A Primer in Dynamic General Equilibrium Analysis
Morten I. Lau a, b, Andreas Pahlke c & Thomas F. Rutherford a, d
a The MobiDK Project, Danish Ministry of Business and Industry, Copenhagen, Denmark
b Institute of Economics, University of Copenhagen, Denmark
c Institut für Energiewirtschaft und Rationelle Energieanwendung (IER), University of Stuttgart, Germany
d Department of Economics, University of Colorado (Boulder, CO)
We are grateful to Glenn W. Harrison, Tobias N. Rasmussen and Niels K. Frederiksen for helpful comments. Contact the authors at email@example.com, firstname.lastname@example.org or email@example.com.
The appeal of dynamic general equilibrium models for policy simulation is apparent, and yet there exist virtually no systematic introduction to these techniques. In this paper we describe the formulation and solution of dynamic general equilibrium models in nonlinear programming (NLP) and mixed complementarity (MCP) formats. Our objective is pedagogic -- we present the essential equations for a few models in a compact and accessible format, along with computer programs which concretely illustrate the models. This approach is of interest to applied economists due to the availability of "off the shelf" software for processing these models (see Rutherford, [1995, 1998a]).
A natural starting point is the classical Ramsey analysis of optimal economic growth under certainty. The model represents a closed economy with perfect competition in all markets, homogenous agents, and a constant rate of technological progress. See Blanchard and Fischer  and Barro and Sala-i-Martin  for general discussions of the Ramsey model. In section 2 we formulate the infinite-horizon Ramsey model as a primal nonlinear program in quantities, two different mixed complementarity problems (MCPs), and a dual nonlinear program in prices. In the primal formulation preferences and technology are represented by utility and production functions, whereas in the dual model they are represented by expenditure and cost functions. The MCP formulations can be interpreted as first-order necessary conditions for the programming models. The complementarity problem associated with the dual nonlinear program is essentially Mathiesen's  formulation of the Arrow-Debreu equilibrium model.
In section 3 we present two methods of approximation for the infinite horizon models. One approximation is appropriate for either of the NLP or MCP formulations, and the other applies only for the MCP formulations. We illustrate the alternative model formulations with a simple Ramsey model based on specific functional forms, and we compare computational precision of alternative termination methods.
Section 4 illustrates how the dynamic equilibrium framework can be extended to address a practical economic problem. Here we consider the implementation of environmental taxes to hasten the transition from conventional to non-conventional energy sources. The model developed in this section is based on stylized data for Denmark.(1)
A familiar representation of the Ramsey model of savings and investment begins with an infinitely-lived representative agent. The closed economy consists of a household with an exogenous labor supply over time. One good is produced in each period using inputs of labor and capital, and output in each period can be either consumed or invested. There is perfect competition in all markets and no taxes. Individuals are assumed to have an infinite horizon, and expectations by private agents are forward-looking and rational. Hence, all agents have perfect foresight because there is no uncertainty. These assumptions imply that the allocation of resources by a central planner who maximizes the utility of the representative agent is identical to the allocation of resources in an undistorted decentralized economy.
We present four alternative formulations of the Ramsey model, all of which produce an identical optimal allocation of resources given common assumptions regarding technology, preferences and initial endowments. Each formulation offers a different perspective into the workings of the Ramsey model. We begin with the most familiar format (primal NLP), and we proceed to two less familiar but convenient complementarity formats, and we conclude with a relatively uncommon dual NLP formulation. We feel that by laying out a set of mathematically-equivalent specifications, the researcher can develop basic insights into the nature of the equilibrium which can be crucial when the time comes to interpret policy results from more complex models.
2.1 Primal Non-Linear Program (NLP) Formulation
The primal NLP formulation is based on an explicit representation of the utility function for the single representative household. The social planner maximizes the present value of lifetime utility for the representative household
The representative agent maximizes utility subject to the constraint that output in period t is either consumed or invested:
where is the annual rate of depreciation, and the initial capital stock in period t=0 is specified exogenously.
