EC3010 Assignment 2016-2017
Laragh Kenny
1. A Simple Real Business Cycle Model
Output in time t is given by:
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Where; is total consumption, is investment, is total factor productivity, is total hours worked, and is the capital stock, which evolves according to;
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1.1 Firms
Defining the real wage and rental price of capital as and , show that firms will hire labour and capital such that;
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The representative firm is perfectly competitive and produces output by renting capital and labour from the household sector. They operate under the consumption function:
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Firms face the static problem of maximising profit π, subject to wage rate and the rental cost of capital .
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In order to solve this maximisation problem for π, we find the partial derivative of Y with respect to L and K.
This yields the following expressions for the real and the rental price of capital:
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1.2 Households
Assume an infinitely lived representative household that seeks to maximise;
Where; , where and are positive constants.
The budget constraint is:
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where is the depreciation rate.
1.2.1 & 1.2.2 Consumption & Labour Supply
Show that;
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Interpret this equation.
Show that;
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Interpret this equation.
In order to determine consumption in the next period, the relationship between consumption and labour supply, and the relationship between wages and labour supply, we will maximise the representative household’s utility function, , subject to the budget constraint.
We use the Lagrangian method to maximise this problem.
First, we define the Lagrangian function:
From this we derive the following three first order conditions (F.O.C.):
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Now we let each partial derivative equal o, and solve for for each case. In the order this yields;
1)
2)
3)
To derive the consumption Euler equation, we substitute in for from Equation 1 into Equation 3.
We then move Equation1 forward by one period (i.e.. t+1), and substitute for in Equation 3.
By rearranging, we produce the expression:
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This expression describes the amount of consumption that the household will have gained in the next period (t+1) as a result of foregoing consumption in the present (t). This describes how much one unit of consumption would be worth to the household in period (t+1) in comparison to period (t). This equation, the Euler equation, is the fundamental equation that drives consumption versus savings models.
If the household decides to give up one unit of consumption in period (t), that unit will be made less valuable by the general discount rate: , but it will be made more valuable by R as the household is able to rent this unit of capital.
The overall effect of the household’s utility of foregoing consumption from one period to the next is ambiguous. It depends on the discount rate, the rate of depreciation of capital and the rental cost of capital available to the household.
The optimum condition is that in which your impatience rate is equal to the discount rate, i.e. impatience is perfectly compensated. The price of consuming today is the interest rate.
To derive the relationship between consumption and labour supply, we combine Equation 1 and Equation 2. This can be rearranged to give the expression:
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This expression indicated the condition that equalises the wage rate and the marginal rate of substitution (MRS) between consumption and leisure. As the household has a unique b-value describing their relative preference for labour versus leisure, the household knows what their wage (W) is at the time (t). They are free to choose the optimal mix of consumption and leisure that corresponds to his wage and his quantifies preference b.
To derive the expression for the relationship between wage and labour supply, we will take Equation 3 and substitute Equation 2 into it twice. First, it directly replaces with . Next, by moving Equation 2 forward by one period to t=2 and substituting the result in for in Equation 3.
This can then be rearranged to yield the expression;
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This describes the rate at which a change in the wage rate from one period to the next changes the amount of leisure time available to the household. The equation tells us that if the wage rate increases in (t=2) above what it was in (t=1), then the household will work more in period (t=2) than they did before, now that it is more rewarding to do so. In doing this, the household will be devoting less time to leisure by working more.
As the wage rate increases, the opportunity cost of leisure also increases. The household is willing to work more because they are being paid more to do so. The effect of a change in W on (1-L) is not one-to-one. It is mediated by the discount rate, the rental cost of capital and the depreciation rate.
If R is high, then the household will be less inclined to work more, even with higher wages. This is because it would decrease the importance of wages to their overall income, relative to rent. The discount rate has exactly the opposite effect, making capital less valuable, thereby increasing the effective value of the household’s wages.
1.3 Market Clearing
For the rest of this question, assume so capital completely depreciates after one period of use.
