Exercise 11.1

Welfare Cost of Lack of Credibility

Problem

Consider a small open perfect-foresight economy populated by a large number of identical infinitely-lived consumers with preferences described by the utility function

\[ \sum_{t=0}^{\infty} \beta^t\ln c_t, \]

where $c_t$ denotes consumption, and $\beta\in(0,1)$ denotes the subjective discount factor. Consumption is a composite good made of traded and nontradable goods, denoted $c^T_t$ and $c^N_t$, respectively, via the aggregator function

\[ c_t = \sqrt{c^T_tc^N_t}. \]

The sequential budget constraint of the representative household is given by

\[ d^h_t = (1+r) d^h_{t-1} + (1+\tau_t)(c^T_t+ p_t c^N_t) -w_t h_t - y^T - s_t, \]

where $d^h_t$ denotes debt acquired in period $t$ and maturing in $t+1$, $h_t$ denotes hours worked, $\tau_t$ is a proportional consumption tax, $w_t$ denotes the real wage in terms of tradables, $p_t$ denotes the relative price of nontradables in terms of tradables, $y^T=1$ is an endowment of tradable goods, and $s_t$ denotes a lump-sum transfer received from the government. The parameter $r$ denotes the real interest rate and is assumed to satisfy $1+r=\beta^{-1}=1.04$. Debt is denominated in terms of tradables. Households are subject to the no-Ponzi-game constraint

\[ \lim_{j\rightarrow\infty}\frac{d^h_{t+j}}{(1+r)^j}\le 0. \]

Assume that the household’s initial debt position is nil ($d^h_{-1}=0$). Households supply inelastically 1 unit of labor to the market each period. Suppose that the law of one price holds for tradables, so that $P^T_t=P^{T*}_t\mathcal{E}_t$, where $P^T_t$ denotes the domestic-currency price of tradables; $\mathcal{E}_t$ denotes the nominal exchange rate, defined as the price of foreign currency in terms of domestic currency; and $P^{T*}_t$ denotes the foreign-currency price of tradables. Assume that $P^{T*}_t$ is constant and equal to unity for all $t$. Let $W_t$ and $P^N_t$ denote the nominal wage rate and the nominal price of nontradables, respectively. Then the real wage and the relative price of nontradables are defined as $w_t = W_t/P^T_t$ and $p_t = P^N_t/P^T_t$.

Firms in the nontraded sector produce goods by means of the linear technology $y^N_t=h_t$, where $y^N_t$ denotes output of nontradables. Firms are price takers in product and labor markets, and there is free entry, so that all firms make zero profits at all times.

The government starts period 0 with no debt or assets outstanding and runs a balanced budget period by period, that is, $s_t = \tau_t( c^T_t+p_tc^N_t)$. The monetary authority pegs the exchange rate at unity, so that $\mathcal{E}_t=1$ for all $t$.

Suppose that nominal wages are flexible and that before period 0 the economy was in a steady state with constant consumption of tradables and nontradables and no external debt.

Compute the equilibrium paths of $c^T_t$, $w_t$, $W_t$, $p_t$, the trade balance, and the current account under two alternative tax policies:

\[ \mbox{Policy 1: }\tau_t = 0, \forall t \]

and

\[ \mbox{Policy 2: } \tau_t = \left\{ \begin{array}{ll} 0&0\le t\le 11\\ 0.3&t\ge 12 \end{array}. \right. \]
Compute the welfare cost of policy 2 relative to policy 1, defined as the percentage increase in the consumption stream of a consumer living under policy 2 required to make him as well off as living under policy 1. Formally, the welfare cost of policy 2 relative to policy 1 is given by $\lambda\times 100$, where $\lambda$ is implicitly given by

\[ \sum_{t=0}^{\infty}\beta^t \ln[ c^{p2}_t(1+\lambda)] = \sum_{t=0}^{\infty} \beta^t \ln c^{p1}_t, \]

where $c^{p1}_t$ and $c^{p2}_t$ denote consumption in period $t$ under policies 1 and 2, respectively.

