ACCOUNTING

 

ACCOUNTING

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CHAPTER 7

1. A higher savings rate will affect the steady state level of income. This is because as savings rate increases, capital stock accumulates and that increases the steady state output. Therefore, steady state income also increases.

The rate of growth however will increase till the economy will reach the next steady state. Once it reaches the next steady state, it will maintain a steady state of growth rate.

2. The golden rule level of capital is denoted as k*. Now, we know that

c* = f(k*) – dk*, where c* is the golden rule level of consumption. Therefore, the golden rule level of consumption is dependent entirely on k*. Therefore, in order to maximize the individual consumption level, it is perfectly justified for the policymaker to choose the golden level of capital k*.

 

3. There is a trade-off between current consumption and future consumption. If current consumption is reduced, saving is done in current period, which translates into investment, which increases capital stock and therefore future consumption. Therefore, when an economy starts above the golden rule level of capital, if it moves towards the golden rule level, addition to the capital stock falls that means current consumption increases. So a policymaker would always choose a higher than golden rule level of capital.

On the other hand, when a policymaker starts below the golden rule level of capital, in order to reach the golden rule level, current consumption has to be sacrificed. Therefore, whether or not the policymaker would choose a level below golden rule would depend entirely on his long-term goals.

 

4. The impact of increase in population can be clearly shown using a diagram. In the diagram below, the initial population growth rate is n. Therefore, the steady state level of capital is k1. Now, the population growth rate increases to n’. It causes the steady state level of capital to fall, to k’. As steady state level of capital falls, the steady state level of income would also fall.

However, the steady state growth rate is given by (n+g) where n is population growth rate and g is the growth rate of efficiency per worker. So when n increases,  the steady state rate of growth also increases.

 

 

 

 

 

 

 

 

	Investment

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. Y = K^.5L^.5

 

1. In order to examine the returns of scale, we replace K and L each by aK and aL respectively, where a is a constant.

 

Y1 = (aK)^.5(aL)^.5 or. Y1 = a K^.5L^.5                     or, Y1 = aY

Therefore, the production function exhibits constant returs to scale.

 

1. y = Y/L = (K/L)^.5 = k^.5     where k = K/L

 

1. We know in steady state, sf(k) = dk

 

Also, from part b, f(k) = k^.5     Therefore, s k^.5 = dk or,  k = (s/d)^.5

Therefore, for country A, k_A =(0.1/0.05)² = 4

And for country B, k_B =(0.2/0.05)² = 16

 

Therefore, y_A = 2 and y_B = 4

 

Also, the consumption is

C_A = (1 – s_A)y_A = 1.8

C_B = (1 – s_B)y_B = 3.2

 

1. If the countries start off with an initial capital stock per worker of 2, then for country A the capital per worker would be 1.414 and consumption would be 1.272. For country B y would be 1.414 and c would be 1.131.

Below, we provide the table for year-to-year change for both country A and B

 

COUNTRY A

YEAR	k	y	c	i	dk	∆k
1	2	1.414214	1.272792	0.141421	0.1	0.041421
2	2.041421	1.428783	1.285905	0.142878	0.102071	0.040807
3	2.082229	1.442993	1.298694	0.144299	0.104111	0.040188
4	2.122416	1.456852	1.311166	0.145685	0.106121	0.039564
5	2.161981	1.470368	1.323331	0.147037	0.108099	0.038938
6	2.200919	1.483549	1.335194	0.148355	0.110046	0.038309

 

 

 

 

COUNTRY B

YEAR	k	y	c	i	dk	∆k
1	2	1.414214	1.131371	0.282843	0.1	0.182843
2	2.182843	1.477445	1.181956	0.295489	0.109142	0.186347
3	2.36919	1.539217	1.231374	0.307843	0.118459	0.189384
4	2.558573	1.599554	1.279643	0.319911	0.127929	0.191982
5	2.750556	1.65848	1.326784	0.331696	0.137528	0.194168
6	2.944724	1.71602	1.372816	0.343204	0.147236	0.195968

As we can see from the table, after 5 years, consumption in country B would be greater than that of country A.

 

2. When a part of the labor force is destroyed, total output will fall, as total output is a direct function of labor and capital Y = F(K, L). However, output per person would increase, as it is given by y = f(k) where k = K/L.
3. As output per person increases, after the war, the economy starts at a point where the level of capital is greater than the steady state level. Therefore, current consumption increases and the economy slowly moves back to the steady state level, i.e. output per person falls. Therefore, the growth of output per person would be slower until the economy moves back to the steady state, after that, the growth rate would be equal to the steady state rate.