1.3.1 Consumption
Show that the savings rate ( is given by:
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and explain why the only constant savings rate is equal to (bonus points for explaining why this is the only savings rate);
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So what does this imply about the volatility of consumption?
Using the expression for the savings rate, we bring it forward one period to;
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We assume that, and substitute into the equation for .
When and defines .
We then proceed to show the savings rate across periods;
• , because
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The final equation above stated is our expression for the relationship between savings rtes across periods. This is conditional on the assumption that;
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Taking the two cases, where and , we can explain why the savings rate must be constant and equal to . The first case would imply that , therefore increases at an increasing without bound as periods pass. Eventually, , contradicting the definition of the savings rate, . Therefore, is an impossible situation. Eventually , and similarly, is impossible.
We can conclude that . If this is the case, then by the above equation, as well. Therefore, the savings rate is constant. Households can now carry over capital from the previous period, as they are less obliged to invest in the current period and can consume more. In effect, is now smaller, i.e. the savings rate will decrease.
1.3.2 Labour Supply
Show that;
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Interpret this equation.
From the previous part of the question, 1.3.1, we know that .
If we rearrange the second two expressions in the equation:
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Additionally, we take the expression;
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Subbing in
Yielding
Then we solve;
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We may interpret the labour supply as a function of labour’s portion of income, the discount rate and the fraction of available time that each household spends working.
The labour supply is decreased by . If capital does not fully depreciate from one period to the next, then each household will have a certain amount of capital in (t+1) that was saved it (t). the extra capital means that the household will have to work less in order to accommodate for their desired consumption in (t+1), relative to the situation in which capital fully depreciates. The supply of labour is therefore smaller when capital does not fully depreciate.
2. A (Slightly) More Complicated RBC Model
Solving the model from the previous question with is more complicated as now savings depends on labour and capital, as capital does not full depreciate. Therefore, the labour supply and the savings rate are not constant, but depend on history and expectations of the future. To solve the model, we assign values to the parameters, linearize the model and use a computer algorithm to solve it.
The model has eight variables (output, consumption, investment, capital, labour, productivity, wages and interest rate) and eight equations:
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The last equation is the law of motion for productivity, where are shocks to productivity at time (t) and represents the persistence of the shocks. is a normally distributed random variable with mean zero and standard deviation .
We choose what are considered reasonable parameter values for quarterly data, drawn from the available evidence (some microeconomic and some macroeconomic).
Defining lower case variables as the percentage deviation of the variable from steady state, these parameters imply that (to a first-order approximation):
2.1 Impulse Responses
Assume the economy starts in steady state ( and ) and is hit by a one-time productivity shock of 1% ( for all . Use equations 1 and 2 to show graphically how the economy responds to the shock over the following 100 periods (equivalent to 25 years). Provide some intuition for these results and in particular try to explain the difference that makes compared to the previous question.
We model the scenario in which the economy is in steady state at (t=0) and is hit by a one-time 1% positive productivity shock. We assume that technology shocks take a first-order autoregressive form with a 0.95 persistence coefficient, and we will use the impulse-response functions to evaluate the shock process for each variable, over 100 periods (25 years). This autoregressive characteristic implies that our one-time productivity shock will continue to have an effect on productivity in the economy over subsequent periods, beginning at 0.01 and steadily decreasing by a factor of 0.95.
Responses over the 100 periods of each variable to the technology shock. Time is on the x-axis and change in the given variable from the steady-state equilibrium is on the y-axis.
Leisure, like consumption, is a normal good. The shock produces a small income effect in labour supply. This can be interpreted as an upward shift, and to the left in the labour supply curve. Agents will demand a higher wage for the same amount of labour. Simultaneously, the productivity shock raises labour productivity, shifting the labour demand curve upward and to the right. Employers will pay a greater wage for the same employee as his output is now greater. The positive effect on wages is ambiguous as per the graph.
