Dynamic Demand For Inputs and Steady State

Question 1

What is the firms pure profit constraint?

Answer 1

π = Y - rK - wL

Question 2

What is pure profit?

Answer 2

income left over after paying the costs of hiring capital and labor

Question 3

Generally what is MPK?

Answer 3

• MPK: β(Y/K)
  
  • Implication for profit max: β(Y/K) = r
  • Consequently β = r(K/Y) (capital share of income)

Question 4

Generally what is MPL?

Answer 4

• MPL: (1-β)(Y/L)
  
  • Implication for profit max: (1-β)(Y/L) = w
  • 1 - β = w(L/Y)

Question 5

Why is pure profit 0?

Answer 5

• Old (K owners): get a share β of total output
• Young (L suppliers): get (1 - β) share
• Zero π is a result of perf comp
  
  • No oligo/monopoly rents

Question 6

What is the MPK condition for CD?

Answer 6

• r = βAk^β-1
  
  • From MPK: β(Y/K)
  • Equivalent to β(Y/K) = r

Question 7

What is the MPL condition for CD?

Answer 7

• w = (1-β)Ak^β
  
  • MPL: (1-β)(Y/L)
  • Equivalent to (1-β)(Y/L) = w

Question 8

What is the relation between the capital labour ratio and r and w?

Answer 8

• Higher k (in eqm) means a lower eqm r and a higher eqm w

Question 9

When n=0, broadly and generally how is next period capital determined?

Answer 9

• Capital-labour ratio this period determines wage rate this period, wt
• wt determines next periods supply of capital kt+1
  
  • Next period supply of L fixed at N
  • therefore wt determines next periods k

Question 10

(n=0) Broadly, kt+1 =

Answer 10

kt+1 = s = at+1 (in closed economy)

Question 11

(n=0) With CD production and CD/2α log utility, kt+1 =

Answer 11

kt+1 = s = (1-α)w = (1-α)(1-β)Atkt^β

Question 12

At t=0 what is the K stock? (n=0)

Answer 12

• At t=0 capital stock is K0 (total assets owned by the elderly at that time); k0 = K0/N

Question 13

What is true of a small k0? (n=0)

Answer 13

• Small k0 - wage w0 in period t = 0 is small: saving by young is limited
  
  • However even limited saving makes K at t = 1 larger than t = 0

Question 14

What is the transition of the economy (n=0)?

Answer 14

• Higher capital stock: higher k: higher wages which results in higher savings by the young
• Diminishing returns to capital means that as capital stock grows, adding another unit has a smaller effect on wages: wage rise becomes smaller
• Thus along the transition path, growth rate in wages gets smaller until it does not change:

Question 15

What is k̅ given CD production and CD/2α log utility?(n=0)

Answer 15

• k̅ = [(1-α)(1-β)A]^1/(1-β)

- From the transition equation where kt = kt+1

Question 16

What is I in s.s.? (n=0)

Answer 16

• Since capital stock is fixed in s.s, ∆k is 0, which implies that national investment = 0

Question 17

In s.s (n=0) what is dissavings by the old and what does this imply about national savings?

Answer 17

• Savings by the young is exactly offset by dissavings by the old
  
  • Saving by each young agent: k
    
    • income of each old: rk̅
  • Consumption of the old: k̅(1+r)
  • Therefore dissaving by the old = rk - k̅(1+r) = -k̅
  • National savings = k̅ + (-k̅) = 0

Question 18

How is the S.S graphed?

Answer 18

• k(x) vs kt+1(y)
• f(k): output
• sf(k): savings/investment
• Population dynamics/deprecation: 1.k line changes in slope

Question 19

What does an increase in the savings rate cause from s.s. (n=0)?

Answer 19

• Increase in savings rate: drop in c: k increases: y increases: new steady state