Show your work please.

For several years, a large U$.$S$.$ gift retailer has been purchasing a pen$-$and$-$pencil set from an overseas manufacturer for $$27$ per set.$$ The retailer$$s policy is to use a $120$ percent markup on such stationery$-$related gifts.

$($a$)$ Recently, a currency exchange$-$rate change has enabled the retailer to buy the pen$-$and$-$pencil set for only $$22$ per set.$$ The retailer is considering passing along this cost decrease to his customers.$$ If the retailer uses his usual markup for stationery$-$related gifts, calculate the new price the retailer would charge for this pen$-$and$-$pencil set.

$($b$)$ A buyer in the retailer$$s organization has taken the initiative to examine the historical data on this pen$-$and$-$pencil set.$$ She obtained data for each of the past $21$ years concerning the number of units sold during the year, the price during the year, and an index of economic activity in the retailer$$s market area during the year, and then carried out a regression analysis on these data.$$ The means for each variable and the regression coefficients from the analysis are as follows:

Variable

Mean

Regression coefficient

Number of units sold per year

$3,103$

$$

Price during the year

$52\mathrm{.}06$

$23\mathrm{.}85$

Index of economic activity for the year

$118\mathrm{.}7$

$106\mathrm{.}3$

Calculate the price elasticity implied by these results.$$ Show your work.

$($c$)$ The buyer found it surprising that the regression coefficient of price had a positive sign.$$ Describe the price$-$quality heuristic, and explain how it might account for this positive price coefficient.

$($d$)$ Given the sign of the price elasticity you calculated in Part $($b$),$ should the retailer go ahead with decreasing the item$$s price to the price that you calculated Part $($a$)?$ Justify your answer.