The price of beef in the United States has been found to depend on the demand (measured by per capita consumption) according to the equation q=342.5/p^0.5314 Find the elasticity. Is the demand for beef elastic or inelastic?

Q = 342.5 / P^.5314

dQ/dP = -342.5 (.5314) P^-.4686/ P^1.063
so
dQ/dP = - 182 P^-1.5316

well, we need to know what P is at the intersection P = q
P^1.5314 = 342.5
1.5314 log P = 2.535
log P = 1.655
P = 45.2
so dQ/dP at intersection = -182(45.2)^-1.5316
= .531

change in percent bought is only half the percent change in price

To find the elasticity of demand, we need to calculate the derivative of the demand equation with respect to the price. Let's differentiate the equation q = 342.5 / p^0.5314 with respect to p.

Using the power rule of differentiation, the derivative of p^n is n * p^(n-1), where n is a constant.

Differentiating the equation q = 342.5 / p^0.5314:

dq/dp = (-0.5314) * 342.5 * p^(-0.5314 - 1)

Simplifying further, we get:

dq/dp = -181.896 * p^(-1.5314)

The elasticity of demand is calculated as the percentage change in quantity (dq/dp) divided by the percentage change in price (dp/dq). However, since the original demand equation is given in terms of per capita consumption, we can't calculate exact percentage changes without additional information.

Nevertheless, we can assess the elasticity by looking at the exponent -1.5314:

- If the exponent is greater than 1, demand is considered elastic.
- If the exponent is less than 1, demand is considered inelastic.
- If the exponent is exactly 1, demand is considered unitary elastic.

In this case, since -1.5314 is less than 1, the demand for beef in the United States is considered inelastic.

To find the elasticity of demand, we need to use the following formula:

Elasticity = (% change in quantity demanded) / (% change in price)

Let's break it down step by step:

1. Determine the current quantity demanded and price of beef. In this case, we have the demand equation: q = 342.5 / p^0.5314.

2. Calculate the derivative of the demand equation with respect to price, which measures the proportional change in quantity demanded for a given change in price:
dq/dp = (342.5 * -0.5314 * p^(-0.5314-1))

3. Find the current quantity demanded q and price p at which we want to calculate elasticity.

4. Calculate the current value of dq/dp using the values of q and p obtained in step 3.

5. To determine the percentage change in quantity demanded, divide the change in quantity demanded by the initial quantity demanded and multiply by 100:
% change in quantity demanded = (delta q / q) * 100

6. Do the same as step 5 for the percentage change in price:
% change in price = (delta p / p) * 100

7. Finally, use the values obtained in steps 5 and 6 to calculate the elasticity of demand using the formula:
Elasticity = (% change in quantity demanded) / (% change in price)

After calculating the elasticity, we can determine the elasticity of demand for beef.

If the elasticity is greater than 1, demand is considered elastic, meaning that a percentage change in price will lead to a larger percentage change in quantity demanded.

If the elasticity is less than 1, demand is considered inelastic, meaning that a percentage change in price will result in a smaller percentage change in quantity demanded.

I apologize, but without knowing the specific quantity demanded and price at which you want to calculate elasticity, I cannot provide a specific answer. However, by following the steps outlined above, you should be able to calculate the elasticity of demand for beef.