It is assume the function is Cobb-Fouglas and exhibits constant returns to scalre in K and L. The subscrip t means the varaibles capital, labor and output can change over time.
\(Y_{t} = \bar{A}K_{t}^{1/3}, L_{t}^{2/3}\)
\(C_{t} + I_{t} = Y_{t}\) For simplification, output can only be use for two purposes, consumption and investment. Meaning; if you are not consuming now to increase your utility, then you are investing to consumer more in the future.
Edogenous Basically everything with a subscrip t. T is for time and it means….
\(Y_{t}\) = Output \(K_{t}\) = Capital
\(L_{t}\) = Labor
\(C_{t}\) = Consumption
\(I_{t}\) = Investment
Parameter: Anything with a bar.
\(\bar{A}\) In the Solow Model A bar stand for everything else the model is not predicting. For example, technology, stablishments of institutions, laws, and rights.
\(\bar{L}\) In this model L bar is a constant, but in more complex model labor would be another model in to itself.
\(\bar{K}_{0}\) This is the starting capital in the model.
\(\bar{d}\) is a fraction of depreciation of capital.
\(\bar{s}\) s tans for savings rate and is a fraction of output, which is the fraction invested.
\(Y_{t} = \bar{A}K_{t}^{1/3}, L_{t}^{2/3}\)
\(Y_{t} = C_{t} + I_{t}\) => \(C_{t} = Y_{t} - I_{t}\)
Labor Force
\(L_{t}= \bar{L}\)
Investment Law of motion of capital.
\(I_{t} = \bar{s}Y_t\)
\(I_{t} = \bar{s}Y_{t}\). The Total ouput used for either consumption or investment implies that \(C_{t} = (1 - \bar{s})Y_{t}\)
\(I_{t} = Y_{t} - C_{t}\) because of the rule of if you are not consuming you are investing. What would happen is investment goes up: Savings feat in in the saving line. The saving rate increases the line would increase and push out the steady state of single capital per worker.
Resource Constrain
\(C_{t} + I_{t} = Y_{t}\)
To solve, write the endogenous variables as function of the parameters of the model. First, Combine the investment allocation equation with the capital accumulation equation.
\[\Delta{K_{t}} = \bar{s}Y_{t} - \bar{d}K_{t}\]
For Example: \(Y_t = \bar{A}K^{3/7}L^{4/7}\).
The steady state of the Solow model is where capital acomulation is zero. Start with the equatin below.
\(\Delta{K_{t}} = I_{t} - \bar{d}K_{t}\). We know \(I_{t} = \bar{s}Y_t\). Then we plug in \(\bar{s}Y_t\) for \(I_{t}\) and we get. \(\Delta{K_{t}} = \bar{s}Y_t - \bar{d}K_{t}\). We know what \(Y_t\) is. \(Y_t\) is \(\bar{A}K^{3/7}L^{4/7}\). So we get:
\[\Delta{K_{t}} = \bar{s}\bar{A}K^{3/7}L^{4/7} - \bar{d}K_{t}\].
After we have the last equation, we set the left hand side of the equation to zero. \(0 = \bar{s}\bar{A}K^{*3/7}L^{*4/7} - \bar{d}K_{t}^{*}\) and we solve for K, which will turn into K*.
Production Function
\(y^{*} = \bar{A}k^{1/3}\)
Steady State
\(\bar{s}\)
present discounted value.
\[PV = \frac{FV}{(1+i)^n}\] 7.2 the us labor market. two importaint ratios: 1. the ratio to employment to the population: 2. the unemplyment rate: is the fraction of the labor force that is unemployed. A person is said to be unemployed if she doesn’t have a job that pays a wage or salary, she has actively looked for such a job during the 4 weeks before the rate was measured, and she is currently available to work.
\[Unemploymnet rate = /frac{numberofunemployed}{the labor force}\]. gross flows: the creating and destruction of jobs. 7.3. supply and demand.
Achange in the labor supply: labor income tax: is (1-t)w. Income, population, work leisure preference, prices of related goods and services, and expectations about the future. Changesin social norms. Changes in tchnology for managing fertility.
A change inthe labor demand are cause by prices of inputs like oil. Regulation making harder for employers to fire employees, change in quantity demanded of the product that the labor produces. Reduce descrimination gainst women.
in the short run the employment rate would stay highg but in the long run employers would be relungtant to hire workers.
