Please answer all questions. Each short question is worth 10% of the grade and each long question is worth 30%. Good luck.

## Short Questions

Suppose that

\(Y(t) = \text{exp}(g_At)F(\text{exp}(g_Kt)K(t), \text{exp}(g_Lt)L(t))\),

where \(F\) exhibits constant returns to scale. Suppose that \(\dot{L}(t)/L(t) = n\) and \(\dot{K}(t) = sY(t)\). Suppose also that \(F\) is not Cobb-Douglas (more specifically, suppose the share of labor is not constant as the effective capital-labor ratio \(\text{exp}(g_K(t))K(t) / \text{exp}(g_L(t))L(t)\) changes). Show that balanced growth, where output grows at a constant rate, is only possible if \(g_K = g_A = 0\).

- Consider the following overlapping generations model with competitive markets. There are \(N\) generations, each of which lives for two periods. Agents from generation \(i\) supply labor at time \(t = i\) and live off capital income at time \(t = i + 1\). The last generation \(N\) simply receives an exogenous rate of return \(\bar{R}\) on their savings at time \(t = N +1\). Is the competitive equilibrium of this economy Pareto optimal? Now consider the same economy with \(N = \infty\). Is the competitive equilibrium still Pareto optimal? Provide an economic intuition (no need for math) for your answer.
- Answer both parts:
- The fact that changes in the policies and institutions of countries has no effect on their long-run growth rate is a challenge to endogenous growth models. True or false?
- Endogenous technological change models imply that product market competition is welfare-reducing because, by reducing monopoly rents, it discourages technological change and economic growth. True or false?

- Consider an economy with two types of labor, \(L\) and \(H\). Whether an increase in the supply of \(H\) induces a change in technology in a direction that is further (relatively) biased towards that factor depends on the elasticity of substitution between \(L\) and \(H\) (where a change in technology relatively biased towards \(H\) increases the wage of \(H\) relative to that of \(L\) at given supplies of \(L\) and \(H\)). True or false?

## Long Questions

**Problem 1.** Consider a variant of the neoclassical economy with preferences at time 0 given by

\(\int_0^{\infty}\text{exp}(-{\rho}t)\frac{c(t)^{1-\theta}-1}{1-\theta}dt\).

Population is constant at \(L\), and labor is supplied inelastically. The aggregate production function is given by

\(F(K,L) = A_KK + A_LL^{1-\alpha}K^{\alpha}\),

where \(\alpha \in (0,1)\) and \(A_K > \rho + \delta\), and capital depreciates at rate \(\delta\). Capital and labor markets are competitive.

- Derive the differential equation system that characterizes the evolution of the capital stock and consumption in equilibrium.
- Show that this economy generates sustained growth without technological change. What determines the asymptotic growth rate in this economy? [Hint: conjecture an equilibrium in which the capital stock asymptotically grows at a constant rate \(g > 0\). Simplify the differential equation system obtained in part 2 under this conjecture. Solve the simplified system and verify that there is an asymptotic equilibrium with a constant growth rate.] What additional condition do we need to impose to ensure that the equilibrium you have just characterized is meaningful?
- What happens if \(L\) grows at a constant rate.
- In what way does this type of growth fail to be a good approximation to the aggregate behavior of OECD countries.

**Problem 2.** Consider the following endogenous growth model. Population at time \(t\) is \(L(t)\) and grows at the constant rate \(n\) (i.e., \(\dot{L}(t) = nL(t)\)). All agents have preferences given by

\(\int_0^{\infty}\text{exp}(-{\rho}t)\frac{C(t)^{1-\theta}-1}{1-\theta}dt\),

where \(C\) is consumption dened over the final good of the economy. This good is produced as

\(Y(t) = \left[\int_0^{N(t)}y(\nu,t)^{\beta}d{\nu}\right]^{1/{\beta}}\),

where \(y(\nu,t)\) is the amount of intermediate good \(\nu\) used in production at time \(t\) and \(N(t)\) denotes the number of intermediate goods available at time \(t\). The production function of each intermediate is

\(y(\nu,t) = l(\nu,t)\)

where \(l(\nu,t)\) is labor allocated to this good at time \(t\). New goods are produced by allocating workers to the R&D process, with the production function

\(\dot{N}(t) = {\eta}N^{\phi}(t)L_R(t)\)

where \(\phi \leq 1\) and \(L_R(t)\) is labor allocated to R&D at time \(t\). So labor market clearing requires \(\int_0^{N(t)}l(\nu,t)d\nu + L_R(t) = L(t)\). Risk-neutral firms hire workers for R&D. A firm who discovers a new good becomes the monopoly supplier, with a perfectly and indefinitely enforced patent.

- Characterize the BGP in the case where \(\phi = 1\) and \(n = 0\). Why does the long-run growth rate depend on \(\theta\)? Why does the growth rate depend on \(L\)? Do you find this plausible?
- Now suppose that \(\phi = 1\) and \(n > 0\). What happens? Interpret.
- Now characterize the BGP when \(\phi < 1\) and \(n > 0\). Does the growth rate depend on \(L\)? Does it depend on \(\theta\)? On \(n\)? Why? Do you think that the configuration \(\phi < 1\) and \(n > 0\) is more plausible than the one with \(\phi = 1\) and \(n = 0\)?