}{\prod_{s=0}^{j}(1+r_{t+s})} +b_{t}=0, {} \end{aligned} $$

(4.47)

and using the fact that the transversality condition implies: $\underset { T\rightarrow \infty }{\lim }\frac {b_{t+T+1}}{\prod _{s=0}^{T}(1+r_{t+s})}=0$, we get,

$$\displaystyle \begin{aligned} b_{t}=\sum_{j=0}^{\infty }\frac{\tau _{t+j}-g_{t+j}}{{}^{\prod_{s=0}^{j}\left( 1+r_{t+s}\right) }}, \end{aligned}$$

so that, in each period, the present value of current and future government budget surplus must be equal to the stock of debt outstanding.

Likewise, it is possible to write Eq. (4.47) as,

$$\displaystyle \begin{aligned} \sum_{j=0}^{\infty }\frac{g_{t+j}}{{}^{\prod_{s=0}^{j}\left( 1+r_{t+s}\right) } }+b_{t}=\sum_{j=0}^{\infty }\frac{\tau _{t+j}}{{}^{\prod_{s=0}^{j}\left( 1+r_{t+s}\right) }}, {} \end{aligned} $$

(4.48)

showing that, each period, the present value of the stream of current and future government expenditures, added to the current stock of debt outstanding, must be equal to the present value of current and future tax revenues.

Alternatively, we could have integrated the government budget constraint towards the past. Then, the stock of public debt at time t could be written:

$$\displaystyle \begin{aligned} b_{t}=g_{t-1}+(1+r_{t-1})b_{t-1}-\tau _{t-1}, \end{aligned}$$

which, plugged into (4.46), allows us to obtain the level of debt outstanding in period t + 1,

$$\displaystyle \begin{aligned} b_{t+1}=g_{t}+(1+r_{t})\left[ g_{t-1}+(1+r_{t-1})b_{t-1}-\tau _{t-1}\right] -\tau _{t}. \end{aligned}$$

Repeating the process a number of times, we get:

$$\displaystyle \begin{aligned} b_{t+1}=\left( g_{t}-\tau _{t}\right) +\sum_{j=1}^{t}\left[ \prod_{s=0}^{j-1}\left( 1+r_{t-s}\right) \left( g_{t-j}-\tau _{t-j}\right) \right] +\prod_{s=0}^{t}\left( 1+r_{t-s}\right) b_{0}, {} \end{aligned} $$

(4.49)

showing that the stock of debt outstanding at the end of period t is the result of capitalizing: (i) the initial stock of debt, and (ii) the government budget deficit or surplus from previous periods. This latter effect can be either negative or positive each time period t, so that each period, the stock of debt can be either above or below initial debt, b ₀.

4.1.2.2 4.3.2.2 Sustainable Steady-State Expenditures and Financing Policies

Steady-state is a dynamic, competitive equilibrium, along which per capita variables remain constant over time. In particular, r _t = r, g _t = g, τ _t = τ must remain constant. We want to characterize steady-state feasible fiscal policies. The steady-state version of the present value government budget constraint, integrated towards the past (4.49), is:

$$\displaystyle \begin{aligned} \begin{array}{rcl} b_{t+1} & =&\displaystyle \left( 1+r\right) ^{t+1}b_{0}+\sum_{j=0}^{t}\left( 1+r\right) ^{j}\left( g-\tau \right) \\ & =&\displaystyle \left( 1+r\right) ^{t+1}b_{0}+\left( g-\tau \right) \left[ \frac{1-\left( 1+r\right) -\left( 1+r\right) ^{t+1}}{1-\left( 1+r\right) }\right] \\ & =&\displaystyle \left( 1+r\right) ^{t+1}b_{0}+\frac{g-\tau }{r}\left[ \left( 1+r\right) ^{t+1}-1 \right] . \end{array} \end{aligned} $$

