(a) A representative firm produces output according to the simple production function, —simply meaning that is a function of — in which the elasticity of real output ( ) with respect to employment ( ) is , which is constant. Assuming that the firm chooses to maximise profits, , with being the firm’s output price and being the nominal wage (per worker), show that the firm’s choice of satisfies
,
and hence derive an expression for the log of nominal unit labour costs, . Label your expression equation (1).
Hint: to choose a variable to maximise a function of that variable, we differentiate the function with respect to that variable, set the derivative to zero, and then solve the resulting equation (called the first-order condition) for the choice variable. For example, suppose we wanted to choose to maximise . First, we get . Then, we solve for , which gives . Strictly speaking, we also need to check the second-order condition (SOC), which is that the second derivative is negative. (However, you do not have to do this in this assignment.) In this example, the SOC is satisfied because .
(b) Based on your equation (1), in a time-series context, state whether or not (“yes” or “no”) you would expect the pairs of variables listed below to have a cointegrating relationship between them and briefly explain your answers.
i. log of nominal unit labour costs, , and log price, log( ;
ii. log of the real wage, log( , and log of nominal output per person employed, log( ;
iii. log of the real wage, log( , and log of real output per person employed log( .
(Question 2 over page)
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QUESTION 2
For this question, use the data file WS.csv, which contains quarterly time series data (in index-number form) from 1992 quarter 1 to 2017 quarter 1 (inclusive) on nominal unit labour costs (ULC) and the consumer price index (CPI).
(a) Present time series plots of , , , and , where is the log of the ‘wage share’, .
(b) Briefly describe the main features of these time series plots.
(c) Using a maximum lag of 8, test for a unit root in each of , , and . Hence report the order of integration of and .
(d) Using a maximum lag of 8, test for a unit root in . Is the result as you would expect based on your equation (1) from Question 1? Explain why or why not.
(e) Using a maximum lag of 8, determine the appropriate lag length from a levels VAR for a time series model of and . Report this lag length.
(f)* Given your results above, specify and estimate an appropriate VAR/VECM model for and . (g) State what evidence (if any) your estimated model from (f) provides of Granger causality between the series, and suggest any economic mechanism(s) that might explain such causality.
QUESTION 3
(a) Considering your equation (1) from Question 1 as relating to labour input ( ), write down a corresponding equation (2) for capital input ( ) simply by replacing the labour-input terms ( , and ) with their capital-input counterparts, , and , where (a constant) is the elasticity of with respect to and denotes the nominal cost of a unit of capital input.
(b) Use your equations (1) and (2) to derive a cointegrating relationship between the log of unit labour costs and the log of unit capital costs, defined as .
(c) Based on your cointegrating relationship derived in (b), write down an appropriate VECM for (the log of unit labour costs) and (the log of unit capital costs) assuming a lag length of two in the corresponding levels VAR.
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(d) In the context of your model in (c), what are the expected signs of each of the two error-correction terms, and why? Outline the economic/market mechanisms that might explain the implied error correcting behaviour.
APPENDIX QUESTION
(a) Tabulate your regression results for the part marked * as a penultimate-page “Appendix A: Results”.
(b) Present your R code as a final-page(s) “Appendix B: R code”.

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