The net effect on employment is ambiguous. The labour supply shifts downwards and labour supply upwards, and it is shown in the graph that labour increased in response to the shock. Therefore, it is clear that the relative magnitude of the labour demand effect is greater. The substitution effect associated with the shock was greater than the income effect, as there was greater incentive to work hard when productivity was higher than the wealth effect on demand for leisure.
A further effect of this greater substitution effect is that the immediate increase in labour is greater than the increase in consumption.
Households smooth the immediate increase in consumption by increasing savings and building future capital reserves. Investment increases at the shock before decreasing, and falls below the steady state during the transition back to the steady level. Capital in the shock has been predetermined, therefore is unaffected in the shock. The increase in investment in the hock causes an increase in capital reserves in the periods that follow, before returning towards steady state levels.
Unemployment increases in the shock period, therefore while capital does not, there is a sharp increase in the real interest rate period. Capital reserves quickly grow after the shock.
ln terms of the aggregate effect on output caused by the productivity shock, the net effect on y is positive and output increases in the shock period. The positive impact shock in productivity with an expansion in labour supply increase output in the economy. Y continues to increase over the periods that follow the shock at a decreasing rate.
According to the RBC model. An economy that witnesses a one-time positive productivity shock will enjoy a simultaneous increase in output, labour, consumption and investment. The increases in these variables are persistent, reflecting the persistence of technology shocks in the model economy.
2.2 The Model and the Data
Use the calibrated model (equations 1 and 2) to see whether this version of the RBC model matches the stylized business cycle facts from the lecture and the textbook. You can ignore the ones related to imports and exports and nominal variables, as this version of the model is a closed economy without monetary effects.
One way to do this is use Excel’s random number generator to draw a long sequence of shocks (perhaps 10,000 periods) from the normal distribution with mean zero and standard deviation of 0.0032. Use these values for and equation1 to generate a time series for . Then use this time series forand equation 2 to generate time series for the other variables. Assume the economy starts in steady state ( and ) and discard the first 100 observations in the generated time series s that the starting values might be thought of as reasonable. Use the generated time series for output etc. to calculate variances and correlations and compare these with the stylised facts.
The model was simulated over 10,000 periods and was subject to a randomly generated shock value in each period. The shocks took on mean, 0, with SD of 0.0032. We assume that when simulated over 10,000 periods, the values for each variable in the model can be interpreted as average models. We can then compare the nature of the modelled variables and relationships between them to the actual values that were observed in real economies. It can then be determined how well the model reflects the actual observed business cycle.
Stated in the tables below are descriptive statistics for output, labour and investment. Also included is a correlation table describing the correlations between each variable. The statistics refer to the simulated model data. The first 100 observations were removed in calculating these statistics.
Investment (I)
Mean
-0.000496083
Standard Error
0.000381557
Median
0.000335844
Mode
#N/A
Standard Deviation
0.038155722
Sample Variance
0.001455859
Kurtosis
0.134604899
Skewness
-0.091373394
Range
0.293426098
Minimum
-0.149276126
Maximum
0.144149972
Sum
-4.960826999
Count
10000
Confidence Level(95.0%)
0.000747929
Income (Y)
Mean
0.001349486
Standard Error
0.000167429
Median
0.001148811
Mode
#N/A
Standard Deviation
0.016742944
Sample Variance
0.000280326
Kurtosis
-0.034500283
Skewness
0.118105768
Range
0.111352983
Minimum
-0.049159477
Maximum
0.062193506
Sum
13.49486069
Count
10000
Confidence Level(95.0%)
0.000328195
Labour (L)
Mean
-4.5482E-05
Standard Error
5.13662E-05
Median
-0.000128236
Mode
#N/A
Standard Deviation
0.005136617
Sample Variance
2.63848E-05
Kurtosis
-0.006954499
Skewness
0.037635683
Range
0.037834737
Minimum
-0.018972679
Maximum
0.018862057
Sum
-0.454820487
Count
10000
Confidence Level(95.0%)
0.000100688
A
K
Y
C
L
R
W
I
A
1.000
K
0.672
1.000
Y
0.989
0.798
1.000
C
0.839
0.971
0.910
1.000
L
0.824
0.131
0.733
0.345
1.000
R
0.550
-0.329
0.318
-0.096
0.898
1.000
W
0.914
0.918
0.965
0.984
0.523
0.076
1.000
I
0.957
0.468
0.904
0.629
0.945
0.749
0.759
1.000
Looking at our stylised facts;
Investment is much more volatile over the business cycle than GDP, typically 3-4 times as volatile. Output has fluctuated either up or down in a business cycle, investment has on average registered a similar fluctuation.
