Part 1: Suppose a firm has a cost structure given by TC = 10Q - 4Q^2 + 4Q^3 and the market price of the commodity is $10. A. Determine the profit maximizing output? B. Determine th... Part 1: Suppose a firm has a cost structure given by TC = 10Q - 4Q^2 + 4Q^3 and the market price of the commodity is $10. A. Determine the profit maximizing output? B. Determine the break-even price? C. What minimum market price does the firm need to continue production? Part 2: Suppose that the total consumption expenditure of Ethiopia in the year 2024 was $10 million and its investment spending was triple of its consumption spending. It is also assumed that the government's spending in the same year was twofold that of the country's net exports. The country's net exports were $3 million in the same year. Having this information: A. Calculate the GDP of Ethiopia. B. If the net income is half of investment spending, how much is the country's GNP in the same year? C. How much is the NNP of Ethiopia in 2024, if depreciation value is half of the government spending. Read more

Steps to Solve

1. Profit Maximizing Output

Profit is maximized where Marginal Cost (MC) equals Marginal Revenue (MR). Since the firm is in a competitive market, MR equals the market price, which is $10. First, find the Marginal Cost (MC) by taking the derivative of the Total Cost (TC) function with respect to $Q$: $$ MC = \frac{d(TC)}{dQ} = \frac{d(10Q - 4Q^2 + 4Q^3)}{dQ} = 10 - 8Q + 12Q^2 $$ Set MC equal to MR (which is the market price): $$ 10 - 8Q + 12Q^2 = 10 $$ $$ 12Q^2 - 8Q = 0 $$ $$ 4Q(3Q - 2) = 0 $$ So, $Q = 0$ or $Q = \frac{2}{3}$. To determine which output level maximizes profit, we can check the second-order condition (the second derivative of the profit function, or the derivative of the MC, must be positive). $$ \frac{d(MC)}{dQ} = -8 + 24Q $$ At $Q = 0$, the second derivative is -8, which is negative, indicating a maximum loss. At $Q = \frac{2}{3}$, the second derivative is $-8 + 24(\frac{2}{3}) = -8 + 16 = 8$, which is positive, indicating profit maximization. Therefore, the profit-maximizing output is $Q = \frac{2}{3}$.

2. Break-Even Price

The break-even price is the price at which the firm makes zero economic profit. This occurs when the price equals the minimum Average Total Cost (ATC). First, we need to find the ATC function: $$ ATC = \frac{TC}{Q} = \frac{10Q - 4Q^2 + 4Q^3}{Q} = 10 - 4Q + 4Q^2 $$ To find the minimum ATC, we take the derivative of ATC with respect to Q and set it equal to zero: $$ \frac{d(ATC)}{dQ} = -4 + 8Q = 0 $$ $$ 8Q = 4 $$ $$ Q = \frac{1}{2} $$ Now, substitute $Q = \frac{1}{2}$ back into the ATC function to find the break-even price: $$ ATC = 10 - 4(\frac{1}{2}) + 4(\frac{1}{2})^2 = 10 - 2 + 1 = 9 $$ So, the break-even price is $9.

3. Minimum Market Price to Continue Production

The firm should continue to produce as long as the market price is greater than or equal to the minimum Average Variable Cost (AVC). First, find the AVC function. Note that $TC = FC + VC$, and in this case $FC = 0$ since there is no constant term, thus $VC = TC$. $$ AVC = \frac{VC}{Q} = \frac{10Q - 4Q^2 + 4Q^3}{Q} = 10 - 4Q + 4Q^2 $$ To find the minimum AVC, we take the derivative of AVC with respect to Q and set it equal to zero: $$ \frac{d(AVC)}{dQ} = -4 + 8Q = 0 $$ $$ 8Q = 4 $$ $$ Q = \frac{1}{2} $$ Substitute $Q = \frac{1}{2}$ back into the AVC function to find the minimum price: $$ AVC = 10 - 4(\frac{1}{2}) + 4(\frac{1}{2})^2 = 10 - 2 + 1 = 9 $$ So, the minimum market price is $9.

4. Calculate GDP of Ethiopia

GDP (Gross Domestic Product) can be calculated using the expenditure approach: $$ GDP = C + I + G + NX $$ Where: $C =$ Consumption expenditure = $10 million $I =$ Investment spending = $3 \times C = 3 \times 10 = $30 million $G =$ Government spending = $2 \times NX = 2 \times 3 = $6 million $NX =$ Net exports = $3 million $$ GDP = 10 + 30 + 6 + 3 = 49 $$ So, the GDP of Ethiopia is $49 million.

5. Calculate GNP of Ethiopia

GNP (Gross National Product) is calculated as: $$ GNP = GDP + \text{Net Factor Income from Abroad} $$ Given that the net income (Net Factor Income from Abroad) is half of investment spending: $$ \text{Net Factor Income from Abroad} = \frac{1}{2} \times I = \frac{1}{2} \times 30 = $15 million $$ $$ GNP = 49 + 15 = 64 $$ So, the GNP of Ethiopia is $64 million.

6. Calculate NNP of Ethiopia

NNP (Net National Product) is calculated as: $$ NNP = GNP - \text{Depreciation} $$ Given that depreciation is half of the government spending: $$ \text{Depreciation} = \frac{1}{2} \times G = \frac{1}{2} \times 6 = $3 million $$ $$ NNP = 64 - 3 = 61 $$ So, the NNP of Ethiopia is $61 million.