Economics W3213. Intermediate MacroProblem Set 3 - Solutions

(1) Human Capital

a) Given that we have an AK production function, Y = AK. In per capita terms, it can be written as y = Ak. Then, thefundamental equation, which provides us with the rate of growth of capital per capita,

k = sA (n+ ) If sA > n+ , then this economy will experience positive growth forever If sA < n+ , then this economy will experience negative growth forever

b) An increase in s will increase the rate of growth if sA > n+ or make it less negative if sA < n+

c) Remember that n = fertility mortality + netmigration. Then, if the fertility rate goes down, n will decrease, andwe have the same consequences as in part b).

d) Too much of a decrease in nmeans that in the long run there could be shortage of labor force to the economy, holdingmortality and net migration constant. At worst, negative population growth rate by way of decreasing fertility rate leadsto 0 population and the country disappears and so, it cannot be unambigously good.

e) Human capital has a number of characteristics:

First, everybody is born with zero human capital or skill. Second, unlike physical capital skills cannot be transmitted from parents to children (when a parent dies, the BMWstays with his children. His skills as a doctor die with him).

Third, human capital accumulation requires time (especially, student time). And finally, lifetimes are finite so there is a finite amount of time within which skills need to be.

This means that there is a limited amount of skills that people in the economy can acquire with a constant technologyand within a lifetime. Once that point is reached, human capital must remain constant at that maximum level. Then, itcannot grow without bounds.

Remember that we can get the AK production function from

Y = AKH(1)

whereH = hL. We obtain the AK production function assuming thatK andH grow at the same rate, since people wantto equate the returns of physical capital to the returns of human capital (if not, there are investment opportunities in oneof the types of capital). However, once H reaches its bound, and it cannot grow more, the production function becomes,

Y = AKH(1)

where H is fixed. In per capita terms,

y = Akh(1)

Call A = Ah(1) which is constant, to get

y = Ak

which is a Cobb-Douglas production function in per capita terms. Then, we go back to the normal Solow Model, so thatin the long run the rate of growth is 0, because we have again diminishing marginal product to capital.

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(2) Technological Progress

(a) The time inconsistency problem emerges in the context of R&D related to invention of cures for diseases for thepoor because the policy or market structure before the invention is no longer optimal after the new drug has beeninvented. R&D involves huge fixed cost. A competitive market structure is not compatible with R&D invention. Perfectcompetition involves pricing at the marginal cost which does not cover average cost if fixed cost is involved in order topay for innovation. To support R&D we need a monopoly market structure where profit maximization yields a pricegreater than the marginal cost. However monopoly has the disadvantage of higher price and lower output comparedto perfect competition, leading to loss of social welfare. Due to this there occurs a divergence between ex-ante andex-post policy. Thus we see that monopoly is needed before invention in order to induce invention. After inventionhowever perfect competition is optimal since it maximizes social welfare. So market structure before and after inventionis different, firms know this and they will not invest in discovering vaccines.

(b) One way to solve the problem is create a fund that would be used for buying the vaccine by non-profit groups atthe monopoly price which can then be distributed among the poor at marginal cost. The firm that develops the vaccinefirst will be compensated for its investment in R&D that was needed to invent the vaccine. Hence firms will have anincentive to develop the vaccine and the poor would be able to procure it at a lower price through some intermediatenon-profit organization. It appears to be the perfect solution for the time inconsistency problem, but unfortunately it isnot. It will always tempting to use the fund to address many other pertinent problems rather than let it sit idle till thevaccine is discovered at some uncertain future date. If companies feel that fund would be spent elsewhere they will nothave incentive to invest in new vaccine.

(3) Convergence

(a) The finding that the growth rate of real per capita GDP shows little relation to the initial level of real per capitaGDP does not conflict with the neoclassical model. The Solow-Swan model predicts conditional, rather than absolute,convergence. Poor countries are probably very different from the rich ones in terms of their rates of saving s, populationgrowth n, depreciation , technology A, and even income share . Hence they have different steady states.

(b) Absolute convergence simply means that poor countries tend to grow faster than rich ones. If there is a negativerelation between income per capita and growth rate, this is absolute convergence. Notice that if poor countries aregrowing faster, eventually they will catch up with the rich ones and then will all grow at the same rate. Thats why itscalled absolute convergence, or simply convergence.

Conditional convergence means that poor countries grow faster than rich ones only if they have similar parameters(saving rate, population growth, technology, and depreciation). Notice that it implies that they have similar steadystates. Hence poor and rich countries will converge only if they have the same steady states.

