Michael Burda - Sciences Po Spire

Appendix
.1 More Details on Retail Trade in General Equilibrium
with Leisure and Retailing Externalities
.1.1 Consumers
The first-order condition for an interior solution22 of a sector i = M, R consumer
is:
(wi/p)φ i U i 1[(wi/p)hi] =V i
1 (T − hi, 1 − T ). (15)
Given T and the real wage wi/p, we can thus compute the optimal shift
length hi, and thus labor supply, of each i-household. Log-linearizing these
first-order conditions gives us equations 1 and 2 in the text, with
and
θi ≡− U i00 (Ci)Ci
U i0 (Ci) > 0, λi ≡−
i V
µ i ≡−
i V11(T − hi, 1 − T )
V i
1 (T − hi, 1 − T ) hi > 0,
11(T − hi, 1 − T ) − V i
12(T − hi, 1 − T )
T ≥ 0.
V i
1 (T − hi, 1 − T )
The elasticity of the marginal utility of consumption θi is simply the Arrow-
Pratt measure of relative curvature of the utility function with respect to
consumption, while λi and µ i measure the (negative of) elasticity of the
marginal utility of solitary leisure with respect to solitary leisure and shop
closing time, respectively. By assumption, the first two elasticities are positive,
while the third one is non-negative. The standard case in which solitary
and common leisure are perfect substitutes in utility corresponds to µ i =0.
.1.2 Manufacturing
Because of the linearity of the production function in manufacturing, the
competitive wage rate in that sector(expressed in units of the manufacturing
good) is simply
wM =1. (16)
22Inada conditions ensure that inequality conditions are never binding, and that the
solutions are interior.
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