2.2 A Complementarity Formulation based on Karush-Kuhn-Tucker Conditions
It is a simple matter to pose a nonlinear program as a complementarity problem: just form the Lagrangian and differentiate. Introducing multipliers for aggregate output and capital stock, the above model produces the following system of first order conditions:
where pt is period t output price and is period t price of capital. As written, we have taken explicit account of the non-negativity constraint for investment and assumed that all other variables will be nonzero. Hence, we do not specify a set of complementarity relations for the output price, the level of output or the capital stock.
2.3 Complementarity Formulation for Constant Returns Models
In order to exploit the complementarity format for economic equilibrium proposed by Mathiesen , we may expand the class of markets represented in the model in order to treat all production activities as constant returns to scale in model inputs. This is possible through the introduction of an additional primary factor, labor. We may then define the instantaneous unit cost function:
where is the rental rate of capital and w is the real wage rate. For example, if we assume that total factor productivity grows at a constant rate, , we have:
This formulation further relies on the existence of closed-form demand functions which express consumption demands as a function of market prices and income,M. We then define:
For example, if we have logarithmic instantaneous utility, we obtain:
or, if utility is isoelastic, , then we have:
Having defined uncompensated demand functions, we now may characterize the equilibrium conditions in terms of three classes of equations: (i) zero profit conditions for all constant returns activities, (ii) market clearance conditions for all goods and factors, and (iii) income balance equations relating factor income to expenditure. The zero profit conditions for macro production, capital accumulation and investment are:
The market clearance conditions for output, labor, capital stock and capital services in each period are:
An income-balance constraint relates the value of expenditure to factor earnings:
Due to homogeneity of cost and demand functions, the solution is not uniquely determined and the model determines only relative prices. A practical normalization is to fix M=1 and omit the income constraint.(3)
We assume non-negativity conditions for investment, . Because of Walras' law and non-negativity of prices, complementary slackness conditions, , arise as a feature of the definition of an equilibrium instead of an equilibrium condition per se.
2.4 Dual Non-Linear Program (NLP) Formulation
In order to represent a given model in dual form, it is necessary that the underlying preferences over consumption from period 0 to the infinite horizon be linearly homogeneous. This restriction allows us to express indirect utility as the ratio of a function of market prices to the present value of income. Define the expenditure function:
It follows that the following nonlinear program has first order conditions which are equivalent to Mathiesen's complementarity formulation (for details, see Rutherford [1998b]):
By Shephard's lemma:
Associating Lagrange multipliers for the three classes of constraints with Y, K and I, it can be seen that first order optimality conditions for the dual nonlinear program correspond to market clearance conditions in the complementarity model.
Numerical models can only be solved for a finite number of periods, hence some adjustments are required to produce a model which when solved over a finite horizon approximates the infinite horizon choices. Most of these methods attempt to impose steady-state choices in the model's terminal period. In this section we present two alternative methods of making this approximation. Both procedures exploit properties of the steady-state growth path in which all quantities grow at the same rate as aggregate labor and all prices decline over time according to the discount rate. A difference between these methods is that one may be used either with the optimization or equilibrium formulation of the model whereas the second can be employed only with the model formulated in a complementarity format.
The method due to Barr and Manne (1967)(4) involves a constraint on terminal investment and an increased utility weight on terminal period consumption. Assuming that the economy is in steady state by the terminal period T, the intertemporal utility function may be written as:
where we define:
The capital stock increases by the exogenous growth rate less depreciation in steady state. The model therefore applies the following constraint on terminal investment:
This approximation of the infinite horizon can be applied in either NLP or MCP formulations of the Ramsey model. Our second method for approximating the infinite time horizon begins from the observation that an optimal solution over the infinite horizon may be decomposed into two distinct optimization problems, one defined over the period t = 0 to t = T, and a second over periods T+1 to the infinite horizon. In a single-sector model, the two subproblems are linked through the period T+1 capital stock. The first of these problems is:
and the second infinite horizon problem is:
The challenge in the decomposition approach is to specify an equation or specific value for . It could seem natural to impose the long-run steady-state level, but in that case the model horizon should be sufficiently long to eliminate terminal effects. Instead we include the level of post-terminal capital as a variable and add a constraint on the growth rate of investment in the terminal period:
This constraint imposes balanced growth in the terminal period but does not require that the model achieves steady-state growth. The advantage of this approach is that we do not need to determine a specific target capital stock, nor a terminal period growth rate.(5) A disadvantage of the approach is that the added constraint is non-integrable so this method is most useful for dynamic general equilibrium models formulated as complementarity problems.