Romers AKmodel does not predict any kind of convergence. In the AK model, if two countries have the same parame-ters A, s, n, , they will be growing at the same rate forever. If one is richer than the other, the poor will never catch up.If, on the contrary, they have different parameters and hence different growth rates, the country with the higher growthrate will grow faster forever. The high-growth country will be eventually richer than the low-growth one, even if it wasinitially poor. Hence, there is no convergence.

(4) Changing savings rates and convergence

(a) When people are poor (that is, when they have low k), they may be willing to sacrifice consumption in order to save,invest, and consume more in the future (like East Asian Tigers 40 years ago). When people are rich, they may prefer toconsume now (like the U.S. now). If this is the case, the saving rate is higher for low k and lower for high k. In otherwords, s(k) is a decreasing function of k, where the first derivative is negative: s0(k) < 0.

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ii. A model with the AK technology and saving as a decreasing function of k will predict convergence. Since s(k) is a decreasing function of k, s(k)A will also be a decreasing function of k. If we assume that s(k) approaches 0 when k grows bigger and bigger, s(k)A must cross (n + ), which is still a constant. This will be a steady state. If a poor and a rich country have the same parameters, they will converge to the same steady state. Therefore, this is conditional convergence.

, where s'(k) < 0.

iii. The Solow-Swan model with a decreasing saving rate still predicts convergence. Now either

, where s'(k) < 0,

or , where s is a constant.

The function s(k)A/k1-, where s(k) is decreasing in k, decreases faster than sA/k1-, where sis a constant.

The two curves s(k)A/k1- and sA/k1- may intersect at the steady state, giving the same steady state whether the saving rate is decreasing or constant (Case 1 below). They may intersect at a point to the left of k*, giving a new lower steady state k** (Case 2), or to the right of k*, giving a higher steady state k*** (Case 3). They may not intersect at all, giving either a lower or a higher steady state (Cases 4 and 5). It all depends on the particular constant saving rate (e.g. s = 0.1 or 0.45) and the particular decreasing function of k (e.g. s(k) = 1/k or 2/(5k2)).

If we are interested in comparing the speed of convergence, we must select two points that

3. I Wanna Be Like You

i. Fertility is usually a decreasing function of k. In general, people in rich countries tend to have fewer children than in poor ones. As a country gets richer (that is, k grows), the wages of its residents also increase. Since raising children requires a lot of humans' time, especially women's time, having children represents a lot of forgone wages and hence a lot of forgone consumption. Having children becomes more expensive in terms of forgone income as a country becomes richer. In addition, poor countries typically do not have well-functioning social security systems and retirement benefits. Hence people in poor countries choose to have more children, so that at least one of them can take care of parents when parents become old or disabled.

ii. Mortality is usually a decreasing function of k. In general, people in rich countries live longer than in poor ones. As a country gets richer, people are able to invest more in healthcare, thereby reducing the mortality rate.

iii. If we define net migration as immigration (moving in) minus emigration (moving out), net migration is usually an increasing function of k. Poor countries tend to send migrants, while rich ones tend to receive them.

iv. A model with the AK technology and a population growth rate as an increasing function of kwill predict convergence. Since n(k) is now an increasing function of k, (n(k) + ) will also be an increasing function of k. If we assume that n(k) grows bigger and bigger when k grows bigger and bigger, (n(k) + ) and sA, which is still a constant, must cross. This will be

a steady state. If a poor and a rich country have the same parameters, they will converge to the same steady state. Therefore, this is conditional convergence.

, where n'(k) > 0.

v. The Solow-Swan model with an increasing population growth predicts faster convergence than with a constant population growth rate. Now (n(k) + ), where n'(k) > 0, increases faster than (n + ), where n is a constant.

Economics W3213. Intermediate MacroProblem Set 4 - Solutions

(1) Development Aid

(a)

The population growth rate may look like this when one considers the opportunity cost of having children as k increases.

Specically, as k increaseswe seewages increase, so that the time invested in raising children by parents (mothers) becomesmore costly. Hence, as k increases we would expect fertility to fall as the return on a unit of time working rises abovethat of raising children. For low levels of k mothers are better of raising children, but around a specic level of capital,say k, this becomes more costly and so they substitute their time from the production of children to the production ofcookies. At this point, fertility falls drastically as all parents (mothers) nearly simultaneously experiece this substitutionof their time. Finally, it is clear that, as one cannot have negative children (and on average any country will have positivechildren), the value of n(k) is bounded below, reaching this bound at, say, k. This discussion produces the graph of n(k)seen, with the first and second kinks at the values k and k, respectively.