 

3. Y = K^.3L^.7

 

1. Per worker production function is Y/L =y = K^.3L^-.3 = (K/L)^.3

or, y = k^.3

 

1. Steady state capital per worker, output per worker, consumption per worker

We know that in the steady state,

∆k = sf(k) – δk

In steady state, ∆k = 0, so

sf(k) = δk

We have from a. f(k) = k^.3

So, s k^.3 = δk

or, k^.7 = s/δ

 

Therefore, k* = (s/δ)^1/.7 = (s/δ)^1.428

This is the steady state level of capital per worker.

 

Therefore,  y* = (s/δ)^.3/.7 = (s/δ)^.428

This is the steady state level of output per worker.

 

c = y* – δk* = (s/δ)^.428 – δ(s/δ)^1.428

This is the steady state level of consumption per worker.

 

1. Given depreciation = 10%. Therefore, the table is reproduced below –

s	k*	y*	c*
0	0	0	0
0.1	1	1	0.9
0.2	2.690734	1.345367	1.076294
0.3	4.800972	1.600324	1.120227
0.4	7.240052	1.810013	1.086008
0.5	9.957015	1.991403	0.995702
0.6	12.91814	2.153023	0.861209
0.7	16.09904	2.299863	0.689959
0.8	19.48106	2.435132	0.487026
0.9	23.04933	2.561037	0.256104
1	26.79168	2.679168	0

 

A savings rate of 1, i.e. 100% saving maximizes output per worker. However, that is practically impossible as it rules out any current consumption.

 

1. MPK = dY/dK = 0.3L^.7K^.7

It can be written as 0.3K^.3L^.7/K = 0.3Y/K

Dividing numerator and denominator by L, we get, 0.3y/k

In the table we substitute the values and find MPK

 

s	k*	y*	c*	MPK
0	0	0	0
0.1	1	1	0.9	0.3
0.2	2.690734	1.345367	1.076294	0.15
0.3	4.800972	1.600324	1.120227	0.1
0.4	7.240052	1.810013	1.086008	0.075
0.5	9.957015	1.991403	0.995702	0.06
0.6	12.91814	2.153023	0.861209	0.05
0.7	16.09904	2.299863	0.689959	0.042857
0.8	19.48106	2.435132	0.487026	0.0375
0.9	23.04933	2.561037	0.256104	0.033333
1	26.79168	2.679168	0	0.03

 

The table shows that marginal product of capital steady declines.

 

 

 

 

 

 

 

 

 

 

 

6. If Canada experiences a fall in population growth rate, it would increase its steady state capital per worker (from k1 to k2) and hence output per worker.

 

	Investment

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The growth rate of output is the sum of population growth rate and rate of technological progress. So as population growth rate falls, total output growth rate falls as well.

However, the growth rate of output per person does not depend on the population growth rate, but it depends on the growth of technological progress. So growth rate of output per person does not change.

 

 

 

CHAPTER 8

 

1. In steady state level, the income per worker is determined by only the growth of technological progress.
3. According to Mankiw, if the economy is operating with less capital than in the Golden Rule steady state, then MPK – d > n + g. And if the economy is operating with too much capital, then MPK –d < n+ g. So in order to know whether an economy is operating, we need to know the growth rate of output (n + g), which can be found from the real GDP growth rate and net marginal product of capital (MPK −d), which can be found from the capital stock, depreciation of capital and capital income.

 

 

 

2. y = k^.5

We know, with technological progress an population growth, the equation of change is

∆k = sf(k) – (δ + n + g)k

in steady state, ∆k = 0, so sf(k) = (δ + n + g)k

f(k) = k^.5

So,

s k^.5 = (δ + n + g)k

or, s = (δ + n + g) k^.5

or, s = (δ + n + g) y

therefore,

y = s / (δ + n + g)  This is the steady state value of y                               (1)

1. we have, Sd = .28 (developed country) SL = .1 (less-developed country)

nd = 0.01(developed country)       nL = 0.04  (less-developed country)

g = 0.02     ∂ = 0.04

Plugging the values in (1),

yd = 4

yL = 1

 

1. The country’s government can take an active population control policy, like China, education people and reducing the birth rates. However, to most of the countries it would be too restrictive and intrusive. Therefore, another way would be to try and increase the rate of growth for technological progress by investing more in research and development.

 

 

5. a. The rate of growth of total income: It is given by n + g, so efficiency of labor does not affect the rate of growth of total income. Therefore, it will not be affected by education.
6. The level of income per worker: Except education, nothing is different between the countries. SO assume, the equilibrium level of income per worker in the less educated country is y1 = Y/E1L where E is the level of income and the equilibrium level of income per worker in the more educated country is y2 = Y/E2L. Now, if everything else is same, y1 = y2 = y*. Therefore, Y/L in the less educate country must be less than Y/L in the more educated country to offset the effect of E2 > E1.
   The level of income per worker will be more in the country with more educated labor force.
7. The real rental price of capital. The real rental is determined by MPK, which is unaffected by the level of education according to the growth model. Therefore, it will not change between countries.
8. The real wage. The real wage per unit of labor can be expressed as wE where w is the nominal wage and E is the level of education. So, the level of real wage will be more in the country with more educated labor force.