Figure 1 illustrates the terminal effect on investment for both terminal conditions using a single-sector Ramsey model. The model is calibrated to the data shown in Table 1, and the scenario which provides the basis for these experiments is a 20% reduction in the capital stock compared to the steady-state level. We first solve the model over a 100 year horizon and assume that these values as a close approximation to the infinite-horizon saddle point path. We then compare computational results for the two terminal approximation methods by solving the two models from 1990 for successively longer horizons. In order to produce data for Figure 1, we compute equilibria through a 15 year horizon using both models. The model labeled "NLP" is based on the method due to Barr and Manne  with a constraint on terminal investment. The model labeled "MCP" is the new procedure we propose. Deviation from the "true" saddle point path is smaller for the model based on the post-terminal capital target (MCP) than for the model with a constraint on terminal investment (NLP). Several other variables exhibit deviations from corresponding infinite horizon values, but investment is typically the most sensitive item.
Figure 2 reports on the relationship between "average" approximation error and model horizon. This is based on a weighted-sum of deviations in investment over the full model horizon excluding the terminal period. The weights used in this calculation correspond to the present discounted value of future output. One interpretation of this diagram is that by using the new termination method, an MCP model can have significantly fewer periods and still obtain a comparable precision. An MCP model with a ten year horizon produces roughly the same average precision as an NLP model with a 17 year horizon.
These results provide a practical argument for dynamic modeling in a complementarity format: terminal approximation methods are more precise, so a model does not need to include as many periods to approximate the infinite horizon saddle path.
The dynamic adjustment behavior in the Ramsey model is illustrated in this section using a specific policy application. In particular, the model in the previous section is extended to represent a small open economy in which energy is imported, and a carbon emission limit is introduced by the year 2000.
The model is calibrated to a data set with parameter values shown in Table 2. We assume initially that financial assets are perfectly mobile internationally, and that the domestic financial market is completely integrated in the world financial market. The domestic interest rate is therefore equal to the interest rate in the world market.
Two competitive production sectors are included in the model: a manufacturing sector exposed to international competition and an energy sector producing carbon-free energy. The manufacturing sector operates on the world market, and the domestic manufactured good is a perfect substitute for the foreign produced manufactured good. The technology in the manufacturing sector is represented by a nested CES function with value added and energy in the upper nest. Value added is in turn composed of capital and labor in the lower nest.
There is no initial production of energy in the domestic economy since the marginal cost of producing carbon-free energy is assumed to be twice the world market price of energy. Hence energy is initially imported. The production of carbon-free energy is represented by a backstop technology requiring inputs of knowledge and manufactured goods. The economy is endowed with a fixed number of ideas which are supplied to a research and development (R&D) sector. Using these ideas and manufactured goods the R&D sector produces blueprints (knowledge capital) which can increase the capacity in the domestic energy sector. The domestic energy sector buys blueprints from the R&D sector in order to produce carbon-free energy. We assume that the initial maximum capacity in the domestic energy sector is 10% of the domestic demand for energy. When the production of carbon-free energy takes place, domestic producers of energy are able to increase the production capacity by 10% each year due to learning by doing.
Households receive income from the supply of labor, capital and the fixed number of ideas. Utility is obtained from the consumption of manufactured goods across different time periods. Hence, the labor supply is exogenous.
