What Is The Own Price Elasticity Of Demand For P 100 5Q

What Is The Own Price Elasticity Of Demand For P 100 5q At Q

1) What is the own-price elasticity of demand for p = 100 – 5q, at q = 10; at q = 5; at q = 12? 2) What is the own-price elasticity of demand for p = a – bq? (Show the elasticity as an expression depending on the value of q.) 3) Show the budget line for p1 = 5, p2 = 10, and m = 150 in a figure. (pi = price of good i; m = income). 4) For the numbers in 3), what can you say about the purchases of goods x1 and x2 if the bundles (x1, x2) available are , a) (16; 7) b) (8: 8) c) (10; 10) d) (14; ) For the situation in 3), show graphically what if p1 a) increases to 8; b) decreases to 4. c) for the increase of p1 to p1 = 8, can the consumer still afford any of the consumption bundles in 4)? 6) For p1 = 5 and p2 = 10, what happens of income increase from m = 150 to m = 180? Show the change in a figure.

Paper For Above instruction

The concept of price elasticity of demand (PED) is fundamental in understanding consumer behavior and market response to price changes. It measures the responsiveness of quantity demanded of a good to a change in its price, quantified as the percentage change in quantity demanded divided by the percentage change in price. This paper delves into the calculation of own-price elasticity at specific quantities, general forms of elasticity expressions, and the implications of income and price changes on consumer choice and purchasing power, supported by graphical analysis.

Calculating Own-Price Elasticity at Specific Quantities

The demand function provided is p = 100 – 5q, which indicates a linear demand curve. The own-price elasticity of demand can be calculated using the formula:

Elasticity (E) = (dQ/dP) * (P/Q)

where dQ/dP is the derivative of quantity with respect to price. Since p = 100 – 5q, rearranging for q gives q = (100 – p)/5, and its derivative with respect to p is dQ/dP = -1/5.

At each quantity, we need to find the corresponding price and substitute into the elasticity formula. When q = 10, p = 100 – 5*10 = 50. When q = 5, p = 100 – 25 = 75. When q = 12, p = 100 – 60 = 40. The corresponding elasticity calculations are:

• At q = 10:
• E = (-1/5) (50 / 10) = (-1/5) 5 = -1
• At q = 5:
• E = (-1/5) (75 / 5) = (-1/5) 15 = -3
• At q = 12:
• E = (-1/5) (40 / 12) ≈ (-1/5) 3.33 ≈ -0.6667

The elasticities indicate that demand is elastic at q = 5, unit elastic at q = 10, and inelastic at q = 12.

Elasticity as an Expression for General Demand Function

For a general demand function of the form p = a – bq, the own-price elasticity can be derived as:

E = (dQ/dP) (P/Q) = (-1/b) (P/Q)

Rearranged, the elasticity depends on the current price and quantity:

E(q) = -(1/b) * (p / q)

Since in this demand function p = a – bq, substituting p results in:

E(q) = -(1/b) ((a – bq) / q) = -(a / (b q)) + 1

This expression shows how elasticity varies with the quantity q, given the parameters a and b.

Graphical Representation of Budget Line

The budget line with prices p1 = 5, p2 = 10, and income m = 150 can be expressed as:

• x2 = (m / p2) – (p1 / p2) * x1

Substituting the values:

x2 = (150 / 10) – (5 / 10) x1 = 15 – 0.5 x1

This line intercepts the x2-axis at 15 and the x1-axis at 30 (since when x2 = 0, x1 = 30). Graphically, this line slopes downward, representing trade-offs between the two goods.

Analysis of Consumer Purchases

Considering the bundles (16,7), (8,8), (10,10), and (14, ?), along with the budget line, we analyze affordability and consumer preferences. For example, at the bundle (16,7), total expenditure is:

Expense = p1 x1 + p2 x2 = 5 16 + 10 7 = 80 + 70 = 150, which exactly exhausts the income, indicating affordability.

Similarly, (8,8): 58 + 108 = 40 + 80 = 120, which is under the budget, leaving some income unused, which might influence consumer choices. The bundle (10,10): 510 + 1010 = 50 + 100 = 150, fully spending income. The last bundle (14, ?) would be computed similarly, but the question mark indicates incomplete data.

Impact of Price Changes on Purchase Choices

If p1 increases to 8, the budget line shifts inward, reducing feasible bundles to those satisfying 8x1 + 10x2 ≤ 150. Conversely, decreasing p1 to 4 shifts the budget line outward, allowing more purchase options. The consumer's ability to afford specific bundles depends on these price shifts, and graphical illustration can show the change in feasible combinations.

For instance, with p1 = 8:

x2 = (150/10) – (8/10) x1 = 15 – 0.8 x1

This lower intercept compared to the original indicates reduced affordability.

At p1 = 4:

x2 = 15 – 0.4 * x1, expanding the feasible set.

Regarding whether the consumer can still afford certain bundles if p1 increases to 8, only those bundles satisfying 8x1 + 10x2 ≤ 150 remain affordable; thus, some previously affordable bundles may no longer be accessible.

Effect of Income Increase

An increase in income from m = 150 to m = 180 shifts the budget line outward parallel to the original, allowing higher maximum consumption of both goods. The new budget line is:

x2 = (180 / 10) – (5/10) x1 = 18 – 0.5 x1

Graphically, this expands the feasible region, enabling the consumer to achieve higher utility levels or purchase more of both goods.

The shift demonstrates the consumer's enhanced purchasing power and potential changes in consumption patterns, assuming preferences remain consistent.

Conclusion

Understanding the variations in demand elasticity and budget constraints provides insight into consumer responsiveness and market dynamics. Specific calculations reveal how demand reacts to price changes at different quantities, and graphical analysis illustrates how shifts in prices and income influence consumer choices. These analytical tools are essential for marketers, policymakers, and economists to predict market behavior and design effective strategies.

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