Question 1. Impact of government debt in a RA framework

\(\Sigma^\infty_{t=0}\beta^tu(c_t)\)

(a) Household budget constraint in t

\(c_t + s_t = y_t + (1+r_t)s_{t-1} - \tau_t\)

\(c_t= y_t + (1+r_t)s_{t-1} - \tau_t - s_t\), \(s_t = k_t + b_t\)

(b) Government budget constraint in t

\(g_t + (1+r)b_{1-t} = \tau_t + b_t\)

(c) in period s, there is a tax cut of 100 goods \(\tau^N_s = \tau_s - 100\). With no change in \(g_s\) how does \(b_s\) change?

\(\tau^N_s = \tau_s - 100\)

\(\therefore b^N_s = b_s + 100\)

\(s^N_t = k_t + b^N_s\)

(d) Government will retire the newly issued debt in some period l (l>s). How much should \(\tau_l\) increase?

Tax cut means \(g_s + (1+r)b^N_s\)

\(g_s + (1+r)b^N_s = (1+r)(b^N_s-b_s)=100(1+r)\)

\(\tau_l\) should increase by \(100(1+r)^t\),\(100(1+r)^{l-s}\)

(e) Household’s lifetime budget constraint

\(c_1 = y_1 - \tau - s_1 + (1+r)s_0\)

\(c_2 = y_2 - \tau - s_2 + (1+r)s_1\)

\(c_3 = y_3 - \tau - s_3 + (1+r)s_2\)

\(c_4 = y_4 - \tau - s_4 + (1+r)s_3\)

\(s_3 = \frac{c_4 - y_4 +\tau + s_4}{1+r}\)

\(s_2 = \frac{c_3 - y_3 +\tau + s_3}{1+r} - \frac{[c_4 - y_4 +\tau +s_4]}{(1+r)^2}\)

\(s_1 = \frac{c_2 - y_2 +\tau + s_2}{1+r} - \frac{[c_3 - y_3 +\tau +s_3]}{(1+r)^2} + \frac{[c_4 - y_4 +\tau +s_4]}{(1+r)^3}\)

\(\therefore \sum^\infty_{t=0} \frac{c_t}{(1+r)^t} = \sum^\infty_{t=0}\Bigg(\frac{y_t}{(1+r)^t} - \frac{\tau}{(1+r)^t}\Bigg)\)

(f) How the consumption, savings, capital of holdings of the household would change following the tax cut in s?

In period s there is a tax cut of 100 goods, \(\tau^N_s = \tau_s - 100\)

The household is infinitely lived and aware that the tax cut must be repaid as there is no change in \(g_s\).

Assuming the household is perfectly rational they will not change their consumption and instead save the tax cut in preparation of the increase in taxes.

Question 2. Ramsey model with productive government

\(\Sigma^\infty_{t=0}\beta^tlog(c_t)\)

\(y_t = Ak^\alpha_tG^{1-\alpha}_t\)

\(\Delta k_{t+1} = i_t - \delta k_t\)

\(\Delta G_{t+1} = g_t - \delta G_t\)

(a) Resource constraint

\(C_t = Ak^\alpha _tG_t^{1-\alpha}-k_{t+1}+(1-\delta)kt-G_{t+1}+(1-\delta)G_t\)

(b) Choice variables for social planner’s problem. First Order Conditions. Equations of steady state?

\(max\Sigma^\infty_{t=0}\beta^tlog(c_t)\) with respect to \(k_{t+1}G_{t+1}\)

\(U = \Sigma^\infty_{t=0}\beta^t[log(Ak^\alpha _tG_t^{1-\alpha}-k_{t+1}+(1-\delta)k_t-G_{t+1}+(1-\delta)G_t)] + \beta^{t+1}[log(Ak^\alpha _{t+1}G_{t+1}^{1-\alpha}-k_{t+2}+(1-\delta)k_{t+1}-G_{t+2}+(1-\delta)G_{t+1})]\)

\(\frac{\partial U}{\partial G_{t+1}} = \beta^t\frac{1}{c_t}(-1)+\beta^{t+1}[(\frac{1}{c_{t+1}})(1-\alpha)G^{-\alpha}_{t+1}Ak^\alpha_{t+1}+(1-\delta)]\)

\(= \beta^t[\frac{1}{c_t}(-1)+\beta(\frac{1}{c_{t+1}})(1-\alpha)G^{-\alpha}_{t+1}Ak^\alpha_{t+1}+(1-\delta)]\)

\(\beta^t[\frac{1}{c_t}(-1)] = -\beta^t[\beta(\frac{1}{c_{t+1}})(1-\alpha)Ak^{\alpha}_{t+1}G^{-\alpha}_{t+1}-(1+\delta)]\)

\(-\beta^t[\frac{1}{c_t}] = -\beta^t[\beta(\frac{1}{c_{t+1}})(1-\alpha)Ak^{\alpha}_{t+1}G^{-\alpha}_{t+1}+(1-\delta)]\)