 

CHAPTER 10

 

1. The fiscal policy has a multiplied effect on national income because there is an income consumption spiral. As a result of an expansionary fiscal policy, the national income increases in the first round. That, in turn, causes the consumption to increase. As consumption increases, expenditures increase even more and more income is generated. This spiral goes on. According to the Keynesian cross, the net effect of an expansionary fiscal policy is given by ∆G/(1 – MPC) where MPC is the marginal propensity to consume.

 

 

Ms

Ms’

r

• The theory of liquidity preference assumes there is a fixed supply of real money balances which is an exogenous policy variable chosen by a central bank. The price level is also an exogenous variable. The demand for money supply is a downward sloping curve. This is given in the figure below.

 

 

 

 

 

 

 

 

 

 

As the money supply increases, the money supply line shifts to right. This causes the equilibrium market clearing rate of interest to fall.

 

3. The IS curve is sloped downwards because it shows the inverse relationship between the rate of interest and investment. As interest rate falls, it is beneficial for investors to borrow more money. So level of investment also rise. The IS curve captures this inverse relationship.
4. The LM curve is upward sloping because it shows the relationship between interest rate and income. A higher level of income leads to a higher interest rate. This is because as income rises, the demand for real money balances also rises. Therefore, to bring the money market back in equilibrium, the rate of interest must rise. The LM curve captures this positive relationship and therefore is upward sloped.

 

 

 

 

1. An increase in government purchases.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

An increase in government purchases causes the planned expenditure function shift upward by the amount of increase in G. The resulting increase in income is much more, from Y1 to Y2, due to the multiplier effect.

 

1. An increase in taxes.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

An increase in tax would shift the expenditure line down, by the amount of the tax. The resultant fall in income (Y2-Y1) would be more than that, due to the multiplier effect.

 

 

1. An equal increase in government purchases and taxes.

 

An increase in G causes Y to increase by ∆G/(1 – MPC) while an increase in taxes causes Y to fall by ∆T*MPC/(1 – MPC).

Therefore, the net effect is
∆G/(1 – MPC) – ∆T*MPC/(1 – MPC) = 1/(1 – MPC) * (∆G – ∆T*MPC)

As ∆G = ∆T, the net effect is

1/(1 – MPC) * (∆G – ∆G*MPC) = ∆G

Therefore Y would increase by the increase in G.

 

 

3. a. In this case, for one dollar increase in G, consumption increases by (1 – t)MPC. Because, according to the problem, t is the marginal tax rate. Therefore, the effect on Y is less.
4. We have seen that a government purchase causes Y to change by ∆G/(1 – MPC). In this case, intuitively, the multiplier would be ∆G/(1 – t)MPC.
5. The per dollar tax would increase the slope of the IS curve.

 

 

 

 

 

 

 

 

 

5. (M/P)^d = Y − 100r (1)

M is 1,000 P is 2.

1. Therefore, M/P = 500 = real money supply.

			M/P

 

 

 

 

 

 

 

 

 

 

 

 

1. Equilibrium interest rate can be solved from (M/P)^d = Y − 100r à 500 = 1000 – 100r à r = 5
2. M is now 1200 but P is 2. So, (M/P)^d = 600. Plugging the value in (1), r = 4. i.e. the equilibrium interest rate falls.
3. Plug r = 7 in (1)

(M/P)^d = Y – 100*7

Y = 1000, P = 2

so, M = 600.

 

Therefore, money supply has to be reduced to 600.

 

 

 

 

 

 

 

 

 

 

CH 11.

Questions For Review

1. The aggregate demand curve is negatively sloped because it reflects the negative relationship between the aggregate price level in the economy and national income. As the price level increases, given money supply doesn’t change, the income falls.

 

2. When taxes are increases the income falls, by the tax rate multiplier. For one unit of tax increase , the national income falls by

∆T × [ – MPC/(1 – MPC)]

 

Where ∆T is the change in tax, MPC is marginal propensity to consume. The movement can be shown using a diagram. The tax increase would cause the IS curve to shift to the left. This would cause the rate of interest to fall and output to fall as well. As rate of interest falls, investment would rise. However the rise in investment would be offset by the fall in income, so consumption would fall.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. A decrease in money supply reduces real money balances therefore, the LM curve shifts leftwards, with no change in the IS curve. This causes the income to fall and the interest rate to rise. Consumption also falls as income falls. Investment falls too, because the interest rate rises.

 

 

 

 

 

					Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4. Falling prices will increase income.

