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... 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 the firm needs to continue production? 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

To find the profit-maximizing output, we need to find the output level where Marginal Cost (MC) equals Marginal Revenue (MR). Given the market price is $10, MR = $10. First, find the Total Cost (TC) equation, which is given as $TC = 10Q - 4Q^2 + 4Q^3$. We need to find MC by taking the derivative of TC with respect to Q.

$$MC = \frac{dTC}{dQ} = 10 - 8Q + 12Q^2$$

Now set $MC = MR$:

$$10 - 8Q + 12Q^2 = 10$$

$$12Q^2 - 8Q = 0$$

$$4Q(3Q - 2) = 0$$

This gives us two possible solutions for Q: $Q = 0$ or $Q = \frac{2}{3}$. Since $Q = 0$ doesn't make sense in this context, we have $Q = \frac{2}{3}$. To check if this is indeed a point of profit maximization, we should find the second derivative of the cost function, that is:

$$\frac{d^2TC}{dQ^2} = -8 + 24Q$$

Plug in $Q = \frac{2}{3}$

$$-8 + 24(\frac{2}{3}) = -8 + 16 = 8 > 0$$

Thus, $Q = \frac{2}{3}$ is the minimum point and thus not the point of profit maximization . Consider $Q = 1$ that is $10 - 8 + 12 = 14$ and $Q = 0.5$ that is 7

2. Break-Even Price

The break-even price is the price at which the firm makes zero economic profit. This occurs where Price (P) equals Average Total Cost (ATC).

$$ATC = \frac{TC}{Q} = \frac{10Q - 4Q^2 + 4Q^3}{Q} = 10 - 4Q + 4Q^2$$

To find the minimum ATC, we take the derivative of ATC with respect to Q and set it to zero.

$$\frac{dATC}{dQ} = -4 + 8Q = 0$$

$$8Q = 4$$

$$Q = \frac{1}{2}$$

Plug $Q = \frac{1}{2}$ into ATC:

$$ATC = 10 - 4(\frac{1}{2}) + 4(\frac{1}{2})^2 = 10 - 2 + 1 = 9$$

So, the break-even price is $9.

3. Minimum Market Price for Continued Production

The minimum market price for the firm to continue production in the short run is the minimum Average Variable Cost (AVC). First, we identify variable cost (VC) from $TC = 10Q - 4Q^2 + 4Q^3$. The variable cost is everything that depends on $Q$. Therefore $VC = 10Q - 4Q^2 + 4Q^3$

$$AVC = \frac{VC}{Q} = 10 - 4Q + 4Q^2$$

To minimize AVC, we take the derivative with respect to Q and set it to zero:

$$\frac{dAVC}{dQ} = -4 + 8Q = 0$$

$$8Q = 4$$

$$Q = \frac{1}{2}$$

Plug $Q = \frac{1}{2}$ into AVC:

$$AVC = 10 - 4(\frac{1}{2}) + 4(\frac{1}{2})^2 = 10 - 2 + 1 = 9$$

The minimum market price for continued production is $9.

4. GDP Calculation

GDP (Gross Domestic Product) is calculated as: $GDP = C + I + G + NX$, where C is Consumption expenditure, I is Investment spending, G is Government spending, and NX is Net Exports.

Given: Consumption (C) = $10 million Investment (I) = 3 * C = 3 * $10 million = $30 million Net Exports (NX) = $3 million Government Spending (G) = 2 * NX = 2 * $3 million = $6 million

$$GDP = 10 + 30 + 6 + 3 = 49 \text{ million}$$

5. GNP Calculation

GNP (Gross National Product) = GDP + Net Income from abroad

Net Income = 0.5 * Investment Spending = 0.5 * $30 million = $15 million

$$GNP = GDP + \text{Net Income} = 49 + 15 = 64 \text{ million}$$

6. NNP Calculation

NNP (Net National Product) = GNP - Depreciation

Depreciation = 0.5 * Government Spending = 0.5 * $6 million = $3 million

$$NNP = GNP - \text{Depreciation} = 64 - 3 = 61 \text{ million}$$