Answer

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--- title: "Exercise 11.1" subtitle: "Welfare Cost of Lack of Credibility" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a small open perfect-foresight economy populated by a large number of identical infinitely-lived consumers with preferences described by the utility function $$ \sum_{t=0}^{\infty} \beta^t\ln c_t, $$ where $c_t$ denotes consumption, and $\beta\in(0,1)$ denotes the subjective discount factor. Consumption is a composite good made of traded and nontradable goods, denoted $c^T_t$ and $c^N_t$, respectively, via the aggregator function $$ c_t = \sqrt{c^T_tc^N_t}. $$ The sequential budget constraint of the representative household is given by $$ d^h_t = (1+r) d^h_{t-1} + (1+\tau_t)(c^T_t+ p_t c^N_t) -w_t h_t - y^T - s_t, $$ where $d^h_t$ denotes debt acquired in period $t$ and maturing in $t+1$, $h_t$ denotes hours worked, $\tau_t$ is a proportional consumption tax, $w_t$ denotes the real wage in terms of tradables, $p_t$ denotes the relative price of nontradables in terms of tradables, $y^T=1$ is an endowment of tradable goods, and $s_t$ denotes a lump-sum transfer received from the government. The parameter $r$ denotes the real interest rate and is assumed to satisfy $1+r=\beta^{-1}=1.04$. Debt is denominated in terms of tradables. Households are subject to the no-Ponzi-game constraint $$ \lim_{j\rightarrow\infty}\frac{d^h_{t+j}}{(1+r)^j}\le 0. $$ Assume that the household's initial debt position is nil ($d^h_{-1}=0$). Households supply inelastically 1 unit of labor to the market each period. Suppose that the law of one price holds for tradables, so that $P^T_t=P^{T*}_t\mathcal{E}_t$, where $P^T_t$ denotes the domestic-currency price of tradables; $\mathcal{E}_t$ denotes the nominal exchange rate, defined as the price of foreign currency in terms of domestic currency; and $P^{T*}_t$ denotes the foreign-currency price of tradables. Assume that $P^{T*}_t$ is constant and equal to unity for all $t$. Let $W_t$ and $P^N_t$ denote the nominal wage rate and the nominal price of nontradables, respectively. Then the real wage and the relative price of nontradables are defined as $w_t = W_t/P^T_t$ and $p_t = P^N_t/P^T_t$. Firms in the nontraded sector produce goods by means of the linear technology $y^N_t=h_t$, where $y^N_t$ denotes output of nontradables. Firms are price takers in product and labor markets, and there is free entry, so that all firms make zero profits at all times. The government starts period 0 with no debt or assets outstanding and runs a balanced budget period by period, that is, $s_t = \tau_t( c^T_t+p_tc^N_t)$. The monetary authority pegs the exchange rate at unity, so that $\mathcal{E}_t=1$ for all $t$. Suppose that nominal wages are flexible and that before period 0 the economy was in a steady state with constant consumption of tradables and nontradables and no external debt. 1. Compute the equilibrium paths of $c^T_t$, $w_t$, $W_t$, $p_t$, the trade balance, and the current account under two alternative tax policies: $$ \mbox{Policy 1: }\tau_t = 0, \forall t $$ and $$ \mbox{Policy 2: } \tau_t = \left\{ \begin{array}{ll} 0&0\le t\le 11\\ 0.3&t\ge 12 \end{array}. \right. $$ 2. Compute the welfare cost of policy 2 relative to policy 1, defined as the percentage increase in the consumption stream of a consumer living under policy 2 required to make him as well off as living under policy 1. Formally, the welfare cost of policy 2 relative to policy 1 is given by $\lambda\times 100$, where $\lambda$ is implicitly given by $$ \sum_{t=0}^{\infty}\beta^t \ln[ c^{p2}_t(1+\lambda)] = \sum_{t=0}^{\infty} \beta^t \ln c^{p1}_t, $$ where $c^{p1}_t$ and $c^{p2}_t$ denote consumption in period $t$ under policies 1 and 2, respectively. ## Answer _No solution has been posted yet. If you have a solution, we invite you to share it in the comment box at the bottom of this page._