4.1 Introduction of Carbon Emissions Limits
To illustrate the transitional effects from the initial steady state to a new steady state we analyze the economic impacts of introducing restrictions on carbon emissions. In particular, we assume that carbon emissions are proportional to the use of imported energy, and the carbon emission limit by year 2000 corresponds to the emission level in 1990. The emission limit is then gradually reduced by 1% in each successive year beyond year 2000. A market for emission rights is then introduced and the consumers receive the tax revenue that accrues from the introduction of marketable permits.
The policy reform is announced in 1997, which is the base year in the model. To assess the impacts of the reform, the simulated policy change is compared with a baseline simulation corresponding to the initial steady state. The figures below illustrate the percentage change for a given variable compared to the Business as Usual (BaU) scenario during the first 53 years after the reform is announced.
Figure 3 shows that the price of energy (P) increases significantly when the policy reform is implemented in 2000. The increasing price of energy stimulates investment in carbon-free energy, and the domestic production of energy eventually begins in 2006. During the following 6 years the price of energy is stabilized since domestic producers of energy buy blueprints from the R&D sector to increase the capacity in the carbon-free energy sector. The production of carbon-free energy is not profitable during the initial years of production. However, the domestic producers know that the price of energy will increase to a level above the marginal cost of producing carbon-free energy in the medium term, and they chose to increase the capacity by 2006. Eventually, domestic producers of energy have to rely solely on learning by doing to increase the capacity in the energy sector and the price of energy increases to a level above the marginal cost of producing carbon-free energy. The price of energy begins to fall in the long run as the capacity constraint becomes less binding, and the new steady state level is reached by year 2027. Hence the demand for energy (Q) decreases significantly in the short to medium term, and the demand for energy in the new steady state falls by 32.7% compared to the BaU scenario.
The investment decision is independent of the saving decision in this open economy framework with an exogenous interest rate. Moreover, the marginal cost of investment is constant in this first model. These assumptions imply that the capital stock adjusts immediately to structural changes, and the demand for capital decreases in the short to medium term. The capital stock reaches the new steady state level when the price of energy is stabilized, and the demand for capital (K) in the long run is 6.9% lower than the baseline scenario (see Figure 4).
The production of manufactured goods (Y) reflects the dynamic paths of the three factor inputs in the manufacturing sector. Hence the domestic production of manufactured goods is reduced throughout. GDP is determined by domestic value added, the value of emission rights, and the return to knowledge. GDP continues to fall after the domestic production of manufactured goods has reached its new steady state level by 2026 since the value of emission rights decreases until the emissions limit is zero.
Aggregate consumption (C) is determined by lifetime income. The private demand for manufactured goods decreases immediately to the new steady state level, because consumers are forward looking and capital markets are perfect. Since the public sector is not included in the model the stock of net foreign assets is determined by private savings and investments (J). According to Figure 5, the balance of payments (A) improves in the short to medium term due to the permanent fall in the level of aggregate consumption and the reduced level of investment. Due to the intertemporal budget constraint, the balance of payments will eventually become negative, but this happens after 2050.
This model is not well suited to describe short term effects of structural reforms given
the immediate adjustment with respect to capital and consumption. Indeed, dynamic adjustment in the Ramsey model depends crucially on assumptions with respect to capital markets and demand elasticities. To make the transition path from the initial steady state to the new steady state more realistic, we consider alternative specifications of imperfect capital markets.
4.2 Balance of Payments Constraints
What happens if there is a balance of payments constraint such that exports equal imports in each period? This constraint implies that capital flows are restricted and the domestic interest rate is therefore endogenous. This implies changes in the time path of investment, production and consumption.
Figure 6 illustrates that the real interest rate increases immediately after the policy reform is announced. It falls below the initial level when the capital stock begins to adjust to the increasing costs of production, and remains below the initial level in the medium to long term. Eventually, in the new steady state the interest rate is equal to the time preference rate. Factor demands and the domestic production of manufactured goods therefore remain unchanged in the new steady state compared to the Ramsey model with perfect capital markets.
The policy reform has almost the same effects on the price of energy and the demand for energy compared to the previous model. Investment falls immediately after the policy reform is announced because the real interest rate increases initially. It also falls because the reduced demand for energy has a negative effect on capital productivity. Figure 7 shows that the demand for capital is decreasing in the medium to long term. Hence investment is decreasing during the transition to the new steady state.
