Economics Section 📈 📉

 if the price was higher than \(P_0\), then many suppliers would be willing to sell the product, thus quantity demanded for supply would be higher than \(Q_0\),  but fewer people would be willing to buy it hence, quantity demanded would be lower than \(Q_0\). That's what the graph shows.

Changes in demand and Supply

On the left,  If a new technology is discovered for example, and it makes producing things easier and faster, then suppliers would produce more quantity, \(Q_1\). but since nothing has changed with the buyers, they'd have to decrease the prices to prevent an excess of stock, hence new equilibrium \(E_1\). It would initially benefit buyers more, according to this model anyway. An increase in tax would have the opposite effect (consumers would pay more for less quantity) and the supply curve would shift to the left.

On the right,  If there is an economic downturn or increase in demand for a rivalry product, the quantity demanded of a product would decline and suppliers would have to supply less which would lead to equilibrium \(E_1\). If the population increases or the economy improves however, the quantity demanded would rise and the demand curve would shift to the right .

So the question now is what is the most efficient combination from this graph? According to this model, the 2 will be more productive if each person focuses on something they have a comparative advantage in (i.e., Henry only does fishing and Martha only collects fire logs), as it would result in less opportunity costs,  this happens at point B and as you can see, B is higher than any point produced by the line which shows when the two share the tasks instead . This model, like many other micro economics models, also applies to firms and economies as well, and it implies a firm or an economy will be overall more productive if we all did what we do best, even when there are other people who can do it better than us, really makes you think huh?.

Consider the following indifference curve between good x and good y. From the graph we can see that if a person wants more of good x, they have to sacrifice good y. At A, to keep the same level of utility, you have to sacrifice a lot of good Y in order to get a little of good X. At B you have to sacrifice a lot of good Y to get a little of good X. This is because when 1 commodity is scarce to a person, they have to pay a lot more to get it. \(U_2(x,y)\) is a utility function which represents higher levels of utility for the same goods.

Slope of indifference curves

The slope of the indifference curve is the marginal rate of substitution (MRS), the slope becomes flatter from left to right. On a single indifference curve, change in utility is 0, hence \(dU=0\rightarrow {dU\over dx}dx+{dU\over dy}dy=0 \) (which means change in utility can happen by either changing good X or good Y or both)  $$dU=MU_xdx+MU_ydy=0$$(MU=marginal utility, the derivative of utility) $$MU_ydy=-MU_xdx$$ $${dy\over dx} ={-MU_y\over MU_x} =MRS$$ At A MRS > 1 and at B MRS < 1. 

When the cost C increases, the IC curve shifts upwards..The further the line is from the origin, the higher the cost.

\( slope={wages\over rent }\) ,a steeper slope implies a higher degree of substitution between rent and wages. Isocost curves help firms determine the most cost effective combination of labour and capital, like how much labour to employ and how much capital to buy.

From the graph we can see that the average product of labour, \( Y\over L \) starts by increasing initially, then it begins to decline as labour as labour increases, why does this happen? This is partly because of the "stepping on toes effect", imagine if there was only 1 employer in a restaurant, the restaurant would barely function or serve any customers, but if there were 5 employees it would obviously be more productive, but if it was 100 employees, most employees would not be much help because there is limited space and limited capital for them to actually do work, they would be a liability, hence the diminishing marginal product of labour . There's also the fact that if output grows, population will grow, hence labour will grow hence average product of labour will decrease in Malthusian(& Thanos) economics.

Relationship between Average Product and Marginal Product of Labour

MP cuts AP at its maximum or minimum. When MP=AP the average doesn't change, when MP>AP then AP is increasing, When MP < AP, then AP is decreasing. The relationship between capital(K) and output(Y) is fairly similar.

The feasible set is all points inside the curve, for more of good Y, we decrease good X. The higher good Y is, the higher the marginal product of good X, meaning the opportunity cost of X increases.   If \(P_X\) increases, the slope \(P_X\over P_Y\) increases, making the curve steeper. The slope \(P_X\over P_Y\) is the marginal rate of transformation, that is, the rate at which good Y is transformed to good X and Visa Versa. If income increases budget line goes up.

The Indifference curve we choose will be the one with the highest possible tangency, hence our Equilibrium will be at \(E_1\), where Marginal rate of transformation=Marginal rate of substitution. Point A is also a possible point, but not the most preferred since it is on a lower indifference curve. If the consumer salary were to decrease, the feasible set would shift down, and the consumer would achieve less utility, as they would have to consume less of both goods, hence the new equilibrium would change to \(E_2\)

Currently in 2024, R27 is the lowest hourly wage in South Africa. This would be the reservation wage, we assume no one will do any work for anything below that. Now, the more effort a worker is putting in, the more you have to increase their wage to get more effort out of them, if their effort is low however, a little increase is enough to get more effort out them. The slope of this graph is also the marginal rate of transformation, i.e., transforming money to effort vice versa.

Shift in best response function.

