Question. Solve the following questions.

Assume that a country's production function is Y = AK^0.3 L^0.7. The ratio of capital to output is 3, the growth rate of output is 3 percent, and the depreciation rate is 4 percent. Capital is paid its marginal product.

a.What is the marginal product of capital in this situation?

b.If the economy is in a steady state, what must be the saving rate?

c.What is the marginal product of capital if the economy reaches the Golden Rule level of capital?

d.What must the saving rate be to achieve the Golden Rule level of capital?

The share price, in pence, of a certain company is monitored over an 8-year period. The results are shown in the table below: Time (years) 0 2 3 4 5 6 7 8 Price 100 131 183 247 330 454 601 819 1,095 [(x; -7)2=60 _(V;-7)2 =925,262 [(; -D)();- 5)=7,087 An actuary fits the following simple linear regression model to the data: yi = a+ Bx; te; i=0,1,...,8 where {e; } are independent normal random variables with mean zero and variance o. (i) Determine the fitted regression line in which the price is modelled as the response and the time as an explanatory variable. [2] (ii) Obtain a 99% confidence interval for: (a) B , the true underlying slope parameter (b) o, the true underlying error variance. [5] (iii) (a) State the "total sum of squares" and calculate its partition into the "regression sum of squares" and the "residual sum of squares" (b) Use the values in part (iii)(a) to calculate the "proportion of variability explained by the model" and comment on the result. [5] (iv) The actuary decides to check the fit of the model by calculating the residuals. (a) Complete the table of residuals (rounding to the nearest integer): Time (years) 0 2 3 4 5 6 7 8 Residua 132 -21 -75 -104 -75 25 (b) Use a dotplot of the residuals to comment on the assumption of normality. (c) Plot the residuals against time and hence comment on the appropriateness of the linear model. [7] [Total 19]A random sample of n observations is taken from a normal distribution with variance o'. The sample variance is an observation of a random variable S- . Show that: (i) E($2) =02 [2] 204 (ii) var($4) = n-1 [2] [Total 4]