Given the relative changes in the demand for energy and capital, the domestic production of manufactured goods falls gradually during the transition towards the new steady state. Hence GDP is decreasing throughout. GDP does not change in the long run compared to the model with perfect capital markets since the real interest rate is equal to the time preference rate in steady state.
Private demand for manufactured goods increases during the first couple of years after the policy reform is announced because the real interest rate increases. This follows from the Euler equation which determines the rate by which consumption changes. Consumption increases when the (growth adjusted) real interest rate is higher than the time preference rate and vice versa. The demand for manufactured goods begins to fall three years after the policy reform is announced because the real interest rate is smaller than time preference rate in the medium to long term. According to Figure 8 the level of consumption in the long run decreases by 6.5% compared to the BaU scenario.
It is apparent from these results that the assumption with respect to capital mobility has a significant impact on the effects of structural policy reforms in this framework. Short run effects with respect to investment and consumption are less pronounced when the domestic interest rate is endogenous.
4.3 Liquidity Constraints
Following McKibbin and Sachs  we assume that some consumers are liquidity constrained. In particular, we assume that 50% of the consumers are liquidity constrained such that they are unable to borrow money from financial institutions. This implies that aggregate consumption is determined as a fixed proportion of current income and as a fixed proportion of financial and human wealth. Hence, the model has some Keynesian features. Two types of households are therefore included in the model. The two types of households have identical endowments with respect to labor, capital and marketable permits, but forward-looking households are endowed with all the ideas needed to increase the capacity in the carbon-free energy sector.
The investment decision does not change compared to the Ramsey model with perfect markets because the real interest rate is exogenous, and the domestic production of manufactured goods is therefore the same. However, private demand for manufactured goods is decreasing during the years of transition since current income is falling. Figure 9 shows the aggregate level of consumption in the model with liquidity constrained households (Hayashi) and in the model with perfect capital markets (Ramsey). Hence, the aggregate level of consumption is higher in the short run and lower in the long run compared to the Ramsey model with perfect capital markets. In other words, short run effects in this model occur only with respect to saving.
4.4 Quadratic Adjustment Costs
What happens if there are real costs of installing capital? Following Uzawa  capital installation costs depend on the rate of gross investment relative to the existing capital stock. Given the level of investment, the cost of new capital decreases when the capital stock increases and vice versa. That is, the installation cost function is given by
where Jt is net investment and reflects the speed of adjustment. This formulation implies that rapid changes in the capital stock are costly and that the speed of adjustment is reduced when installation costs increase. It follows from this specification that net investment is included in the intertemporal market clearance condition for capital and that gross investment is included in the market clearance condition for output.
Figure 10 illustrates the relative changes in investment when capital installation costs are included (Uzawa) compared to the stylized Ramsey model without installation costs (Ramsey). Dynamic adjustment with respect to capital is more smooth when installation costs are included in the model. Investment falls immediately after the policy reform is announced because domestic producers of manufactured goods know that it will become more costly to produce goods when the policy reform is implemented. Net investment is eventually 1% smaller in the long run than net investment in the Ramsey model without installation costs. Hence the domestic production of manufactured goods and GDP are virtually not affected in the long run by introducing quadratic installation costs.
Private demand for manufactured goods falls immediately to the new steady state level, which is 0.4% smaller than the level of consumption in the Ramsey model without installation costs. Hence the introduction of capital installation costs has a very small negative welfare effect.
4.5 Putty-Clay Adjustment Costs
Another approach to modeling adjustment costs for the capital stock is based on the partial putty-clay approach (Phelps ). We assume that the elasticity of substitution between a fixed proportion (90%) of old capital and other factor inputs is 0, and that the elasticity of substitution between the residual fraction (10%) of old capital and other primary factors is 1. In turn, replacement of new capital for old capital leads to a successive increase in the elasticity of substitution between primary factors. Hence adjustment costs occur because the production technology in the domestic manufacturing sector is rigid in the short run. This approach leads to quite different results with respect to investment decisions by domestic producers of manufactured goods and energy compared to the previous specifications.
