(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)
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.
(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.
(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.
(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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