But b _t+1 must satisfy the transversality condition (4.22):

$$\displaystyle \begin{aligned} \lim_{t\rightarrow \infty }\frac{1}{{}^{\left( 1+r\right) ^{t}}}b_{t+1}=0, \end{aligned}$$

which is equivalent to:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lim_{t\rightarrow \infty }\frac{1}{{}^{\left( 1+r\right) ^{t}}}\left[ \left( 1+r\right) ^{t+1}b_{0}+\frac{g-\tau }{r}\left[ \left( 1+r\right) ^{t+1}-1 \right] \right] & =&\displaystyle 0,\Leftrightarrow \\ \left( 1+r\right) \lim_{t\rightarrow \infty }\left[ b_{0}+\frac{g-\tau }{r} \left[ 1-\left( 1+r\right) ^{-t-1}\right] \right] & =&\displaystyle 0,\Leftrightarrow \\ b_{0}+\frac{g-\tau }{r}\left[ 1-\lim_{t\rightarrow \infty }\left( 1+r\right) ^{-t-1}\right] & =&\displaystyle 0,\Leftrightarrow \\ b_{0} & =&\displaystyle \frac{\tau -g}{r}. \end{array} \end{aligned} $$

Only if $b_{0}=\frac {\tau -g}{r}$ will the transversality constraint hold. In fact, any steady-state policy involving a period-by-period budget surplus in an amount τ − g ≥ rb ₀ will be feasible. In the absence of initial debt outstanding, the only feasible policy would be one of maintaining a balance government budget forever. With strict inequality, the transversality condition will be violated, with $\underset {t\rightarrow \infty }{\lim }\frac {1}{{ }^{\left ( 1+r\right ) ^{t}}}b_{t+1}<0$. That would be a feasible, albeit suboptimal steady-state fiscal policy, since the government could have afforded running larger deficits at some point, presumably leading to an increase in consumers’ welfare.

4.1.3 4.3.3 Appendix 3: The Ricardian Proposition Under Non-distortionary Taxes in Discrete Time

We use now the discrete time representation to show that the Ricardian proposition holds in the Cass–Koopmans economy under non-distortionary taxation. The proposition states that the way how government expenditures, g _t, are financed is irrelevant, provided that the bond issuing policy associated to each alternative financing strategy be feasible, i.e ., that the transversality condition (4.19) holds. This implies that the distribution of resources between consumption and savings implied by the competitive equilibrium mechanism is independent of the way how savings are split into government bonds and firm’s stock.

The irrelevance of government financing is known as the Ricardian doctrine: consumers are indifferent between paying higher taxes today and holding a lower stock of debt in their portfolio, or paying less taxes today, but being forced to hold more debt in their portfolios. As we have seen in previous sections, in the absence of uncertainty and with perfect capital markets, the sequence of single-period budget constraints can be integrated in a time aggregated present value budget constraint, this being the only constraint faced by the consumer. A similar consideration holds for the sequence of single-period government budget constraints. Joint consideration of both intertemporal constraints leads to the Ricardian proposition.

We now proceed to integrating the consumer budget constraint towards the future. For simplicity, we assume zero population growth (n = 0). From (4.13) we obtain:

$$\displaystyle \begin{aligned} a_{t+1}=(1+r_{t})a_{t}+\left( \omega _{t}-\tau _{t}-c_{t}\right) . {} \end{aligned} $$

(4.50)

Analogously,

$$\displaystyle \begin{aligned} a_{t+2}=(1+r_{t+1})a_{t+1}+\left( \omega _{t+1}-\tau _{t+1}-c_{t+1}\right) . {} \end{aligned} $$

(4.51)

Plugging (4.51) into (4.50), we get:

$$\displaystyle \begin{aligned} \frac{a_{t+2}}{(1+r_{t+1})(1+r_{t})}=\frac{\omega _{t+1}-\tau _{t+1}-c_{t+1} }{(1+r_{t+1})(1+r_{t})}+\frac{\omega _{t}-\tau _{t}-c_{t}}{(1+r_{t})}+a_{t}. \end{aligned}$$

