Question

4. One of the earliest contributors to mathematical economics was J.H. von Thünen (1783-1850) He believed that his discovery of a formula for "aggregate natural wage income per year" was so signif icant that he had it inscribed on his tombstone! The formula he came up with isw (cy)2, where w is aggregate natural wage income, c is aggregate consumption expenditures, and y is real aggregate in come, i.e., real GDP. Let w be aggregate wage income, and note that all of aforesaid variables are an nual values. In von Thünen's model the real rate of interest is defined by the formula r(w [y-w]/w while aggregate savings s(w) is defined by s(w) w-c def Interpret von Thünen's definition of the real rate of interest. Interpret von Thünen's definition of aggregate savings. (a) (c) Show that von Thünen's formula for aggregate natural wage income, to wit, w' (Cy) is the solution to the problem of maximizing the total return on aggregate savings. That is, show that w (cy)is the solution of the maximization problem max r(w)s(w) Determine the effect of an increase in real aggregate income on aggregate natural wage income, and provide an economic interpretation of the result Determine the effect of an increase in aggregate consumption expenditures on aggregate natural wage income, and provide an economic interpretation of the result Show that the elasticity of aggregate natural wage income with respect to real aggregate income is equal to 0.5 Show that the elasticity of aggregate natural wage income with respect to aggregate con- sumption expenditures is equal to (d) (e) (f) (g)

Answer 1

a) Von Thunen real rate of interest as the percentage difference in real aggregate income and the aggregate wage income. Note that on taking real aggregate income, Thunen is adjusting for the inflation in the economy. His idea is that the total income is higher than what the labor earns as wages. This extra income in economy should account for the interest rate that is generated in the economy. In order to increase the rate of interest, wages of the labor can be decreased or if the motive is to decrease the rate, then wages can be increased. The important assumption is that the labor has nothing other than his labor at his exposure. There is no explicit discussion of capital held by households as done in other models like Solow.

b) Aggregate savings follow the basic idea. The part of the aggregate wages that is left after consumption is termed as aggregate savings. These are the savings of the entire economy and just a particular household. The sum of amounts saved by each household make up for the aggregate savings of the economy.

c) The optimization problem at hand is:

$\underset{w}max \ r(w)s(w)$

Substituting

$r(w)=(y-w)/w\\$and $ s(w)=w-c$, we get:

$\underset{w}max\ \frac{y-w}{w}(w-c) = (y-w)-c\frac{y-w}{w}$

Using First order condition with respect to w:

$\frac{\partial{(y-w)}}{\partial w}-c\frac{\partial{(y-w)/w}}{\partial w}=0$

This gives$-1 -c(\frac{-y}{w^2})=0\\\\ \implies cy=w^2\\ \equiv w^*= (cy)^{1/2}$

In order to ensure that this is the argument that gives maximum, we can check the second order conditions:

$$S.O.C.$: \frac{\partial (-1&plus;cy/w^2)}{\partial w}=-2/w^3 <0$

Hence the S.O.C is satisfied. Hence w*=(cy)^1/2 is the solution to the above maximization problem.

d) The effect of increase in real aggregate income on the aggregate natural income can be found by taking the derivative of w^* w.r.t y

i.e. $\frac{\partial w^*}{\partial y}=\frac{\partial (cy)^{1/2}}{\partial y}=c^{1/2}.\frac{1}{2}y^{1/2-1}=c^{1/2}.\frac{1}{2}y^{-1/2}\\\\ =\frac{1}{2}(\frac{c}{y})^{1/2}>0$

Thus an increase in y, leads to increase in w* but with a lower magnitude than the increment in y. Keeping consumption constant, increase in one unit of y, leads to increase in w* in the proportion of $(\frac{1}{y})^{1/2}$

This implies that all the change in income is not directly transferred into aggregate wages. More the aggregate income in the economy, lower would be the increment in the wages of the labors of that economy. Richer countries have lower scope of increasing the wages of its citizens.