Employment is considerably less volatile over the business cycle than GDP, typically 60-80%. While the business cycle supports the data in the GDP is volatile than unemployment, the model found labour to be far less volatile than expected. While data refers to employment, our model refers to labour, which represents both the rate of employment and hours worked. It is possible that hours worked were less volatile than GDP, thus exerting downward pressure on the aggregated volatility of labour.
Consumption and investment are strongly positively correlated with GDP. Our model implies that GDP, consumption and investment all move in the same direction as the business cycle fluctuates, so that they almost share a linear relationship. There are very positive correlations, however there are unobserved variables that influence some or all of Y, C and I in the real economy.
Employment is pro-cyclical and much more strongly correlated with GDP than real wages and productivity. Productivity tends to be pro-cyclical, whereas real wages tend to be very weakly correlated with GDP. While labour is pro-cyclical, wages and productivity are both very strongly positively correlated with output and are more correlated than labour. A reasonably strong positive correlation between labour and output will require an increase in labour input. In our model, productivity shocks are the only source of economic fluctuation. In a real economy, there will be numerous factors independent of productivity that will influence fluctuations in output, such as population growth, population demographics and international trade. Correlation between output and productivity can be expected to be exaggerated in our model. By acknowledging this correlation between productivity and wages, we can explain the exaggerated correlation between output and wages in the model.
Employment is a lagged variable, implying in the next period a stronger correlation will be exhibited with current output than contemporaneous correlation with output. A change in value of output in the economy will have greater effect on that variable for the next period than in present period. In the RBC model, there is no price stickiness in the economy. If labour was observed to be a lagging variable, some wage stickiness may be implied. If our model does not account for any wage stickiness, then it will fail to reproduce labour as a lagging variable, as we have seen.
The model accurately reflects the volatility of investment and output over the business cycle. It correctly identifies lower volatility in labour relative to output. It correctly identifies a strong, positive correlation between output and all of consumption, investment and labour and the high degree of persistence in output, consumption and labour. The model does not accurately reflect the fact that productivity and wages are not only weakly correlated with output, as the model only has a narrow definition of factors that can affect GDP. The model failed to identify employment as a lagging variable.
3. Real Business Cycles
Discuss the main ideas and assumptions underlying the theory of real business cycles as well as the arguments for and against the approach.
Business cycles are a type of fluctuation found in the aggregate economic activity of nations that organise their work mainly in business enterprises. A cycle consists of expansions occurring at the about the same time in many economic activities, followed by similarly general recessions, contraction and revivals which merge into the expansion phase of the next cycle. This sequence of changes is recurrent but not periodic. In duration business cycles vary from more than one year to ten or twelve years; they are not divisible into shorter cycles of similar character with amplitudes approximating their own.
We can study the eleven facts of the stylised business cycle to understand what happens during business cycle. Looking at the first three facts, we know that;
1. Investment is much more volatile over the business cycle than GDP
2. Foreign trade volumes are also more volatile than GDP
3. Employment is considerably less volatile over the business cycle than GDP, typically only 60-80% as volatile.
By using these facts, we study whether, and to what extent, the cyclical components of the economic variables move in the same direction of GDP.
4. Private consumption, investment and imports are strongly positively correlated with GDP. Employment is pro-cyclical and strongly correlated with GDP.
