EC404 MRes MACROECONOMICS MID-TERM TEST 2020
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MID-TERM TEST 2020
EC404 MRes MACROECONOMICS
1. We extend our analysis of growth theory with optimal consumption by introducing the govern- ment sector. In particular, consider the Ramsey growth model. Assume a government sector spends g(t) = g for all periodst on public goods. We treat the spending requirements of the government, g(t), as exogenous. Public goods enter directly in households utility functions such that
U (0) = l0 [u(c(t)) + v(g(t))]e-(ρ -n)tdt
describes the representative household inter-temporal utility function, where u(.) and v(.) satisfy the Inada conditions. For simplicity assume that the rate of technological progress is set to zero. Population grows at rate n and individuals discount the future at rate ρ. To pay for public good provision, the government needs to raise income from taxes and must keep a per period balanced budget. Assuming that τ (t) denotes the lump sum tax the government uses to fund the public good, the individual budget constraint for this problem is then given by
c(t) + ˙(a)(t) + na(t) = w(t) + r(t)a(t) - τ (t),
where a(t) denotes the level of per-capita assets, r(t) the real interest rate on assets and w(t) the wage rate.
(a) (15 marks) Assuming that the utility over per-capita consumption is CRRA with param- eter σ and that f (k) denotes the aggregate production function in intensive form (also satisfying Inada conditions), show that the equations of motion that describe the optimal consumption per capita and capital per capita paths are given by
˙ c(t) \
k(˙)(t) = f (k(t)) - c(t) - nk(t) - g.
(b) (15 marks) Solve for the steady state of this economy and compare it with the version of the model in which the government sector was assumed away. Provide an interpretation for your results.
(c) Now assume that instead of a lump-sum the government levies a proportional tax on asset earnings, τK (t). In this case the individual budget constraint is given by
˙(a)(t) = w(t) + (1 - τK (t))r((t)a(t) + z(t) - c(t) - na(t),
where z(t) = τKr(t)a(t) denotes the government transfer which individuals take as given and a(t) denotes the average asset holdings. Since in equilibrium a(t) = a(t)
(because of the representative agent assumption), the government balances its budget.
i. (10 marks) Show that the equations of motion that describe the optimal consumption per capita and capital per capita paths are given by
˙(c)(t) = 
[(1 - τK )f\ (k(t)) - ρ],
k(˙)(t) = f (k(t)) - c(t) - nk(t).
ii. (15 marks) Solve for the steady state of this economy and compare it with the version of the model in which the government sector used lump-sum taxes. Provide an interpretation for your results.
2. Consider the following decentralised economy where the representative agent makes per capita consumption ct and per capita labour supply lt decisions,
max ct;ltE0 
βtu(ct , 1 - lt)
subject to
at+1 = rtat + wtlt - ct ,
where u(., .) is a per period utility function that satisfied the Inada conditions, β 2 [0, 1] denotes the discount factor, at the level of assets, rt the real return to asset holdings and wt the hourly wage. Assume that all markets operate under perfect competition and aggregate production is given by Yt = z tKt(α)Nt1-α , where Kt denotes the stock of capital and Nt the total labour supply. Note that this formulation implies that per capita production is given by y t = z tkt(α), where α 2 [0, 1] and z t is a random technology shock that is known at the beginning of period tand E0 is the expectation at time t = 0 based on the distribution of z. Business cycles are generated by zt.
(a) (10 marks) Using Bellman’s principle of optimality write the above optimization problem in terms of its value function, V (at, z t).
(b) (15 marks) From the derived Bellman equation and using the envelope theorem, show that the first-order condition that describe the optimal choices of consumption and leisure are given by
u1 (ct , 1 - lt) = βEt[u1 (ct+1 , 1 - lt+1)rt+1jz t]
u2 (ct , 1 - lt) = u1 (ct , 1 - lt)wt ,
where u1 (.) refers to the first derivative of u with respect to candu2 (.) refers to the first derivative of u with respect to l. Explain the economic intuition behind these equations.
(c) (20 marks) Consider the case in which ln(ct)+b
, where > 0 the elasticity of inter-temporal substitution and b > 0. Using the Euler equation for consumption derived earlier, show that the savings rate is constant.
2024-02-02