\(\frac{1}{c_t} = \beta(\frac{1}{c_{t+1}})(1-\alpha)Ak^{\alpha}_{t+1}G^{-\alpha}_{t+1}+(1-\delta)\)

\(\frac{c_{t+1}}{c_t} = \beta(1-\alpha)G^{-\alpha}_{t+1}Ak^\alpha_{t+1}+(1-\delta)\)

\(\frac{\partial U}{\partial k_{t+1}} = \beta^t\frac{1}{c_t}(-1)+\beta^{t+1}[(\frac{1}{c_{t+1}})\alpha Ak^{\alpha-1}_{t+1}G^{1-\alpha}_{t+1}+(1-\delta)]\)

\(\beta^t[\frac{1}{c_t}(-1)+\beta(\frac{1}{c_{t+1}})\alpha Ak^{\alpha - 1}_{t+1}G^{1-\alpha}_{t+1}+(1-\delta)]\)

\(\beta^t[\frac{1}{c_t}(-1)] = -\beta^t[\beta(\frac{1}{c_{t+1}}\alpha Ak^{\alpha -1}_{t+1}G^{1-\alpha}_{t+1}+(1-\delta))]\)

\(-\beta^t[\frac{1}{c_t}] = -\beta^t[\beta(\frac{1}{c_{t+1}}\alpha Ak^{\alpha -1}_{t+1}G^{1-\alpha}_{t+1}+(1-\delta))]\)

\(\frac{1}{c_t} = \beta(\frac{1}{c_{t+1}})\alpha Ak^{\alpha -1}_{t+1}G^{1-\alpha}_{t+1}+(1-\delta)\)

\(\frac{c_{t+1}}{c_t} = \beta \alpha Ak^{\alpha - 1}_{t+1}G^{1-\alpha}_{t+1}+(1-\delta)\)

(c) Two types of capital return \(r_{t+1}\) where \(r_{t+1} = MP_k-\delta\) and \(r_{t+1} = MP_g-\delta\). Morever the steady state level of consumption grows at \(\gamma\). What is \(\gamma\)?

Question 3. Ramsey model with endogenous labour

\(F(K_t,H_t) = K^\alpha _tH^{1-\alpha}_t\),\(\delta = 1\)

\(\sum_{t=0}^{\infty}\beta^t(log(c_t)+\gamma log(l_t))\)

\(l_t + h_t = 1\), \(h_t = 1 - l_t\), \(l_t = 1 - h_t\)

(a) Representative household’s budget constraint in a given period t, formulate utility maximisation problem.

\(c_t = w_th_t + r_ts_{t-1}-s_t\)

\(max\sum_{t=0}^{\infty}\beta^t(log(c_t)+\gamma log(l_t))\), with respect to \(s_t\),\(h_t\)

\(U = \sum_{t=0}^{\infty}\beta^t[log(w_th_t + r_ts_{t-1}-s_t)+\gamma log(1-ht)] +\beta^{t+1}[log(w_{t+1}h_{t+1}+r_{t+1}s_t-s_{t+1}) +\gamma log(1-h_{h+1})]\)

\(\frac{\partial U}{\partial S_t} = \beta^t\frac{1}{c_t}(-1)+\beta^{1+t}\frac{1}{c_{t+1}}(r_{t+1}) = 0\)

\(\beta^t[\frac{1}{c_t}(-1)+\beta\frac{1}{c_{t+1}}(r_{t+1})] = 0\)

\(-\beta^t\frac{1}{c_t} = -\beta^t[\beta\frac{1}{c_{t+1}}(r_{t+1})]\)

\(\frac{c_{t+1}}{\beta c_t} = r_{t+1}\)

\(\frac{\partial U}{\partial h_t} = \beta^t\frac{1}{c_t}(wt) + \beta^{t+1}\frac{1}{l_t}\gamma(-1) = 0\)

\(\beta^t\gamma\frac{1}{l_t} = \beta^t\frac{1}{c_t}(w_t)\)

\(\frac{\gamma c_t}{l_t}=w_t\)

(b) Firm’s profit maximisation problem and market clearing conditions for capital and labour markets.

\(\pi_{max} = K^\alpha_tH^{1-\alpha}_t - r_tK_t - w_tH_t\)

\(\frac{\partial\pi}{\partial K_t} = \alpha\big(\frac{H_t}{K_t}\big)^{1-\alpha} - r_t\)

\(r_t = \alpha\big(\frac{H_t}{K_t}\big)^{1-\alpha}\)

\(\frac{\partial\pi}{\partial H_t} = (1-\alpha)(\frac{K_t}{H_t})^{-\alpha} - w_t\)

\(w_t = (1-\alpha)(\frac{K_t}{H_t})^{-\alpha}\)