5. Labour productivity tends to be pro-cyclical, whereas real wages tend to be very weakly correlated with GDP.
6. In most countries, inflation is positively correlated with GDP, although the correlation is not very strong.
7. Employment is a lagging variable; inflation and nominal interest rates also tend to be lagging variables.
To study the degree of persistence in the economic variables, we compute the coefficient of autocorrelation.
8. There is considerable persistence in GDP and about the same degree of persistence in private consumption.
9. Employment tends to be even more persistent than GDP. Total labour input varies in a pro-cyclical manner, explaining most of the variation in the output gap.
10. Total factor productivity also varies pro-cyclically. The cyclical component of total factor productivity accounts for a large fraction of the total output gap at business cycle peaks and troughs. Most of the cyclical variation in total labour input stems from fluctuations in cyclical unemployment.
11. Average working hours, and to some extent the labour force, also vary periodically.
The Real Business Cycle (RBC) theory attempts to model the business cycle. According to Kydland and Prescott (1982), the business cycle arises out of random fluctuations, as known as shocks, in technology and their impact on the effectiveness of labour and capital. These shocks change the behaviour of buyers and sellers in the market, effectively determining output in the economy.
The basic RBC model follows two main assumptions; the first that fluctuations are accounted for as real rather than nominal shocks, and that individuals and firms respond to these shocks optimally.
The business cycle is mainly driven by fluctuations in the rate of productivity growth. A recession can consequently be interpreted as a sequence of negative productivity shocks, but to which agents have responded optimally and produced the best possible outcome. The employment fluctuations observed during business cycles reflect voluntary movements along individual labour supply curves. RBC model give primacy to technology shocks as the source of these fluctuations in the economy however.
Increases in technology induced extra output through higher capital accumulation and by inducing people to work more. As Kurlat (2013) describes, if people aren’t working because there is a recession, for example, this is an efficient response to the fact that productivity is low in time period ‘today’, i.e. t=0, as people take time off when productivity is low and work harder when productivity is high. The RBC also fails explain the labour market response to technology shocks. The hours worked tends to decline after a positive shock; this goes against the predictions of the RBC model. The high elasticity of labour supply is important in the theory of the RBC, but also in the argument that taxes have large effects.
To explain business cycles, there is no need to postulate nominal and/or rigidities. Due to the fact that RBC models exhibit complete monetary neutrality, so there is no role for monetary policy. Monetary and fiscal policy have no capacity to stabilise the short-run economy under RBC theory, which would consequently lead to minimal government intervention in the market activity of an economy.
The RBC is often criticised, but can also be useful as a baseline model. RBCs follow the theory that economic growth and business cycles can and should be explained with a unified model framework. The basic idea of RBC model is to take a simple model on optimizing agents and see how well it matches the data. The RBC is an important advance in founding business cycle on the behaviour of individuals. The most simple model, the Solow Model, shows perfect competition and describes the behaviour of a set of completely optimising rational agents. This also implies that the labour market clears efficiently, i.e. that labour supply and labour demand are always balanced and in equilibrium. However, the economy cannot always be deemed as perfectly competitive and people are not always rational in their decision-making.
The RBC should be seen as a benchmark, against which more complicated models can be assessed. The RBC can be seen as an effective alternative to the Keynesian system (Whelan, 2016) and was built around the understanding that agents behave optimally, generating real supply and demand curves as a result that are unaffected by nominal prices and money income. The assertion that nominal variables, such as prices, are irrelevant to the business cycle is central to RBC theory. Future wealth has a positive effect on the consumption of households and decreases hours worked as well as decreasing investment. If monetary policies have no effect, i.e. if prices are sticky and the labour market clearing assumption does not hold, then monetary policy does have a role in the stabilisation of the short-run business cycle and the basic theory of the RBC does not hold. It is possible that alternative explanation to these criticisms of the model lies in the possibility that some market imperfections could be missing from the RBD model.