However, we need to consider that n does not only include fertility. Remember that n = fertility mortality +net migration. Empirically we also observe that there exists a negative relationship between the mortality rate andthe level of capital per capita. Rich countries have access to better health care, pharmaceutical products and better foodthat rises life expectancy, and that reduces the mortality rate.

Also, there exists a positive relationship between net migration and the level of income per capita. Rich countries attractmore immigration. Then, we should expect that both the mortality and migration rates decrease n when capital is low,and increase it when k is high.

Thus, if we want the graph to look like that, the effect of the fertility rate should more than offset the effects of themortality and migration rates.

(b)

Yes, a steady state will necessarily exist. The reason is that, if we consider a Cobb-Douglas production function, thesavings curve s f(k)k will have the following limits,

limk!0

sf(k)

k= 1

limk!1

sf(k)

k= 0

Since the maximum and the minimum of n(k) are between 0 and1, then the savings curve will cross the depreciationline at least once.

(c)

No, the steady state may not be unique. We might have three different cases,

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(d)

For cases I and II the unique steady states are stable. If we perturbate the economy to the right or to the left of the steadystate, capital per capita will go back to that steady state.

In case III , k3 and k003 are stable steady states, for the same reason as before. However, k03 is an unstable steady state. Ifme move slightly to the left (right) of that steady state, the economy will converge to k3 (k003 ). Then, the only chance thatthe economy stays forever in the unstable steady state is that the original level of capital per capita is exactly k03.

(e)

In Case III , k3 is a poverty trap. The country will be stuck in that level of capital per capita unless we provide it with ahuge amount of aid.

(f)

The Case III model can be used to justify large increases in foreign development aid as a large enough gift of capital toa country stuck in the poverty trap, k3, can lift it out of the trap and to a level of capital that will ensure that it convergesto the highest steady state in the long-run. Specically, for a country in the poverty trap, the gift of capital must be greaterthan k03 k3.(g)

The first problem with this model is has been discussed in part (a). We need the effect of the fertility rate to morethan offset the effects of the mortality and net migration effects. Empirically we can observe that n(k) does not looklike that. It looks more or less constant when we consider the effects of the other rates.

There is no evidence to support the fact that although fertility does decrease as income per capita increases thedecrease is abrupt and not gradual.

It might be difficult to know which is the necessary level of aid to get rid of the trap. The model forgets that there exists technological progress in the real world. Then, even if the population growthrate looks like that, an increase in total factor productivity might push the savings line to the right, so that we endup in case III .

(2) Development Aid

(a) and (b)

Given this structure for the savings rate, we need first to derive how does the savings curve look like. When the countryis poor, we observe a very small savings rate, that increases suddenly once we become richer. The fundamental equationof Solow-Swan is given by:

k = s(k)f(k)

k (n+ )

where in this case, s is no longer constant (hence, we should write s(k)).

Notice that in the previous graph there are 3 different areas.

In area A, the savings rate is constant but small; therefore, the savings curve in the fundamental equation is de-creasing . This is just like the regular Solow model with a small savings rate.

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In area C, the savings rate is constant but big; again, the savings curve is decreasing. Solow model with highsavings rate.

In area B, the savings rate is increasing with capital. We are going to assume that it increases more than the negativeeffect of the diminishing returns to capital; then, the savings curve is increasing in that area.(Think of this,

s(k)f(k)

k

we know that s(k) is increasing in this area, but f(k)/k is decreasing because of diminishing returns. Then, we areassuming that s(k) grows much faster than the decrease in f(k)/k when capital increases)

As a result, we obtain this graphical representation for the fundamental equation,

Notice that, even if we have the movement in the middle of the savings line, the limits of it are again1 (from the smallsavings rate area) and 0 (from the high savings rate area). Since we consider a positive and bounded n + , the savingsrate will cross for sure the depreciation line at least once. But in this case the steady state might not be unique.

In the case of the dashed and the dotted depreciation line, there exists only one steady sate; a low one in the case of thedotted line and a high one in the case of the dashed one. However, if we consider the solid savings curve, there are threesteady states, a low one, a high one, and one in the middle.

(c)

In the case of the dotted and the dashed line, there is a unique steady state, which is stable. If we deviate the amount ofcapital per capita to the left or to the right of those steady states, the country will converge back to them.

