Question

the supply function of a product is p = 0.4e^2x, where x denotes 1000 units. Find producer's surplus when sales are 2000 units

Answer 1

Answer:

he is incorrect ( make negative)

Answer 2

The producer's surplus p$_{o}$ when sales are 2000 units is 32.96 units.

Given:

The supply function of a product is p = 0.4$e^{2x}$ , where x denotes 1000 units.

To Find:

The producer's surplus when sales are 2000 units.

Solution:

We have been given that the supply function of a product is p = 0.4$e^{2x}$ , where x denotes 1000 units.

We need to find the producer's surplus p$_{o}$ when sales are 2000 units, i.e.

x$_{o}$ = 2000/1000 = 2.

Now, p$_{o}$ =  0.4$e^{2x \exponential_o}$ = 0.4$e^{2*2}$ =  0.4$e^{4}$

⇒ p$_{o}$ / 0.4 = $e^{4}$

⇒ $(\log_{e} e)$ $\frac{p \exponential_o}{0.4}$ = 4

⇒  $($ $\log_{10}\frac{p}{4.0}$ ) /  $\log_{10}e$= 4

⇒ $($ $\log_{10}\frac{p}{4.0}$ ) / (0.4343) = 4

⇒ p$_{o}$ / 0.4 = $10^{1.7372}$

Taking log on both sides,

1.7372 log 10 = log $\frac{p}{0.4}$

⇒ p$_{o}$ = 0.4 x Antilog 1.7372 = 0.4 x 54.6 = 21.84

The producer's surplus p$_{o}$ when sales are 2000 units =  x$_{o}$ p$_{o}$ -$0.4\int\limits^2_0 {e^{2x} } \, dx$ = 2 x 21.84 - [0.2 ($e^{4} - e^{0}$ )] = 32.96

∴ The producer's surplus p$_{o}$ when sales are 2000 units is 32.96 units.

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