In this specification the demand for energy remains relatively high after the carbon emission limit is introduced because a large fraction of the capital stock is immobile in the short run. Figure 11 illustrates that the price of energy increases in the short run to a level above the cost of producing carbon-free energy, stimulating investment in carbon-free energy. Domestic production of energy begins immediately after the reform is introduced in 2000, and domestic producers of energy buy the available stock of knowledge from the R&D sector in that year.
In the putty-clay model investment in the manufacturing sector falls significantly just before the policy reform is implemented. However, investment increases in the following year and is virtually constant thereafter since the price of energy adjusts relatively fast to its long run level. The demand for capital in the long run does not change compared to the Ramsey model because the interest rate is exogenous and because the price of energy in the long run is determined by the marginal cost of producing carbon-free energy. Hence the domestic production of manufactured goods and GDP are not affected in the long run when a gradually increasing elasticity of substitution is introduced.
Private demand for manufactured goods falls immediately to the new steady state level since financial assets are perfectly mobile. The adjustment costs are relatively high compared to the original Ramsey model, and the level of consumption is 8.2% smaller than the level of consumption in the original Ramsey model without adjustment costs.
4.6 An Armington Formulation of International Trade
A popular specification of trade flows is due to Armington  who treats domestic and foreign goods as imperfect substitutes. In our example, we assume that domestic exports and imports have no effect on terms of trade because the model represents a small open economy. Hence the export demand and import supply functions are perfectly elastic. Domestic producers face a falling demand curve in the domestic market since domestic and foreign manufactured goods are imperfect substitutes. For simplicity of interpretation, we assume that the share of domestic and imported manufactured goods is the same across all components of final and intermediate demand.
Despite the assumption that financial assets are perfectly mobile internationally, the real interest rate may change during the transition because the price of domestic manufactured goods is endogenous. Hence changes with respect to investment, production and consumption compared to the original Ramsey model with homogenous domestic and foreign goods are due to the endogenous price of domestic manufactured goods.
Figure 12 illustrates that the real interest rate increases immediately after the reform is announced. The real interest rate decreases when the reform is implemented, and remains below the initial level until prices are stabilized in the new steady state. This implies that factor demands and GDP do not change in the long run compared to the Ramsey model with perfect substitution between domestic and foreign goods.
The policy reform has almost the same impacts with respect to the price of energy and the demand for energy as in the Ramsey model. Due to the increase in the real interest rate, the demand for capital decreases immediately after the reform is announced. In turn, the demand for capital falls because the reduced demand for energy has a negative impact on the productivity of capital. Given the relative changes in the demand for energy and capital, the domestic production of manufactured goods is permanently smaller than the initial level of production. The domestic production of manufactured goods falls by less than the level of production in the Ramsey model due to the falling demand curve in the domestic market. Figure 13 illustrates that the private demand for manufactured goods falls when the policy reform is announced. Due to the Euler equation, private demand for manufactured goods increases slightly after the initial drop in the level of consumption and decreases thereafter until the new steady state level is reached. Hence the adjustment with respect to consumption is not immediate since the price of manufactured goods in the domestic market varies across periods.
Our objectives were to introduce methods for formulation and solution of dynamic computable equilibrium models using nonlinear and complementarity programming. Two methods of approximation for the infinite horizon models were presented. The approximation method that applies only for models formulated in a complementarity format is more precise than the method that may be used with models formulated in either a nonlinear or a complementarity format. These results provide a practical argument for dynamic modeling in a complementarity format, since a model does not need to include as many periods to approximate the infinite horizon saddle path.
We have also illustrated how different representations of adjustment produce different time paths with respect to economic costs when structural reforms are introduced. The classical Ramsey analysis of optimal economic growth under certainty was used to simulate the introduction of carbon emission limits in a small open economy. The Ramsey model with perfect capital markets was unable to represent short term effects of structural reforms given the immediate adjustment with respect to consumption and investment. Hence, five alternative ways of modeling dynamic adjustment were considered:
1. The introduction of a balance of payments constraint implies that the real interest rate is endogenous. This assumption with respect to capital mobility has a significant impact on the effects of structural reforms. The negative short run effects with respect to investment and consumption are less pronounced when the interest rate is endogenous, but the long-run welfare cost is greater.