However, in the case of the solid savings curve, we have

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To the left of k1, savings are greater than depreciation in average per unit of capital, and we have positive growth. Ifwe start at a level of capital to the left of k1, we will eventually end up in k1. To the right, until k2, we have negativegrowth, and we will end up in k1. Therefore, k1 is a stable steady-state with a low level of capital per capita (andGDP per capita).

If you depart from a point to the left of k2, the economy will experience negative growth and will move until k1. Tothe right of k2, we observe positive growth, and the economy will end up in k3 in the long run. k2 is an unstableequilibrium. You will only remain in k2 forever as long as you start exactly in k2. If you move just a bit to the left(right) you will go to k1 (k3).

For k3 we can apply the same argument as with k1. Then, k3 is a stable equilibrium, and it is associated with a highlevel of capital (and GDP) per capita.

(d)

Yes. In the previous case, k1 is a povery trap. Unless the country does not have a high income per capita, it will not go tothe richer steady state, k3. If we push the economy to a point between k1 and k2, the country will go back to the trap, k1.

(e)

The previous case can be used to justify large increases in foreign development aid as a large enough gift of capital to acountry stuck in the poverty trap, k1, can lift it out of the trap and to a level of capital that will ensure that it convergesto the highest steady state in the long-run. Specically, for a country in the poverty trap, the gift of capital must be greaterthan k2 k1.(f)

It might be impossible to know which is the level of aid which is necessary to get rid of the trap. And then, if wedo not give enough capital, we might conclude erroneously that aid does not help.

Unrealistic savings function. It is quite unlikely that savings increase so dramatically in region B. There is no evidence to support the fact that the increase in savings will offset the diminishing product of capital. This derivations rely on the fact that there is no technological progress. With technological progress, we can get ridof the poverty trap (savings curve shifts upward). Is it credible that there is no technological progress?

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(3) Dual Solow

1. Given that the production funtion is a Cobb-Douglas one, then, the fundamental equation is given by

k = sAk-1 (n+ ) = s 10 k0.5 0.1

2. We are going to have two different savings curves in this problem. For countries whose capital per capita is smallerthan 1000, we will have a savings line associated to s = 0.20 and for countries whose capital per capita is biggerthan 1000, the savings line will be associated to s = 0.70. Then,

3. If k < 1000 then the fundamental equation is,

k = 0.2 10 k0.5 0.1

In the steady state, k = 0. Then,

2k0.5 = 0.1k = 400

4. If k > 1000 the fundamental equation is,

k = 0.7 10 k0.5 0.1

In the steady state,

7k0.5 = 0.1k = 4900

5. If the economy is in the steady state k then, if we give the country D = 300, it will have 400 + 300 = 700 units ofcapital per capita. Howeverm 700 < 1000, and the country will still be using the low savings rate. At that point,the savings line is below the depreciation line. Average investment is smaller than average depreciation per unit ofcapital. Then, we have negative growth and the country will go back to k eventually.

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6. In this case, 400 + 700 = 1100 > 1000. Then, the country will be in the area with high savings rate. At that point,we are between 1000 and k. Then, the savings line is above the depreciation line. We have positive growth andwe converge to the high steady state, k.

7. No. We need to give to the country at least 600 units of capital in order to get rid of the poverty trap. Smalldonations will not help the country to grow to a high level of GDP per capita, as we have shown before withD = 300.

(4) Is aid harmful?

First of all, some economists argue that international aid might destroy incentives in poor countries to get more produc-tive by theirselves, since they receive many good for free. That is, aid might keep poor countries in a servant role insteadof in a self-sufficient one. If countries have this free permanent income from abroad, they do not have the incentive toincrease income or to develop other sources of income like capital income.

Second, and most importantly, many people argue that international aid can undermine local producers. For example, ifrich countries send T-Shirts to poor countries, local producers of T-Shirts may go bankrupt. This problem has especiallybeen critical in the case of agricultural products. Many people in poor countries survive with local farms. Food aid tohelp countries through a temporary famine often drives farmers out of business. It is impossible to compete with thefood that wealthy western countries send for free. Also, this type of aid might reduce the price of farmers products,which is also harmful for them.

It has also been argued that international aid may reduce exports and the participation of poor countries in internationaltrade. This is harmful since many research papers show that international free trade might improve long run growth.

Other problem is ineffective penetration and lack of accountability. Big aid money goes to governments rather than thepeople, and there are no constraints imposed on local governments on how to use the aid. This approach enables corrup-tion and encourages irresponsible governance, which can harm long run growth, as many empirical studies suggest. Youshould not only consider that these governors steal the aid, but that they might end up passing laws that make difficultto be more productive, like forbidding farmers to own land.

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