10
PEDRO PABLO ALVAREZ LOIS
Lemma 3 (Capacity Choice). The optimal decision of investment in capital Kt±ι
and capital-labor ratio Xt+ι is given, respectively, by the following Euler equations
(2.23)
⅛u lʌ/ ■ |i’/ - Δt4-2Pt+l (1 - ð)) = ¾1,0 1 Δ⅛+2 (1 - F (¾+ι)) Φt+ι ( 7J+1 I r
I ∖Λt+ι∕ J
and
(2.24)
¾ 0 ) Δt+2Φt+l ∖ Jr +
I ∖Af+l
a(e- 1)∖ Γ,t+1 Γ
' ' / vdF(υ)- dF(v)
^t+l JJy Jvt + 1
where
(2.25)
Φt+ι ≡ Ft+ι -
(1 + ⅞1) Wt+1
∠4⅛+ιΔ"t+1
The first equation states that the optimal capital stock is such that the expected
user cost of capital is equal to its expected revenue, which is given by the dis-
counted increase in profits generated by an additional unit of capital corrected by
the probability of operating such unit. From the second equation one can observe
the trade-off faced by the intermediate firm when choosing the optimal capital-
labor ratio. When increasing the capital-labor ratio, the firm increases its labor
productivity, which is given by AtXp, something that has a favorable effect on its
competitive position in case of excess capacities. However, increasing Xt means
that the maximum level of employment available in period t will be lower, and like-
wise the maximum volume of sales of the firm. The optimal capital-labor ratio will
be such that the two opposite effects on expected profits are equal in the margin.
2.3. Money Supply and Financial Intermediation. In this model, banks’
main task is the provision of liquidity to their customers, the input producing
firms. Banks begin each period with assets and liabilities that consist solely of
the funds deposited with them by the households, Dt. Competition among banks
for these deposits determines the market-clearing gross interest rate, (1 + Bf) ,
which is payable at the end of the period. Banks finance their lending activities
with household deposits, as well as with funds obtained from cash injections, Xt,
made by the monetary authority every period. The asset side of banks’ balance
sheet is composed by loans, Bf, that are supplied to intermediate firms. The bank
charges a gross lending rate equal to (1 + Rt ) . Financial intermediation is assumed
to be a costless activity. With no barriers to entry, competitive forces will ensure
that the equilibrium interest rate on loans equals the rate paid on deposits, that
is, Rt = Rp. Moreover, in equilibrium, the financial intermediaries will supply
Inelastically the total amount of loanable funds at their disposal:
(2.26)
Bf = Dt + Xt
At the end of the period, banks remit ll( = ( 1 + Rt) xtMt as dividends to house-
holds, where xt is the growth rate of money,
(2.27)
Mt+1 - Mt = Xl
t ~ Mt Mt