Question

The following problem is based on the idea of a Malthusian trap. Thomas Malthus, an 18th century British cleric and scholar, argued that as population increases, the limited amount of natural resources will lead societies into a trap of gradually decreasing standard of living, thus negating the effects of any technological progress. We can study this idea using the Solow model framework.

Consider a modified version of the Solow growth model where the aggregate production function in period t is

Yt = F(Kt,Lt,Z) = (Kt^α)(Lt^β)(Z^(1−α−β))

Z represents land and is available in fixed inelastic supply given by nature. Assume that α + β < 1, capital depreciates at the rare δ, the exogenous savings rate is s, and the population growth rate is n. (i.e. everything else is the same as the basic Solow Model with population growth)

(a) Show that the production function exhibits constant returns to scale in (K,L,Z), but decreasing returns to scale in (K, L) only.

We are going to transform the model in per worker form. Let yt = Yt/Lt , kt = Kt/Lt , zt = Z/Lt , it = It/Lt , and

ct = Ct/Lt

(b) Note that the available land Z is fixed, but the amount of land per worker is not. What happens to land per worker zt over time? What is the growth rate of zt?

(c) Write down the equations for output, investment, consumption, and the capital accumulation equation. Then transform all equations in per-worker form (hint: all 5 variables - capital, output, consumption, investment, and land must be in per-worker form).

(d) Use the capital accumulation equation you obtained in (c) to discuss how capital per worker kt will evolve over time. Draw the investment and depreciation (break-even investment) curves. What happens to the investment curve over time? Is there a steady-state?

(e) What happens to capital, output, consumption, investment, and land per worker over time? Find k∗,y∗,c∗,i∗,and z∗ as t→∞.

(f) Derive an expression for the real wage as a function of capital per worker and land per worker only. What happens to the real wage as t → ∞.

(g) Do your answers from parts (e-f) confirm or reject the idea of a Malthusian trap? Briefly discuss your results.

Now suppose that advances in agriculture and technology allow us to use land, labor, and capital more efficiently. In the most general case we can measure those changes in technology by total factor productivity. Let At be total factor productivity. The production function becomes:

Yt = F(Kt,Lt,Z) = (At)(Kt^α)(Lt^β)(Z^(1−α−β))

(h) What must the growth rate of total factor productivity △At/At be equal to in order to negate the effects of the Malthusian trap (i.e. in order to have a standard-looking Solow steady-state with k∗ > 0)?

Hint: Recall that △(xt^a)/xt^a = (a)(△xt/xt) - the growth rate of a variable raised to a power is equal to the power times the growth rate of the variable

(i) Given your answer to (h), is the Malthusian trap something we should worry about in the real world?

Answer 1

a) Here production function indicates (Kt^α)(Lt^β)(Z^(1−α−β))c

A production function F has constant returns to scale if, for any > 1, F ( z1, z2) = F (z1, z2) for all (z1, z2). If, when we multiply the amount of every input by the number , the factor by which output increases is less than , then the production function has decreasing returns to scale . The easiest way to find out if a production function has increasing, decreasing, or constant returns to scale is to multiply each input in the function with a positive constant, (t > 0), and then see if the whole production function is multiplied with a number that is higher, lower, or equal to that constant.

b) Here since the land is fixed but the rate cannot be fixed in all the instances .The per worker production function attempts to model how much a single employee will produce based on either land available or capital invested. The Malthusian model bases it solely on land while the Solow model bases it on only capital.

c) Output equations assign the result of a measurement to a variable, which you can use in other equations just like other variables. A project can include multiple Output Equations documents, each of which can contain multiple output equations and standard equations.

Filters 11=Filter IS(1.1)I

Filters 21=Filter:IS(2,1)I

Filters 22 =Filter:IS(2,2)I

Output filters 21/(1-filters22*filters11)

Equations of investment

Return on investment formula=Net profit/cost of investment*100

I consumption equation C = C + bY shows that consumption (C) at a given level of income (Y) is equal to autonomous consumption (C) + b times of given level of income. ADVERTISEMENTS: Calculate consumption level for Y = Rs 1,000 crores if consumption function is C = 300 + 0.5Y.

capital accumalation equation =  K' = (1–d)K + sY.

d) Capital accumulation can be calculated by measuring:

1. Change in wealth/value of assets in an economy.
2. Level of gross fixed capital formation – depreciation.

Capital per unit of effective labor evolves over time. In particular, it says that when actual investment (sk(t)α) is higher that required investment ((n + g + δ)k(t)), capital per unit of effective labor would be rising. Actual investment is the new investment in the economy.

Investment and Depreciation of Fixed Assets Grossfied capital formation and consumption of fixed assets in 2009 prices 40.000 30,000 20,000 10,000 1995 1996 2000 2002 2004 2005 2006 2010 Source: Central Statistics Office

In an economic sense, an investment is the purchase of goods that are not consumed today but are used in the future to create wealth. In finance, an investment is a monetary asset purchased with the idea that the asset will provide income in the future or will later be sold at a higher price for a profit.

