Why does the tape measure hook move?

Suppose that output (Y ) in an economy is given by the following aggregate production function: Yt = Kt + Nt where Kt is capital

shusha [124]

Answer:

Check the explanation

Explanation:

Yt = Kt + Nt

Taking output per worker, we divide by Nt

Yt/Nt = Kt/Nt + 1

yt = kt + 1

where yt is output per worker and kt is capital per worker.

a) With population being constant, savings rate s and depreciation rate δ.

ΔKt = It - δKt

dividing by Nt, we get

ΔKt/Nt = It/Nt - δKt/Nt ..... [1]

for kt = Kt/Nt, taking derivative

d(kt)/dt = d(Kt/Nt)/dt ... since Nt is a constant, we have

d(kt)/dt = d(Kt/Nt)/dt = (dKt/dt)/Nt = ΔKt/Nt = It/Nt - δKt/Nt = it - δkt

thus, Capital accumulation Δkt = i – δkt

In steady state, Δkt = 0

That is I – δkt = 0

S = I means that I = s.yt

Thus, s.yt – δkt = 0

Then kt* = s/δ(yt) = s(kt+1)/(δ )

kt*= skt/(δ) + s/(δ)

kt* - skt*/(δ) = s/(δ)

kt*(1- s/(δ) = s/(δ)

kt*((δ - s)/(δ) = s/(δ)

kt*(δ-s)) = s

kt* = s/(δ -s)

capital per worker is given by kt*

b) with population growth rate of n,

d(kt)/dt = d(Kt/Nt)/dt =

= $\frac{\frac{dKt}{dt}Nt - \frac{dNt}{dt}Kt}{N^{2}t}$

= $\frac{dKt/dt}{Nt} - \frac{dNt/dt}{Nt}.\frac{Kt}{Nt}$

= ΔKt/Nt - n.kt

because (dNt/dt)/Nt = growth rate of population = n and Kt/Nt = kt (capital per worker)

so, d(kt)/dt = ΔKt/Nt - n.kt

Δkt = ΔKt/Nt - n.kt = It/Nt - δKt/Nt - n.kt ......(from [1])

Δkt = it - δkt - n.kt

at steady state Δkt = it - δkt - n.kt = 0

s.yt - (δ + n)kt = 0........... since it = s.yt

kt* = s.yt/(δ + n) =s(kt+1)/(δ + n)

kt*= skt/(δ + n) + s/(δ + n)

kt* - skt*/(δ + n) = s/(δ + n)

kt*(1- s/(δ + n)) = s/(δ + n)

kt*((δ + n - s)/(δ + n)) = s/(δ + n)

kt*(δ + n -s)) = s

kt* = s/(δ + n -s)

.... is the steady state level of capital per worker with population growth rate of n.

3. a) capital per worker. in steady state Δkt = 0 therefore, growth rate of kt is zero

b) output per worker, yt = kt + 1

g(yt) = g(kt) = 0

since capital per worker is not growing, output per worker also does not grow.

c)capital.

kt* = s/(δ + n -s)

Kt*/Nt = s/(δ + n -s)

Kt* = sNt/(δ + n -s)

taking derivative with respect to t.

d(Kt*)/dt = s/(δ + n -s). dNt/dt

(dNt/dt)/N =n (population growth rate)

so dNt/dt = n.Nt

d(Kt*)/dt = s/(δ + n -s).n.Nt

dividing by Kt*

(d(Kt*)/dt)/Kt* = s/(δ + n -s).n.Nt/Kt* = sn/(δ + n -s). (Nt/Kt)