2. Keynesian features can be considered by assuming that some households are liquidity constrained. The investment decision does not change compared to the Ramsey model with perfect capital market because the interest rate is exogenous. Hence short run effects in this model occur only with respect to saving. The aggregate level of consumption is higher in the short run and lower in the long run compared to the Ramsey model with perfect capital markets.
3. Dynamic adjustment is smoother with respect to investment compared to the Ramsey model when quadratic installation costs are included in the model. However, the private demand for manufactured goods falls immediately to the new steady state level since capital markets are perfect. Moreover, the introduction of quadratic installation costs has a very small impact on private welfare compared to the Ramsey model.
4. The introduction of putty-clay adjustment costs implies that a large fraction of capital is immobile in the short run. The price of energy increases significantly after the carbon emission limit is implemented since the demand for energy remains high in the short run. The price of energy is stabilized at the new steady state level relatively fast compared to the other versions of the Ramsey model. Hence dynamic adjustment with respect to investment is fast compared to the Ramsey model. Private demand for manufactured goods falls immediately to the new steady state level when the reform is announced.
5. Finally, we assumed that domestic and foreign goods were heterogenous and that export and imports had no effect on terms of trade. Domestic producers of manufactured goods therefore face a falling demand curve in the domestic market. The negative effects on investment due to the carbon emission limit are therefore less pronounced compared to the Ramsey model with homogenous goods. However, the impact on consumption compared to the Ramsey model is small because the real interest rate does not change much during the transition to the new steady state.
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Table 1. Values of Parameters in Closed Economy Ramsey Model
|Value Share of Capital||0.36|
|Annual Time Preference Rate||0.05|
|Annual Steady State Growth Rate||0.02|
|Annual Rate of Depreciation||0.07|
Table 2. Values of Parameters in Open Economy Ramsey Model
|Annual Steady State Interest Rate r||0.05|
|Annual Rate of Depreciation||0.07|
|Annual Steady State Growth Rate||0.02|
|Initial Stock of Capital Kstock||3.0|
|Value Share of Capital||0.27|
|Value Share of Energy||0.05|
|Intertemporal Elasticity of Substitution||0.5|
|Elasticity of Substitution between Energy and Value Added||0.5|
|Elasticity of Substitution between Capital and Labor||1.0|
|Elasticity of Substitution in R&D Sector||0.0|
|Elasticity of Substitution in Carbon-free Energy Sector||0.0|
|Marginal Cost of Producing Carbon-free Energy bscost||2.0|
|Initial Bound on Backstop Capacity bsinit||0.1|
|Annual Growth Rate wrt \maximum Backstop Capacity bsgrow||0.1|
|Number of R&D Ideas rdidea||0.005|
|Share of Liquidity Constrained Households||0.0/0.5|
|Extant Capital Share xkshr||0.0/0.9|
|Elasticity of Substitution between Domestic Goods and Imports||/2.0|
|Elasticity of Transformation between Domestic Goods and Exports||/2.0|
The equations in this document have been formatted with HTeX. All figures have been produced using GNUPLOT together with Corel Presentations (version 8) running under Windows NT.
1. Computer programs used to produce computational results in this paper These programs have been developed using the GAMS subsystems CONOPT, PATH and MPSGE.
2. That is, total factor productivity growth at rate requires Harrod-neutral labor productivity at rate:
3. Note that in a model with multiple consumers, price normalization involves fixing the income value of only one consumer.
4. See also Eckhaus and Parikh , Chakravarty  and Manne .
5. Although the present paper focuses solely on models with exogenous labor growth rates, there are clear advantages of this terminal constraint for models with endogenous growth where the post-terminal growth rate is not determined ex ante, such as in Rutherford and Tarr .