Economics 3213Answers to Problem Set 1(1) When you Wish Upon a Star(a) (i) In 2007 the five richest countries in the world in terms of GDP per capita at purchasing powerparity (PPP) are Luxembourg, Norway, Singapore, Hong Kong, and the United States.(ii) In 2007 the five poorest countries in the world are the Democratic Republic of Congo, Guinea-Bissau, Burundi, Madagascar, and Niger.(iii) The average person in Luxembourg is almost 200 times richer than the average person inthe Democratic Republic of Congo.(b) (i) The countries that have grown the most from 1960 to 2007 are Equatorial Guinea, Taiwan, China, the Republic of Korea, and Botswana.(ii) The countries that have experienced the most negative growth are the Democratic Republic of Congo, Zimbabwe, Central African Republic, Niger, and Senegal. (c), (d), (e), (f), (g)<...>(h)

Since explaining why some countries are rich and others are poor is basically what economics is all about, the following few paragraphs cannot be considered exhaustive.

The majority of the countries that are rich today were already richer than others a hundred years ago. These are the countries in Western Europe and North America that benefited the most from theIndustrial Revolution in the 18th and 19th centuries and had the most advanced technology at the time.The most notable exception is East Asian "Tigers" that grew spectacularly after World War II. Manypeople agree that stable governments, legal systems, and the protection of property and intellectualproperty rights benefited economic growth in the rich countries.

We can analyze poor countries in two categories. The first one consists of former socialist countries (the former Soviet Union, China, India, and satellite countries), and the second are equatorial countries. Since World War II to the late 1980s ex-socialist countries adopted a system of planned oradministrative command economy that was fundamentally different from market capitalism. Since it isvirtually impossible even for the most well-intentioned central planning authority to have enoughinformation about all economic activities in a country, central planning inevitably led to an inefficientallocation of resources. Since the government was the main decision-maker, many economic decisionswere based on political or military grounds. Widespread inefficiencies often necessitated rationing inconsumption and led to a breakdown in the incentives system. Labor productivity and consequentlycapital productivity fell. Therefore, these economies have been brought down to low wealth levels.

The second set of poor countries is tropical countries. Although many of them were also socialists at some point, these countries can blame more differentiated causes. Some of them are unfavorable climatic conditions, such as hot weather that makes working very difficult; tropical diseases, such as malaria, which weaken the labor force; distance from expanding markets in the post-WWII period; and a particularly exploiting attitude of European colonialism in these regions. Remember that the U.S., Australia, New Zealand, and some other rich countries were also colonies. However, favorable climate in those countries allowed Europeans to settle and therefore set up sustainable institutions, such as local or municipal governments. In countries with particularly hostile climate, such as Congo, Europeans did not settle and set up institutions that allowed them to extract natural resources. All these features reduced labor productivity, then capital productivity, and therefore the growth potential of these economies.

(2) Figaro and Cleo(a) There are many definitions of poverty. One is the situation of having "few" resources to make aliving. What "few" means, however, varies. Some people say that "few" means less than one dollar aday; others use two dollars or three dollars as the main definition. Any quantitative definition of poverty will be arbitrary. Most people, however, agree that being unable to provide for sufficient calorie intake, basic clothing, shelter, and basic health care qualifies as absolute poverty.(b) Growth is, essentially, the only way to achieve poverty reduction. Aggregate economic growth and poverty reduction tend to go hand-in-hand, as growing economies provide opportunities for the poor to increase their income. (c) Growth tends NOT to be associated with inequality. Even if inequality increases, it usually does notincrease enough to offset the growth of mean income. The lower part of the distribution shifts to theright when the mean shifts to the right. Hence poverty falls.