$\frac{sn}{\delta +n-s}.\frac{Nt}{Kt}$

using K/N = k

$\frac{s}{\delta +n-s}.\frac{n}{kt}$

plugging the value of kt*

$\frac{sn}{\delta +n-s}.\frac{(\delta + n -s)}{s}$

n

thus, Capital K grows at rate n

d) Yt = Kt + Nt

dYt/dt = dKt/dt + dNt/dt = s/(δ + n -s).n.Nt + n.Nt

using d(Kt*)/dt = s/(δ + n -s).n.Nt from previous part and that (dNt/dt)/N =n

dYt/dt = n.Nt(s/(δ + n -s) + 1) = n.Nt(s+ δ + n -s)/(δ + n -s) = n.Nt((δ + n)/(δ + n -s)

dYt/dt = n.Nt((δ + n)/(δ + n -s)

dividing by Yt

g(Yt) = n.(δ + n)/(δ + n -s).Nt/Yt

since Yt/Nt = yt

g(Yt) = n.(δ + n)/(δ + n -s) (1/yt)

at kt* = s/(δ + n -s), yt* = kt* + 1

so yt* = s/(δ + n -s) + 1 = (s + δ + n -s)/(δ + n -s) = (δ + n)/(δ + n -s)

thus, g(Yt) = n.(δ + n)/(δ + n -s) (1/yt) = n.(δ + n)/(δ + n -s) ((δ + n -s)/(δ + n)) = n

therefore, in steady state Yt grows at rate n.

5 0

3 years ago

Amherst City provides a defined benefit pension plan for employees of the city electric utility, an enterprise fund. Assume that

Sever21 [200]

Answer:

Dr Service cost 245,000

Cr Interest 166,400

Cr Cash 411,400

Dr Plan assets - pension 411,400

Cr Service cost 245,000

Cr Interest 166,400

Explanation:

Preparation of the journal entries to record annual pension expense for the enterprise fund of Amherst City

Since we are Assuming that the plan investments was $184,300 while the service cost component is the sum of $245,000, and interest on the pension liability is the sum of $166,400 for the year this means the Journal entries to record annual pension expense for the enterprise fund of Amherst City will be:

Dr Service cost 245,000

Cr Interest 166,400

Cr Cash 411,400

(245,000+166,400)

Dr Plan assets - pension 411,400

(245,000+166,400)

Cr Service cost 245,000

Cr Interest 166,400

4 0

3 years ago

Inâ€‹ 2011, Kendallâ€‹ Ford, an automobileâ€‹ dealership, spentâ€‹ $20,000 on a new car lift for its repairâ€‹ shop, $2,000 on a

Lilit [14]

Answer:

$522,000

Explanation:

The computation of the Kendall Ford's total investment spending in 2018 is shown below:

= Dealership spent + repairing cost + unsold cars and trucks were valued i.e closing cost - unsold cars and trucks were valued i.e opening cost

= $20,000 + $2,000 + $900,000 - $400,000

= $522,000

The $600,000 would be ignored and the rest cost are taken for the computation

4 0

2 years ago

XYZ Company makes one product and has calculated the following amounts for direct labor: AH x AR = $84,000; AH x SR = $83,000; S

Molodets [167]

Based on the various cost rates and hours for XYZ Company, the labor efficiency variance is $2,000 unfavorable

<h3>What is the labor efficiency variance?</h3>

This can be found as:

= (Actual hours x Standard rate) - (Standard hours x Standard rate)

Solving gives:

= 83,000 - 85,000

= $2,000 unfavorable

Find out more on the labor efficiency variance at brainly.com/question/27404735

#SPJ1

5 0

1 year ago

Strategic PlanningImagine that IBM has decided to diversify into the telecommunications business to provide online cloud-computi

rosijanka [135]

Answer:

IBM could either diversify by the strategy of market penetration, which consists in increasing the market share in a particular sector (in this case, cloud computing) through more marketing efforts.

Or it could integrate horizontally, acquiring a possible competitor that is more advanced in the cloud-computing business. Or even a start-up with good prospects, because with the amount of capital that IBM has, it could more easily expand the start-up operation as a new internal business division.

4 0

2 years ago

Why does the tape measure hook move?​

Why does the tape measure hook move?