After T substitutions,

$$\displaystyle \begin{aligned} \frac{a_{t+T+1}}{{}^{\prod_{s=0}^{T}\left( 1+r_{t+s}\right) }}=\sum_{j=0}^{T} \frac{\omega _{t+j}-\tau _{t+j}-c_{t+j}}{{}^{\prod_{s=0}^{j}\left( 1+r_{t+s}\right) }}+a_{t}, \end{aligned}$$

and, by continuous substitutions,

$$\displaystyle \begin{aligned} \lim_{T\rightarrow \infty }\frac{a_{t+T+1}}{{}^{\prod_{s=0}^{T}\left( 1+r_{t+s}\right) }}=\sum_{j=0}^{\infty }\frac{\omega _{t+j}-\tau _{t+j}-c_{t+j}}{{}^{\prod_{s=0}^{j}\left( 1+r_{t+s}\right) }}+a_{t}=0, \end{aligned}$$

since the transversality condition holds. From this expression, we get:

(4.52)

showing that each period, the stock of assets held by the consumer, plus the present value of his/her current and future labor income, must be equal to the present value of current and future consumption, plus the present value of taxes.

Combining the budget constraints for the government (4.48) and the consumer (4.52), both integrated towards the future, we get:

so that the present value of the stock issued by the firm, plus the present value of the sequence of current and future labor income, must be equal to the present value of public and private consumption:

showing that the feasible sequences of consumption are those whose present value remains below the sum of current holdings of firm’s stock plus the present value of labor income, net of the present value of public consumption. In this latter expression, neither bonds nor taxes appear, either at a given point in time, or in present value form. That means that consumer’s decisions are affected by the level of current and future government expenditures, but not by the way how these are financed, be that by issuing debt or through lump-sum taxes.

4.4 Exercises

Exercise 1

In the discrete time version of the Cass–Koopmans economy, show the inefficiency of the competitive equilibrium mechanism by showing that the implied allocation of resources does not coincide with the one that is obtained from the planner’s problem when there is a government that purchases g _tunits of the consumption commodity which are ‘thrown to the sea’, financing them with a lump-sum tax and debt issuing. Start by showing that the competitive equilibrium allocation can be obtained as the solution to a representative agent problem. Repeat the exercise for an economy in which the government uses a consumption tax at a rate τ ^c, and taxes all income at the same tax, τ ^y. Identify the several reasons for inefficiency of the competitive equilibrium allocation in this economy.

Exercise 2

In the discrete time version of the Cass–Koopmans economy, show the inefficiency of the competitive equilibrium mechanism by showing that the implied allocation of resources does not coincide with the one that emerges from the planner’s problem when there is a government that purchases g _t units of the consumption commodity which are returned to consumers as a lump-sum transfer, financing them with a consumption tax, a capital income tax, and issuing debt. Identify the several reasons for that inefficiency. Explain why it is not possible to characterize the competitive equilibrium allocation through a representative agent problem in this economy.

Exercise 3

In the deterministic version of the Cass–Koopmans economy, suppose that the production function has constant returns to scale and the utility function is U(c _t) = ln c _t. There is a government that implements a lump-sum transfer to consumers, financed by a consumption tax. Suppose that the government keeps constant the consumption tax rate along the transition, so that it is the level of the lump-sum transfer that gets adjusted over time.

• Set up the set of equations characterizing the resource allocation under the competitive equilibrium mechanism

• Notice that, since Walras’ law holds, it is not necessary to impose the budget constraint of the consumer and yet, the competitive equilibrium allocation satisfies that equation. Notice that the system made up by the Euler equation and the equation characterizing equilibrium in the market for the consumption commodity also characterizes the time paths for consumption and capital. Hence, the time path for both variables is independent of fiscal policy instruments. Once consumption and capital have been determined, the level of the lump-sum transfer can be solved from the government budget constraint.

• Choose a set of parameters and calculate implied steady state values for consumption, capital and output.

• Choose arbitrary initial conditions for the stock of capital and consumption and get time series for these variables using the two non-linear conditions: the global resources constraint and the Keynes–Ramsey condition. Check that the resulting time series will be explosive, because of not having imposed any stability condition.

• With the same initial conditions, solve the model using the linear approximation. Check that the solution is again unstable.

• Characterize the stability condition for the model under the chosen parameterization. Generate time series for the relevant variables using the stability condition and either one of the equations in the linear approximation. Check that we obtain convergence, no matter which of the equation in the linear approximation is used. Also, check that the other equation in the linear approximation holds.

