Question

In my paper about climate change and grape growers’ revenue/profit in the European Union (see PPT file) I specify a log-linear equation and find that

ln (%) = 0.221 ∙ -./0 − 0.006 ∙ -./0! + ⋯ , where Temp is measured in oC.
a) Based on these coefficients, calculate the profit maximizing temperature (which may be slightly different from what is provided in the Table due to rounding). Provide a precise number with 3 decimals!

b) Now assume I had used a linear equation and had found the same coefficients, that is

%=0.221∙-./0− 0.006∙-./0! +⋯.

What is the profit maximizing temperature now?

c) Based of the original log-linear equation ln(%) = 0.221 ∙ -./0 − 0.006 ∙ -./0! + ⋯, what is the profit % if a plot of land experiences a (growing season) temperature of 24oC. What is the profit if temperatures increase to 26oC? What is the percentage decrease in profits?

Answer 1

a) In () = 0.221Temp - 0.006Temp2+...

(Hence is never equal to zero)

To maximize Profit, we need d/d(Temp) = 0

Differentiating the equation wrt Temp on both sides and ignoring terms with powers 3 and greater(because their coefficients are low), we get:

(1/)*(d)/d(Temp)) = 0.221 - 0.006*2*Temp

d/d(Temp) = *(0.221 - 0.012Temp) = 0

Hence, either =0, or 0.221 - 0.012Temp = 0

But is never equal to zero and hence 0.221 - 0.012Temp = 0

Hence Temp = 0.221/0.012 = 18.417 C

Hence 18.417C maximizes profit.

b) = 0.221 - 0.006Temp2+.....

For profit maximization, d/d(Temp) = 0

Hence differentiating the equation wrt Temp, we get,

d/d(Temp) = 0.221 - 0.012Temp=0

Hence Temp = 18.417 C

Hence 18.417C maximizes profit in this case too.

c) In () = 0.221 Temp - 0.006 Temp2+.....

Hence = e0.221 Temp - 0.006 Temp2+....

If Temp = 24 C, then

= e0.221*24 - 0.006*24*24+....

= e1.848

= 6.347

If

Temp = 24 C, then

= e0.221*26 - 0.006*26*26+....

= e1.69

= 5.419

Hence the percentage decrease in profits = ((24 - 26)/24))*100

Hence the percentage decrease in profits = ((6.347 - 5.419)/6.347))*100

Hence the percentage decrease in profits = 14.621%

Hence profits decrease by 14.621%