Exercise 8.3

A Two-Period TNT Economy with Production

⬅ Return

Problem

Consider a two-period economy populated by identical households with preferences defined by the utility function

\[ \ln c_1 + \ln c_2 \]

where \(c_1\) and \(c_2\) denote consumption in periods 1 and 2, respectively. Suppose that consumption is a composite of tradable and nontradable goods given by

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

for \(t=1,2\), where \(c^T_t\) and \(c^N_t\) denote consumption of tradables and nontradables in period \(t\), respectively. Let \(p_t\) denote the relative price of nontradables in terms of tradables. Household are endowed with 2 units of labor per period, which they supply to the market inelastically. Let \(w_t\) denote the wage rate in period \(t\) expressed in terms of tradables. Households are the owners of firms, from which they receive profits each period. Assume also that households start period 1 without any assets or liabilities and can borrow or lend in the international credit market at a zero interest rate.

Tradable and nontradable goods are produced using labor as the sole input, via the technologies

\[ y^T_t = \sqrt{h^T_t} \]

and

\[ y^N_t = \sqrt{h^N_t}, \]

where \(y^i_t\) and \(h^i_t\) denote output and employment in sector \(i\) in period \(t\) for \(i=T,N\) and \(t=1,2\). Firms in both sectors behave competitively. They pay the wage \(w_t\) per unit of labor and distribute profits to households.

  1. Write down the household’s utility maximization problem.

  2. Derive the optimality conditions associated with the household’s problem.

  3. Write down the firms’ profit maximization problems and derive the associated optimality conditions.

  4. Write down the market clearing conditions in this economy.

  5. Calculate the equilibrium values of consumption of tradables, consumption of nontradables, the wage rate, the relative price of nontradables, the trade balance, and the current account in periods 1 and 2. The answer to this question are 12 numbers, but show your work. Provide intuition for the temporal behavior of the relative price of nontradables and the current account.

Answer

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