• Solve the model using the stability condition and either the global resources constraint or the Keynes–Ramsey condition. Check that the obtained time series are stable in either case. Check that the condition not being used, is not satisfied by the set of time series we have obtained.

• Solve the model using the two non-linear conditions: the global resources constraint and the Keynes–Ramsey condition, after having imposed the stability condition at t = 0, to compute c ₀, c ₀ = ϕk ₀. Check that the solution is unstable.

• Solve the model using conditions $k_{t}-k_{{\textit {ss}}}=\left ( k_{0}-k_{{\textit {ss}}}\right ) \mu _{2}^{t}$, $c_{t}-c_{{\textit {ss}}}=\left ( c_{0}-c_{{\textit {ss}}}\right ) \mu _{2}^{t}$, after having imposed the stability condition at t = 0 to compute c ₀. Check that the implied time series are stable. Check that stability arises even without imposing stability at t = 0.

In the next two exercises, the ratio of government expenditures to output (G/Y), rather than government expenditures themselves, are kept constant.

Exercise 4

In the discrete-time version of the Cass–Koopmans economy, consider a Cobb–Douglas production function with constant returns to scale and a utility function with a constant intertemporal elasticity of substitution of consumption. The government tax consumption and income from the representative agent, using the revenues to purchase the single good produced in the economy.

• 1. Characterize analytical expressions for steady state values for consumption, the stock of capital, output and government expenditures. Show in a graph how these values depend on each of the two tax rates. Let us assume that the tax rate on consumption is initially 0.2 while the income tax rate is 0.15. Compute the welfare long-term gains or losses, in terms of consumption, from a permanent change in either one of the two tax rates.

• 2. Characterize different combinations of tax rates on consumption and income that can be used to finance in steady-state the same ratio of government expenditures/output than with τ ^c = 0.2 and τ ^y = 0.15. Compute steady-state values obtained under each of these fiscal policies for each variable in the economy. What is among them the tax policy that maximizes utility while maintaining the ratio of expenditures/output? (second-best policy)

• 3. Assume now that initial tax rates are τ ^c = 0.2and τ ^y = 0.15. Characterize the short- and long-term effects from a change in each of the two tax rates.

• 4. Assume that initial tax rates are τ ^c = 0.2 and τ ^y = 0.15. The government modifies tax policy while maintaining ratio of government expenditures/output constant in steady-state. Characterize the short- and long-term effects from an increment in each tax rate. Do ratio of government expenditures/output remain constant during the transition to the new steady-state? What type of change in tax rates should the government introduce to maximize time-aggregate discounted utility while maintaining steady-state government expenditures/output ratio constant?

Exercise 5

In the discrete-time version of the Cass–Koopmans economy, consider a Cobb–Douglas production function with constant returns to scale and a utility function with a constant intertemporal elasticity of substitution of consumption. Consider a time discount parameter of β = 0.99, a depreciation rate δ = 0.025, zero population growth, n = 0, output elasticity with respect to capital of 0.33 and an intertemporal elasticity of substitution of consumption 1/σ = 1∕3. The government tax consumption and income from the representative agent, using the revenues to purchase the single good produced in the economy.

• 1. Let us assume that the tax rate on consumption is τ ^c = 0.2 while the tax rate on income is τ ^y = 0.30. Characterize steady-state levels for private and public consumption, the stock of capital, output and utility. What is the composition of aggregate demand in this economy?

• 2. Starting from the previous situation, let us assume that the government reduces the tax rate on income to τ ^y = 0.25, while adjusting the tax rate on consumption so that the steady-state level of tax revenues/output ratio remains constant. Characterize and provide an interpretation for the short- and long-term effects of fiscal policy on private consumption, output, investment, public expenditures, the stock of capital, and the utility level. What are the computed welfare gains when (a) only long-term effects are taken into account, (b) when short-term effects are also taken into account?

• 3. So long, we have assumed that the government tax at the same rate income from labor and from renting capital. Let us now assume that both revenue sources are taxed differently, while maintaining the consumption tax in the model. Characterize the steady-state in this economy and discuss the effects of a reduction in government expenditures. (Hint: write and solve the decentralized competitive general equilibrium problem).