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If the probability of getting fired were to increase, and the wage remained constant, the worker would put in more effort meaning the curve \(B_0\) would shift to \(B_2\). On the other hand, If the unemployment benefits were increase and wage remained constant, the reservation wage would rise, and the curve \(B_0\) would shift to \(B_1\), hence less effort from workers.

The firm's Isocost curve shows that employer is willing to pay more to get higher effort \(slope={effort\over wage}\) which is the marginal rate of substitution, it shows the amount of effort per wage the firm is willing to pay. The curve \(I_1\) means the business pays more for the same amount of effort, and \(I_2\) means they pay less for the same amount of effort. Combining the firm Isocost and the Workers best response we get:

We find that the Equilibrium in the labour market is found at the tangency point where Marginal Rate of Substitution=Marginal Rate of Transformation . At this point, there is not too much pay from the employers and not too little effort from the workers.

\(y=L(x)=x\) is when a country is perfectly egalitarian, meaning everyone receives the same income. The area between \(y=L(x)\) and \(y=x\) measures how much the income distribution differs from absolute equality.

The Gini index measures the income distribution among the inhabitants of a country, The Gini index is given by \(A\over {A+B}\) from the graph below, where A is the area of inequality and B is the income distribution. \(A\over {A+B}\) = \( {\int_0^1 {x-L(x)}dx}\over{1\over2} \) = \( 2\int_0^1 {x-L(x)}dx \)

Here are top 10 countries with the highest Gini indexes (data source:https://www.statista.com/). For comparison, USA has 39.8% and Russia has 36.0% .

It appears a lot of Southern African countries have high inequalities.

The second equation is the capital accumulation : \( \dot{K}=sY-δK\) , Which means change in capital stock \( \dot{K}\) , equals the amount of investments(sY), minus depreciation(δK).(we assume the only use of investment in this economy is to acquire capital).

To write capital accumulation in per worker terms. We use the method of taking logarithms on both sides then find the derivatives. For capital per worker: $$ k={K\over L} \Rightarrow log(k)=log(K)-log(L) $$ $$ \Rightarrow {\dot{k}\over k}={\dot{K}\over K}-{\dot{L}\over L} $$ and for output per worker: $$ y=k^{α} \Rightarrow log(y)=αlog(k) $$ $$ \Rightarrow{\dot{y}\over y}=α{\dot{k}\over k} $$ $$ \Rightarrow{\dot{k}\over k}={sY-δK\over K}-{\dot{L}\over L} $$ $$ \Rightarrow {sY\over K}-δ-n $$

\({\dot{L}\over L}=n\) because we assume the labour force is constant and change in the labour force is proportional to the population growth n. $$\therefore \dot{k} ={sy}-(δ+n)k $$ is our change in capital per worker. It tells us that change in capital per worker is positively affected by the investment per worker, sy, and negatively affected by depreciation per worker, δk, and population growth, nk.

Solving the Solow model: using output per worker and capital accumulation per worker equations, we find the Equilibrium or steady state where change in capital per worker, \(\dot{k}=0\) $$y=k^{α}$$ $$ \dot{k} ={sy}-(δ+n)k $$ $$\Rightarrow 0={sy}-(δ+n)k $$ $$\Rightarrow sk^{α}=(δ+n)k $$ $$\Rightarrow ({s\over {n+ δ}})=k^{1-α} $$ $$\Rightarrow ({s\over {n+ δ}})^{1\over{1-α}}=k $$ $$\Rightarrow ({s\over {n+ δ}})^{α\over{1-α}}=k^{α}=y $$

$$\therefore y=({s\over {n+ δ}})^{α\over{1-α}} $$

A graphical representation:

The Solow diagram consists of two curves,investment per person, sy  and the amount of new investments per person required to keep the amount of capital per worker constant (n + δ)k . The steady state is where the 2 curves meet, where sy=(n + δ)k. the difference between the 2 curves is change in capital, \(\dot{k}=sy-(n + δ)k\). Suppose an economy begins \(k_0\), then the economy will grow, between \(k_0\) and \(k_1\). but the closer it gets to \(k_1\)(the steady state), the slower it will grow, then eventually stop \(k_1\). So basically the economy converges to the steady state.(that is 1 flaw of the basic solow model however, it does not explain sustained economic growth. the model has many refined variations however, which account for some of the flaws basic solow model. )

Here's one example of the steady state of one of the solow variants:

$$y(t)=({s_k\over {n+ δ+g}})^{α\over{1-α}}({µ\over g}e^{ψµ})^{1\overγ}A(t) $$

According to this steady state, the things that help improve an economy are:

• High levels of investment(\(s_k\)).
• High levels of technology(\(A(t)\))(Technology is the key to sustained economic growth, as long as technology keeps improving, the economy will keep growing, it will not be stuck in the steady state, like with the basic Solow model. According to this model, AI development will help improve our lives... ).
• High numbers skilled labour(µ) (that can effectively use said technology).
• Spending more time acquiring skills\(e^{ψu}\)(education).
• Being an open economy that is able to import(m) capital goods to improve productivity.
• Low levels of capital depreciation(δ)(depreciation is high when there is poor maintenance... *cough* Eskom *cough*).
• Steady population growth rates(g).