Wage Rigidity This is in he short run. Bcause of wage contract. When wages fail to adjust to clear the labor market. Wages remains unchaged at its original level. If wages are rigid for some reason and don’t fall to clear the labor market, the result is an even larger reduction in employment. Also, labor supply exceeds labor demand which means people would like to work than are able to find jobs.
Different kinds of unemployment: The natural rate of unemployment is the rate that would prevail if the economy was in neither a boom nor a recession.
Cyclical unemployment is the difference between the actual rate and the natural rate and is associated with short run fluctuations, such as occur in booms and recessions.
The Natural Rate of Unemployment is broken down into two smaller parts: frictional & structural.
Frictional unemployment is when workers are changing jobs. Job Searching. Changing sectors. Aquiring skills.
structural unemployment results from the labor market institutions that match up workers and firms in the labor market. Is going to put a gap between labor supply and laor demand. Minimum wage would do thisand union workers. Increasing wages.
Actual unemployment = (frictional + Structural) + cyclical.
Bathtub model \[E_t + U_t = \bar{L}\]
Bathtub model.
\[\Delta{U}_{t+1} = \bar{s}E_{t} - \bar{f}U_t\] \(\Delta{U}_{t+1}\) = Change in unemployment.
\(\bar{s}E_{t}\) = Employed people who lose their jobs.
\(\bar{f}U_t\) = Unemployed people who find new jobs.
\(\bar{s}\) = job separation rate. the percentage of employees who left the organisation during the reporting period.
\(\bar{f}\) = The probability a job seeker will find a job in a given month. Job finding rate.
\(E_{t} = \bar{L} - U_t\): Employed people is equal to the labor force - unemployed people.
There is a steady state and is where the change in unemployment equals 0, \(\Delta{U}_{t+1} = 0\).
To solve: \(\Delta{U}_{t+1} = \bar{s}E_{t} - \bar{f}U_t\) \(0 = \bar{s}E_{t} - \bar{f}U_t\)
\(0 = \bar{s}(\bar{L} - U_t) - \bar{f}U_t\). Take out the \(U_t\).
\(0 = \bar{s}\bar{L} - (\bar{f} + \bar{s})U_t\). Solve for \(U_t\).
\(U^* = \frac{\bar{s}\bar{L}}{\bar{f} + \bar{s}}\)
Change it per capital: This would give you the natural rate of unemployment.
\[u^*\equiv \frac{U^*}{\bar{L}} = \frac{\bar{s}}{\bar{f}+\bar{s}}\]
The only way to change the natural rate of unemployment is to change the job finding rate \(\bar{f}\) or the job separation rate \(\bar{s}\). Job separation rate can be change by policies which makes employers harder to fire people.
Present discount value: \[PV = \frac{FV}{(1+i)^n}\]
If there is a string of payments every time period use this formula:
PV = FV * (1-[1/(1+i)]) / (1 - (1/(1+i))).
Example: Annual w = 63,000. You work for 45 years. interest rate at 3%.
63000 * (1-((1/(1+.03))^45)) / (1 - (1/(1+.03)))## [1] 1591019The supply and demand of college workers has increase. The demand has had a greater increase that offsets the supply of workers, thus wages for college workers has also increase. This is due to the skill biased technical change. For example, the use of technology, that can make people more productive.
The labor market is arguably the most imporatint market in an economy. The tools of supply and demand allow us to understand the basic changes in the labor market that have occurred in the US since 1950, in cluding the increase in the employment population ratio and the rising return to education.
The labor markets are typically characterized by large quantities of job creatin and job destruction that result in much smaller overall changes in employment. Most unemployed workers find new jobds relatively quickly in the US, and most unemployment is accounted for by people out of work for long spells.
Adverse shocks (like the oil shock and productivity slowdown of the 1970s) as well as inefficinet labor market institutions appear to play important roles in explaining the relatively high unemployment rates and low hours worked in Europe.
Because the labor market is so important, problems like unemployment merit serous responses by society. Designing the right safety net requires balancing the needs for social insurance against the disincentives associated with teh insurance.
Present discounted values help us compoare fiancial payments recived at different times.
The rising return to a collage education is one of the key facts about the labor market. This college wage premium has risen from about 50 percent in 1963 to around 100 percent i recent years. Another way of viewing this fact is that wage inequality between college graduates and high school graduates has increased, mirroring a broader increase i income inequality. Possible explanations include skill biased technical change and globalization.
CH7. Questions: what does it mean by wage rigity, and can you give us some examples
CH8