Time Allocation Under Autarky and Free Trade in the Presence of Time-Consuming Consumption

Abstract

This chapter examines the impact of incorporating a Gossenian−Beckerian consumption time constraint in a simple general equilibrium model with representative agents. In the closed economy case, the conventional theory is shown to remain more or less intact with the conventional transformation curve being replaced by the generalized transformation curve. In the open economy case, while trade remains welfare improving, the sources of trade gainfulness differ from those in conventional trade models. In particular, the conventionally defined exchange (consumption) and specialization (production) gains vanish. There are, however, positive gains from time reallocation (away from production toward consumption) and specialization associated with this time reallocation. The model produces results which are similar to those obtained from trade theory with an endogenous labour supply.

Appendix 6.1: Determination of Equilibrium Output and Commodity and Factor Prices

The optimal consumption bundle (${C}_1^{\ast }$, ${C}_2^{\ast }$) exists uniquely in view of Corollary 6.1. Once (${C}_1^{\ast }$, ${C}_2^{\ast }$) has been determined, we can work out the equilibrium amount of time devoted to consumption of each good ${L}_i^{C\ast }$ = a _i ${C}_i^{\ast }$ and the equilibrium total time expended on consumption L^C∗ ≡ ${L}_1^{C\ast }$ + ${L}_2^{C\ast }$. More generally, given that a country allocates L^C ≡ ${L}_1^C$ + ${L}_2^C$ units of labour time to consumption, let L _M ≡ $\overline{L}$ − L^C be the amount of labour time devoted to manufacturing. We define the production possibility set

$$ {\boldsymbol{S}}_{\boldsymbol{Q}}\equiv \left\{\left(Q1,Q2\right)\in {\mathbb{R}}_{+}^{\mathbf{2}}: Qi\le Fi\left( Li, Ki\right),\kern0.5em L1+L2\le {L}_M\ \mathrm{and}\ K1+K2\le \overline{K}\right\}. $$

The upper boundary of the set is called the transformation curve and can be represented by the function Q₂ = ψ(Q₁; L _M , $\overline{K}$). The graph of this function is a concave, downward sloping curve.

We are particularly interested in the specific curve Q₂ = ψ(Q₁; ${L}_M^{\ast }$, $\overline{K}$) where ${L}_M^{\ast }$ ≡ $\overline{L}$ − L^C∗. Given L^C∗, this curve generates the relevant relative supply curve Q₂/Q₁, as an increasing function of their relative price p ≡ P₂/P₁. Since we know ${C}_2^{\ast }$/${C}_1^{\ast }$ = ${X}_2^{\ast }$/${X}_1^{\ast }$, we can pin down the equilibrium relative supply, ${Q}_2^{\ast }$/${Q}_1^{\ast }$ = ${X}_2^{\ast }$/${X}_1^{\ast }$, under autarky. This point on the relative supply curve determines the equilibrium relative price p* ≡ ${P}_2^{\ast }$/${P}_1^{\ast }$. Without loss of generality, let ${P}_1^{\ast }$ = 1 so that p* = ${P}_2^{\ast }$.

Given the relative price p*, we can now work backward to find the equilibrium factor prices, the wage rate w* and the rental rate r*(both in terms of good one), by using the conditions that the price of each good is equal to its unit cost (see, e.g. Woodland 1982).

$$ {c}_1\left(w,r\right)=1 $$

$$ {c}_2\left(w,r\right)={p}^{\ast }. $$

Assuming that factor intensities differ between the two goods, the above two equations uniquely determine the equilibrium factor prices w* and r*.