Economic Growth and Dynamic Optimization - The Comeback -

1. Economic Growth and Dynamic Optimization- The Comeback - Rui Mota – rmota@ist.utl.pt Tel. 21 841 9442. Ext. - 3442 April 2009

2. Solow Model – Assumptions Can capital accumulation explain observed growth? How does the capital accumulation behaves along time and what are the explanatory variables? Consumers: Receive income Y(t) from labour supply and ownership of firms consume a constant proportion of income

3. Solow Model – Assumptions Labour augmenting production function: Constant returns to scale Positive and diminishing returns to inputs: Inada (1964) conditions: Ensures the existence of equilibrium. Example of a neoclassical production function: Cobb-Douglas: Intensive form:

4. Solow Model – Dynamics Labour and knowledge (exogenous): Dynamics of man-made Capital Fraction of output devoted to investment Dynamics per unit of effective labor - actual investment per unit of effective labour - break-even investment.

5. Solow Model – Balanced Growth Path t How do the variables of the model behave in the steady state?

6. Solow Model – Central questions of growth theory Only changes in technological progress have growth effects on per capita variables. Convergence occurs because savings allow for net capital accumulation, but the presence of decreasing marginal returns imply that the this effect decreases with increases in the level of capital. Two possible sources of variation of Y/L: Changes in K/L; Changes in g. Variations in accumulation of capital do not explain a significant part of: Worldwide economic growth differences; Cross-country income differences. Identified source of growth is exogenous (assumed growth).

7. Dynamic Optimization: Infinite Horizon Optimal control: Pontryagin’s maximum principle Find a control vector for some class of piece-wise continuous r-vector such as to : Control variables are instruments whose value can be choosen by the decision-maker to steer the evolution of the state-variables. Most economic growth models consider a problem of the above form.

8. Pontryagin’s Maximum Principle – Usual Procedure Step 1 – Construct the present value Hamiltonian Step 2 – Maximize the Hamiltonian in w.r.t the controls Step 3 – Write the Euler equations Step 4 – Transversality condition

9. Pontryagin’s Maximum Principle – With discount Step 1 – Construct the current value Hamiltonian Step 2 – Maximize the Hamiltonian in w.r.t the controls Step 3 – Write the Euler equations Step 4 – Transversality condition

10. Dynamic Optimization: Cake-Eating Economy What is the optimal path for an economy “eating” a cake? Optimal System: Transversality condition: subject to

11. Dynamic Optimization: Cake-Eating Economy

12. Dynamic Optimization: Cake-Eating Economy Explicit Solution: From the dynamics of consumption Resource stock constraint: The remaining stock of cake is the sum of all future consumption of cake, i.e., In the planning horizon, all the cake is to be consumed, i.e, The optimal strategy is to consume a fixed portion of the cake

13. Assignments • Firm supply