The output of the representative firm in period t is given by Yt
= At √ Kt where At is productivity and Kt is capital in period t.
Denote the marginal product of capital in period t by MPKt and note
that MPKt = At 2 √ Kt . There are two periods with productivity A1
and A2 for the first and second period, respectively. K1 is the
capital stock of period 1 and is the outcome of past decisions. K2
is the capital stock of period 2 and it is determined by K2 = (1 −
d)K1 + I where d is the depreciation rate and I is the firm’s
period 1 investment. At the end of period 2, the undepreciated
capital (1 − d)K2 is sold at a price of 1 per unit of capital. In
period 1 the firm decides how much to invest, taking as given the
interest rate R and its expectations about future productivity A2.
The firm’s goal is to maximize the intertemporal value of the firm
which is equal to the discounted stream of profits: V = Π1 + Π2 1+R
where Π1 = Y1 − I and Π2 = Y2 + (1 − d) ∗ K2. The firm’s period 1
capital stock is K1 = 10, the depreciation rate is d = 0.15, the
interest rate is R = 0.3 and productivity is given by A1 = 3 and A2
= 3.
1a
Calculate first period profits. Calculate second period output
and second period profits. Calculate the value of the firm V .
b
The firm makes an important technological discovery which will
increase its future productivity but does not affect its current
productivity, i.e. A2 increases and A1 remains constant. Describe
how investment will be affected by this change. What about first
period output and first period profits? How about second period
profits and the firm’s value?
c
A financial journalist comments that if a firm’s profits drop,
then its value must also fall. Does this statement hold under all
circumstances?
The output of the representative firm in period t is given by Yt = At √ Kt where At is productivity and Kt is capital in
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