Depreciation is a non-cash expense that reduces the value of an asset as a result of wear and tear, age, or obsolescence over the period of its useful life.

The break-even point (BEP) in economics, business—and specifically cost accounting—is the point at which total cost and total revenue are equal, i.e. "even". There is no net loss or gain, and one has "broken even", though opportunity costs have been paid and capital has received the risk-adjusted, expected return. In short, all costs that must be paid are paid, and there is neither profit nor loss.

e) Here K*Y*C*I =CY

This is the loss of capital equipment due to depreciation. Depreciation can occur due to the machines wearing out, getting lost or breaking down. Capital can also become obsolete through advances in technology. Capital consumption can also occur due to a shift in demand.

A positive occurs when actual output is greater than potential output. This will occur when economic growth is above the long run trend rate (e.g. during an economic boom). It will involve firms asking workers to overtime. With a positive output gap, there will be inflationary pressures.

Higher interest rates are thought to affect consumer spending through both substitution and income effects. Higher interest rates lower consumption through the substitution effect, because current consumption becomes expensive relative to saving--households reduce their spending today in favor of spending tomorrow.

Inflation occurs when the supply of money increases relative to the level of ... With this idea in mind, investors should try to buy investment .

f) The quality of capital per worker is a measure of how much capital exists in an economy and how good that capital is. Imagine, for example, the difference between an economy where bakers are using wooden spoons to mix their cakes and one in which they use electric mixers.The steady state is defined as a situation in which per capita output is unchanging, which implies that k be constant. This requires that the amount of saved output be exactly what is needed to (1) equip any additional workers and (2) replace any worn out capital.

The per worker production function attempts to model how much a single employee will produce based on either land available or capital invested . . As you increase the land for the employee to work in the Malthusian model or the capital invested in the employee in the Solow model, the worker's productivity increases.

Agricultural land per worker (hectares) in India was reported at 0.74149 in 1994, according to the World Bank collection of development indicators .

g) Here Yt = F(Kt,Lt,Z) = (At)(Kt^α)(Lt^β)(Z^(1−α−β))

Malthus argued that society has a natural propensity to increase its population, a propensity that causes population growth to be the best measure of the happiness of a people: "The happiness of a country does not depend, absolutely, upon its poverty, or its riches, upon its youth, or its age, upon its being thinly, or fully inhabited, but upon the rapidity with which it is increasing, upon the degree in which the yearly increase of food approaches to the yearly increase of an unrestricted population

Well-functioning factor markets (markets used to exchange the services of land, labor, and capital) are an essential condition for the competitiveness and sustainable development of agriculture and rural areas. At the same time, the functioning of the factor markets themselves is influenced by changes in agriculture and the rural economy. Such changes can be the result of progress in technology, globalization and European market integration, changing consumer preferences, and shifts in policy. Changes in the EU’s Common Agricultural Policy (CAP) over the last decade have particularly affected rural factor markets.

h) Total factor productivity is a measure of economic efficiency and accounts for part of the differences in cross-country per-capita income. The rate of TFP growth is calculated by subtracting growth rates of labor and capital inputs from the growth rate of output.

so k X(t -k) (7) k=1 Then we have the following expression for △E(t): △E(t)= ... d 2X()tWkX(t-k) (8) k= k= Note that △X(t - d) = X(t + 1 - d) - X(t-d)(d = 0,1,2,......, n). ... d 2XT(t)Wk△X(t-k) k=1 k=1 =ηT(t) Ө η(t) Where ηT(t)={△XT (t),△XT (t - 1), ..X T

i) The Malthusian trap or population trap is a condition whereby excess population would stop growing due to shortage of food supply leading to starvation.

Research indicates that technological superiority and higher land productivity had significant positive effects on population density but insignificant effects on the standard of living during the time period 1–1500 AD.In addition, scholars have reported on the lack of a significant trend of wages in various places over the world for very long stretches of time.[2][10] In Babylonia during the period 1800 to 1600 BC, for example, the daily wage for a common laborer was enough to buy about 15 pounds of wheat. In Classical Athens in about 328 BC, the corresponding wage could buy about 24 pounds of wheat. In England in 1800 AD the wage was about 13 pounds of wheat.n spite of the technological developments across these societies, the daily wage hardly varied. In Britain between 1200 and 1800, only relatively minor fluctuations from the mean (less than a factor of two) in real wages occurred. Following depopulation by the Black Death and other epidemics, real income in Britain peaked around 1450–1500 and began declining until the British Agricultural Revolution.Historian Walter Scheidel posits that waves of plague following the initial outbreak of the Black Death throughout Europe had a leveling effect that changed the ratio of land to labor, reducing the value of the former while boosting that of the latter, which lowered economic inequality by making employers and landowners less well off while improving the economic prospects and living standards of workers. He says that "the observed improvement in living standards of the laboring population was rooted in the suffering and premature death of tens of millions over the course of several generations." This leveling effect was reversed by a "demographic recovery that resulted in renewed population pressure.