(3) Jiminy Cricket(a) Returns to scale refer to the relationship between inputs taken all together and production. We saythat a production process exhibits constant returns to scale (CRS) whenever an increase in all inputs inthe same proportion implies an increase in output in the same proportion. For example, if the production function is Y=F(A, K, L), and we double K and L, we will also double output:2Y=F(A, 2K, 2L). Notice that technology (A) does not have to double to obtain double product. This is because it is a nonrivalgood and hence cannot be appropriated by a single entrepreneur. Returns to scale only refer to rival factors of production that cannot be used in two places at the same time.Returns to capital instead refer to the effect on production of the increase of capital alone, keeping labor constant. We say that returns to capital are diminishing if successive increases of capital cause smaller and smaller increases of production. The idea is that the same number of workers has to work with more and more machines, and hence the productivity of additional machines gets lower and lower.(b) In principle, it is always possible to replicate a production process and to double output. We canbuild a factory identical to the one we have, hire the same number of workers, and we will get twice asmuch output. In this sense the assumption of constant returns to scale is sensible.Diminishing returns to capital can be justified as follows. Consider a very simple case in which one unit of capital (e.g. a computer) is given to each worker. His or her productivity compared with the case in which he or she used no machines is much higher. If we now give another machine to each of theseworkers (e.g. another computer), it is likely that the workers will not be able to use this second unit ofcapital as effectively as the first one. Think of a secretary with two computers, a cook with two kitchens, and so on. The increase in output caused by the second machine is lower than the one associated with the first machine. Since we can repeat this empirical reasoning forever, it is sensible to assume that the contribution of extra capital to production (capital productivity) is a decreasing function of the capital already present in the economy.(c) To demonstrate that a function exhibits constant returns to scale (CRS), it is sufficient to demonstrate that output doubles whenever capital and labor both double at the same time:(2K)0.3(2L)0.7 = 20.3K0.320.7L0.7 = 20.3+0.7K0.3L0.7 = 2K0.3L0.7 = 2Y.(Capital and labor can be multiplied by a variable λ rather than 2 for a more general proof.)To demonstrate diminishing returns to capital, first think of what would happen if capital had constantreturns instead of diminishing. Each additional unit of capital would bring equal boost to output. Bydoubling capital alone we would be able to double output. But since capital, in fact, has diminishingreturns, doubling capital alone less than doubles output:(2K)0.3L0.7 = 20.3K0.3L0.7 < 2K0.3L0.7 = 2Y because 20.3 = 1.23... .(Alternately, one can show that the second derivative of the production function with respect to capital is negative.)

(d) Following the same reasoning as in (c),(2K)α(2L)1-α = 2αKα21-αL1-α = 2α+1-αKαL1-α = 2KαL1-α = 2Y. (Substitute λ for 2 for a more general proof.)(2K)αL1-α = 2αKαL1-α < 2KαL1-α = 2Y because 2α < 2 for any 0 < α < 1. (Can also be proved by taking the second derivative of Y with respect to K, and showing that it is negative.)

(4) Stromboli(a) The equation says that the increase in capital (net investment) is equal to total savings (which is gross investment in a country with no government and no foreign sector) minus whatever is used to replace capital that broke down δ and whatever is needed to equip newly born people (n) with the same level of capital that the people already alive have. In other words, we take whatever we saved, replace the broken machines, and give capital to new people. What is left increases capital per person or capital-labor ratio k.(b) In the long run growth cannot last forever because of diminishing returns. As the capital stock grows bigger, it is less and less productive; eventually, this will bring us to a situation where all the savings have to be used only to replace old capital that broke down and to equip new people. Nothing is left to increase the capital stock per person.

(5) MonstroY = AK0.3L0.7; s = 0.11, δ = 0.1, n = 0.01(Notice that the coefficient A representing technology is not necessarily 1.)(a) Substitute an expression for Y into the definition of y:

y = Y / L = AK0.3L0.7 / L = AK0.3L0.7-1 = AK0.3L0.7-1 = AK0.3L-0.3 = AK0.3/ L0.3

= A(K/L)0.3 = Ak0.3

(b) Remember the fundamental equation of the Solow-Swan model:

Substituting numerical values, we obtain:

Δk/k = 0.11Ak0.3/k – (0.1 + 0.11) = 0.11Ak-0.7 - 0.11

(c) In the steady state Δk/k = 0. Hence, to calculate the steady state value of k, we need to impose thatthe expression we wrote in (b) equals 0 and solve for k:

0 = 0.11Ak-0.7 – 0.11

0.11 = 0.11Ak-0.7

1 = Ak-0.7

A = 1/k-0.7 = k0.7

Problem 5.(a)

y =Y

L=AK0.5L0.5

L= Ak0.5

(b)

γk = sf(A, k)

k− (n+ δ) =

= sAk−0.5 − (n+ δ) =

= 0.3Ak−0.5 − 0.15

(c)

0.3Ak−0.5 − 0.15 = 0

0.3Ak−0.5 = 0.15

0.3A = 0.15k0.5

k0.5 = 2A

k∗ = 4A2

(d)Capital-labor ratio k suddenly increases. Output and savings per capita (y

and sy) also suddenly increase but less than proportionately because of decreas-ing returns to capital. As the amount of capital per person lost to depreciation(δk) and to the newly born (nk) instead grows in the same proportion as capital,actual savings will not be enough for replacement and for the newborn. As aresult, part of the capital stock cannot be maintained. The country will expe-rience negative growth. Every year more capital will break down than built.This will go on until all the additional capital per person is destroyed, andthe economy settles to the initial steady state level. To phrase it another way,when a country is at its steady state, this means that savings is perfectly offsetby depreciation and population growth. Additional capital, due to diminishingreturns, would produce less than enough to offset depreciation and populationgrowth. Therefore, the donation by the World Bank pushes the economy to astate where more capital depreciates each period than is created, until the econ-omy converges back to the steady state. This process can be shown graphically,with the growth rate represented by the difference between the savings and de-preciation lines. Note that the growth rate becomes negative immediately afterthe donation of capital.

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