1 . We want to estimate, from output data. the output elasticities of capital and labor. For this a suitable candidate is the following Cobb-Douglas production function y = (i 1 . . . ,
where output (1 ) is produced through the combination of low and high skilled labour(L and H) and capital (K). Total factor productivity. depends on whether the firms is located in a city (C = 1) and a random shock (). ß, ‘, , O(- O are parameters.
(a) [5 marks] Using your own words describe what does account for.
(b) [5 marks] Show that one can transform the production function in such way that it is possible to perform OLS estimation.
(c) [10 mars] Suppose that = ‘y = O = O. write the OLS objective function and
derive the OLS estimator ( and Ô1j) of n and t9,. Show your steps.
(i) [5 marks] Give conditions on the distribution of íj such that the OLS estimator is unbiased
(d) [5 marks] Using your data compute mean. standard deviation, minimum and maximum for output, high skilled labour. low skilled labour and capital. Provide these in a table and report the number of observations in your sample.
(e) [5 marks] Produce and provide a scatter plot of your transformed dependent and the log of capital. Does output increase with capital? (For full marks plots must be properly labelled.)
(f) [10 marks] Test at 5% significance level whether average output per worker is higher in cities. report your results. Compute and report average high and low-skilled labour and capital in cities and elsewhere. Use this to explain what could be behind the “City effect” observed on averages. if any.
(g) [5 marks] Estimate the following model and report your estimates on a well-presented table including standard errors in parenthesis and sample size.
log(Y) = (k) + ølCi + U.jwhere itt, 1 = 1, . . . , N are independently and identically distributed mean zero shocks.
(j) [5 marks] Test at the 5% significance level Ji0 : = () against H : ()
(h) [10 marks] Estimate the transformed model in (b) by OLS. Report the estimates
on a well-presented table including standard errors in parenthesis and sample
size. Interpret the estimates of 13, y, and n. Are all three statistically significant
at the 5% level?
(i) [5 marks] Using your estimates from (h), test. at 5% significance level, whether there is a “City effect” after controlling for differences on labor and capital inputs. What does this test suggest for your findings in question (g)?
2(j) [5 marks] Imaging that a firm that operates in a city increases capital by 5%.
How much will the expected output increase? What if the same increase in capital were to be made by a firm operating outside a city?
(k) [5 marks] Show that under constant returns to scale, i.e. ß + ‘ + ( = 1, it i5possible to write the estimating function with one less parameter. Estimate one of the three possible parametrizations under constant returns to scale and report your estimates in a table.
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