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Question 3 (10\%) When Sam applies N\mathrm{N} pounds of fertilizer to an acre...
May 7, 2024
a Solution
a
To maximize profits, Sam should use the amount of fertilizer where the marginal revenue product (MRP) equals the marginal cost (MC) of fertilizer. The MRP is the additional revenue from selling the extra corn produced by using an additional pound of fertilizer, which can be calculated as the marginal product (MP) of fertilizer times the price of corn. The MC is the price of fertilizer
b
The optimal amount of fertilizer is found where the derivative of the profit function equals zero. The profit function is π(N)=3(1−N/200)N−pN\pi(N) = 3(1-N/200)N - pN. Taking the derivative with respect to NN and setting it to zero gives us the optimal NN
c
Solving the derivative of the profit function dπ(N)dN=3(1−N/200)+3N(−1/200)−p=0\frac{d\pi(N)}{dN} = 3(1-N/200) + 3N(-1/200) - p = 0 gives us the optimal amount of fertilizer to use
d
Simplifying the derivative, we get 3−3N100−p=03 - \frac{3N}{100} - p = 0. Solving for NN gives us N=300−100p3N = \frac{300 - 100p}{3}
a Answer
Sam should use 300−100p3\frac{300 - 100p}{3} pounds of fertilizer per acre to maximize profits.
Key Concept
Profit Maximization
Explanation
Sam maximizes profits by equating the marginal revenue product of fertilizer with its marginal cost.
b Solution
a
Tam's marginal product is twice that of Sam's, so his profit function is π(N)=6(1−N/200)N−pN\pi(N) = 6(1-N/200)N - pN
b
Taking the derivative of Tam's profit function with respect to NN and setting it to zero gives us the optimal NN for Tam
c
Solving the derivative dπ(N)dN=6(1−N/200)+6N(−1/200)−p=0\frac{d\pi(N)}{dN} = 6(1-N/200) + 6N(-1/200) - p = 0 gives us the optimal amount of fertilizer for Tam
d
Simplifying the derivative, we get 6−6N100−p=06 - \frac{6N}{100} - p = 0. Solving for NN gives us N=600−100p6N = \frac{600 - 100p}{6}
b Answer
Tam should use 600−100p6\frac{600 - 100p}{6} pounds of fertilizer per acre to maximize profits.
Key Concept
Profit Maximization with Different Productivity
Explanation
Tam maximizes profits by equating his marginal revenue product, which is twice that of Sam's, with the marginal cost of fertilizer.
c Solution
a
To determine if Tam's output is more than, less than, or exactly twice as much as Sam's when both are maximizing profits, we compare the optimal amounts of fertilizer each uses
b
Since Tam's land produces twice as much corn for any amount of fertilizer, if he uses the same amount of fertilizer as Sam, his output would be exactly twice. However, the optimal amount of fertilizer for Tam is different from Sam's because of the difference in marginal products
c
By comparing the formulas for the optimal amount of fertilizer for Sam and Tam, we see that Tam uses less fertilizer than twice the amount Sam uses because the marginal product of fertilizer decreases as more is used
c Answer
Tam's output will be less than twice as much as Sam's when both are maximizing profits.
Key Concept
Diminishing Marginal Product
Explanation
Even though Tam's land is more productive, the diminishing marginal product means that the additional output he gets from each extra unit of fertilizer is less than twice that of Sam's, leading to less than double the total output when both maximize profits.
Solution
a
Initial Demand Calculation: To calculate Miss Chan's initial demand for X1X_1 and X2X_2, we use the budget constraint and the Cobb-Douglas utility function
The budget constraint is $P_{X1}X_1 + P_{X2}X_2 = I$, where $P_{X1}$ and $P_{X2}$ are the prices of goods $X_1$ and $X_2$, and $I$ is the income. Initially, $P_{X1} = P_{X2} = \$1$ and $I = \$100$.
The Cobb-Douglas utility function is $U = X_1^{1/2}X_2^{1/2}$. To maximize utility subject to the budget constraint, we set the ratio of the marginal utilities equal to the ratio of the prices: $\frac{MU_1}{P_{X1}} = \frac{MU_2}{P_{X2}}$.
Substituting the marginal utilities, we get $\frac{0.5X_1^{-1/2}X_2^{1/2}}{1} = \frac{0.5X_2^{-1/2}X_1^{1/2}}{1}$. Simplifying, we find $X_1 = X_2$.
Using the budget constraint $X_1 + X_2 = 100$, and since $X_1 = X_2$, we find $X_1 = X_2 = 50$.
b
New Demand Calculation: After the price change, we need to recalculate the demand for X1X_1 and X2X_2 with the new price of X2X_2
The new budget constraint is $X_1 + 2X_2 = 100$. Using the same method as before, we set the ratio of the marginal utilities equal to the ratio of the prices: $\frac{0.5X_1^{-1/2}X_2^{1/2}}{1} = \frac{0.5X_2^{-1/2}X_1^{1/2}}{2}$.
Simplifying, we find $X_2 = \frac{1}{2}X_1$. Substituting into the budget constraint, we get $X_1 + 2(\frac{1}{2}X_1) = 100$, which simplifies to $X_1 = 100/2 = 50$ and $X_2 = 25$.
c
Compensating Variation (CV) Calculation: CV is the amount of money needed to reach the initial utility level at the new prices
The initial utility is $U = 50^{1/2} \times 50^{1/2} = 50$. To find the CV, we need to solve for the income that allows Miss Chan to reach this utility level with the new prices.
We set the utility function equal to the initial utility and solve for the new income ($I'$) using the new prices: $U = (I'/1)^{1/2} \times (I'/2)^{1/2} = 50$.
Solving for $I'$, we get $I'^2/2 = 2500$, which simplifies to $I' = \sqrt{5000} \approx \$70.71$. The CV is the difference between the new income and the initial income: $CV = \$100 - \$70.71 \approx \$29.29$.
Answer
Initial demand for X1X_1 and X2X_2 is 50 units each. After the price change, demand for X1X_1 remains 50 units, and demand for X2X_2 is 25 units. The compensating variation (CV) is approximately \$29.29.
Key Concept
Consumer demand and compensating variation in response to price changes
Explanation
The consumer adjusts her demand for goods in response to price changes to maximize utility, and the compensating variation is the income needed to maintain initial utility after a price change.
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