The following data show how sales of a product increased over a six$-$year period:

$\backslash$table$[[$Year$,1994,1995,1996,1997,1998,1999],[$Sales $($units$),32,47,74,109,163,242]]$

$2$

a$.$ Estimate the average rate of growth of sales over the period.

b$.$ How well does the trend pattern fit?

c$.$ Draw a graph of the data and appropriate relationship.

d$.$ Forecast sales in $2000$ and $2001,$ stating any assumptions.

$3.$ The following data relate to the sales of a product over an eight$-$month period:

$\backslash$table$[[$Month$,$Jan.,Feb.,Mar.,Apr.,May,Jun.,Jul.,Aug.$],[$Sales $($units$),56,72,70,65,68,75,66,67],[$Price $($E$),75,65,59,69,69,49,59,59]]$

a$.$ Investigate whether sales are affected more by the level of price or by the change in price of the product.

b$.$ Interpret the regression coefficient of the explanatory variable.

c$.$ Draw an appropriate graph of the data and relationship.

d$.$ Forecast sales in September if price is $65.$

$4.$ KAD has estimated the following demand relationship for its product over the last four years, using monthly observations:

$ln{Q}_t=4\mathrm{.}932-1\mathrm{.}238ln{P}_t+1\mathrm{.}524ln{Y}_{t-1}+0\mathrm{.}4865ln{Q}_{t-1}$

$(2\mathrm{.}54)$

$R^2=0\mathrm{.}8738$

where $Q=$ sales in units, $P=$ price in $,Y$ is income in $,000,$ and the numbers in brackets are t$-$statistics.

a$.$ Interpret the above model.

b$.$ Make a sales forecast if price is $9,$ income last month was $25,000$ and sales last month were $2,981$ units.

c$.$ Make a sales forecast for the following month if there is no change in price or income.

d$.$ If price is increased by $5$ per cent in general terms, estimate the effect on sales, stating any assumptions.

$5.$ Data are collected from eight firms regarding their costs in the last year, each firm having the same level of fixed assets:

a$.$ Estimate an appropriate cost relationship.

b$.$ A firm of the same type is aiming to produce $48,000$ units next year; estimate its total costs, stating any assumptions that you need to make.

$\backslash$table$[[$Total costs $($$ million$),245,260,272,265,248,282,295,288],[$Output $(\text{'}000),31,35,38,36,32,40,44,42]]$

c$.$ Estimate average costs for the firm next year.

d$.$ What can be said about returns to scale for the firms in this industry?

e$.$ Estimate the marginal cost of the last unit produced by the firm next year.

f$.$ If the firm is currently charging $$8,000,$ estimate the break$-$even level of output for the firm.

g$.$ Estimate the effect on costs for each firm in the industry of an increase in output of $5,000$ units.

$6.$ A survey of costs and output is carried out for eight firms in an industry that have different plant sizes. The results are:

$\backslash$table$[[$Output $($x$100),50,134,90,178,84,200,152,108],[$Costs $($$$000),23,56,42,75,38,84,64,48]]$

h$.$ Estimate an appropriate cost function in the industry.

i$.$ Estimate costs for a firm producing $12,000$ units.

j$.$ What can be said about returns to scale in the industry? Give your reasoning.

k$.$ Estimate average costs at $12,000$ units.