 

CASE 1: SHIFT IN IS :

As prices fall, real wealth increases, which increases demand and would shift the IS curve to the right, this causes the rate of interest to rise, investment to fall and income to rise.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CASE 2: SHIFT IN LM:

As prices fall, real money balance increase, which would shift the LM curve to the right, this causes the rate of interest to fall, investment to rise and income to rise.

					Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CH11.

Problems

 

1. When the central bank increases money supply, LM curve shifts to the right, with no change in the IS curve. Interest rate falls, investment increases and income and consumption rise.

 

 

 

 

 

 

 

PLEASE TURN OVER

					Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. The IS curve shifts to the right, interest increases, investment falls, output and consumption both rise.

 

		r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. If the givt increases taxes, the IS curve will shift to the right by the amount of the tax multiplier, interest would fall, investment would rise, income and consumption both would fall.

 

			r
	r1
		r2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. If the govt increases government purchases and taxes by equal amounts, the net change in the IS curve would depend on the impact of the two policies.

The change in Y due to government purchases  = ∆G/(1 – MPC)

The change in Y due to taxes  = ∆T *MPC/(1 – MPC)

 

The net impact is ∆G /(1 – MPC) – ∆T *MPC/(1 – MPC)

Now, ∆G = ∆T by the question

 

so,the net impact is ∆G /(1 – MPC) – ∆G *MPC/(1 – MPC)

= ∆G [(1-MPC) / (1 – MPC)]

= ∆G

 

Therefore, it would shift the IS curve to the right by the amount of the change in govt purchases.

 

 

 

 

 

 

 

PLEASE TURN OVER

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. C = 200 + 0.75(Y – T)

I = 200 – 25r

G = 100

T = 100

 

1. IS equation: Y = C(Y – T) + I(r)+ G

 

Substituting the values and simplifying,

Y = 1700 – 100r.

 

THE GRAPH IS AT THE END OF PART C

 

 

1. LM equation: (M/P) = Y – 100r, M = 1000

 

M/P =  1000/2 = 500

LM: 500 = Y – 100r.

or, Y = 500 + 100r

1. In eqlm, IS = LM,

 

1700 – 100r = 500 + 100r

r = 6

 

When r =6, Y = 1100

 

 

		r
	6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. If  G increase from 100 to 150, IS: Y = 200 + 0.75(Y – 100) + 200 – 25r + 150, or, Y = 1900 – 100r

Therefore, is shifts to the right by (1900 – 100r) – (1700 – 100r) = 200

1. If M becomes 1200, L eqn: 600 =  Y – 100r, or, Y = 600 + 100r

LM shifts to the right by (600 + 100r) – (500 + 100r) = 100

1. If prices rise to 4, LM would be: 250 = Y – 100r or, Y = 250 + 100r

The new interest rate can be found from the IS and LM equation

1,700 – 100r = 250 + 100r, r = 7.25

Y = 975

1. The AD curve shows the relationship between output and rice level.

We have, IS = Y = 1700 – 100r    or, Y – 1700 = 100r

LM = (M/P) = Y – 100r                or, (M/P) – Y = 100r

 

Therefore, 1700 – Y = Y – (M/P),                        M = 1000

 

Y = 850 + 500/P

 

 

When P = 5, Y = 950

When P = 10, Y = 900

When P = 50, Y = 860

 

 

				AD
							Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

If (d) govt purchases increase, the AD curve would shift to the right.

 

Because, when G increases, the IS curve is Y = 1900 – 100r

LM: 100r = Y – (1,000/P).

 

Solving, Y = 950 + 500/P, so AD shifts to the right by 100.

 

 

If (e) money supply increases, the AD curve will also shift to the right.

 

 

 

6. Increase in money supply:

 

In short run, the increase in money supply would shift the LM curve to the right. That would reduce the rate of interest, increase investment and increase the output beyond the potential GDP. Therefore, in long run price would increase, real balances would fall and the LM curve would shift to the left and it will come back to its original position.

 

					Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Increase in govt purchase:

 

An increase in govt purchase in the short run would shift the IS curve to the right, which will increase the output beyond the potential output, but also increase the interest rate. In long run, the increased interest rate would crowd out the output and price will rise. This would shift the LM curve to the left. LM would shift to left till the initial output is reached. However, the prevailing interest rate in the economy would be higher.

 

 

 

 

 

 

PLEASE TURN OVER

 

					Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Increase in taxes:

 

In short run, the IS curve will shift to left, causing the interest rate to drop and income to fall. As output falls below the potential level, the price would fall and the LM curve would shift to right, restoring the potential output in the long run but at a lower interest rate level.

 

 

					Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8. M/P = L(r, Y – T)

 

1. No, the money demand function is independent of the government purchase, so it should not alter.
2. Yes, as money demand is depends on the amount of tax (T), so when T increases, M/P increases and vice versa, leading to a rightward shift of the IS curve.