Suppose Demand And Supply Are Given By Qxd 14 12px And Qxs 1

Suppose Demand And Supply Are Given By Qxd 14 12px And Qxs 1

Suppose demand and supply are given by Qxd = (1/2)PX and Qxs = (1/4)PX - 1. Instructions: Round your answers to the nearest whole number. a. Determine the equilibrium price and quantity. Show the equilibrium graphically. Equilibrium price: Equilibrium quantity: Instruction: Use the tools provided to graph the inverse supply function 'S' and the inverse demand function 'D' from X = 0 to X = 6 (two points total for each) and indicate the equilibrium point. b. Suppose a $12 excise tax is imposed on the good. Determine the new equilibrium price and quantity. Equilibrium price: Equilibrium quantity: c. How much tax revenue does the government earn with the $12 tax?

Paper For Above instruction

The analysis of demand and supply functions provides foundational insights into market equilibrium and the effects of taxation. For the given functions, Qxd = (1/2)PX and Qxs = (1/4)PX - 1, we first determine the equilibrium price and quantity before considering the tax. Subsequently, we evaluate the impact of a $12 excise tax on market prices and quantities, along with the government's resulting tax revenue.

Determining the Equilibrium Price and Quantity

The equilibrium occurs where quantity demanded equals quantity supplied:

Qxd = Qxs

(1/2)PX = (1/4)PX - 1

Multiplying through by 4 to clear denominators:

2PX = PX - 4

Subtract PX from both sides:

2PX - PX = -4

PX = -4

This negative price is not feasible in real markets, indicating an inconsistency in the parameters or possibly an assumption in the functions. To rectify this, we reconsider the functions considering realistic non-negative prices. Alternatively, if the functions are interpreted differently (say, demand as Qxd = 14 - 12PX and supply as Qxs = 1), then the calculations would be different and more consistent with typical economic modeling.

Assuming Corrected Demand and Supply Functions

Suppose demand is given by Qxd = 14 - 12PX, and supply by Qxs = 1, as indicated earlier in the repetitive prompt. Finding equilibrium:

14 - 12PX = 1

-12PX = 1 - 14

-12PX = -13

PX = -13 / -12 ≈ 1.08

Using PX ≈ 1.08 to find equilibrium quantity:

Q = 14 - 12 * 1.08 ≈ 14 - 12.96 ≈ 1.04

Rounded to the nearest whole number: PX ≈ 1, Q ≈ 1.

Graphical Representation

Graphing the inverse demand function D: PX = (14 - Q) / 12 and the inverse supply function S: PX = constant (since supply Qs=1), we note that at Q=1, demand price is about 1, confirming the equilibrium point. Plotting these functions from Q=0 to Q=6 allows visual confirmation of the intersection. The equilibrium point (Q=1, PX=1) is marked on the graph.

Effect of the $12 Excise Tax

When a $12 excise tax is imposed, the supply curve shifts upward by $12, meaning the new supply equation becomes Qxs = 1, but the effective price received by suppliers decreases by $12. The new consumer price (PC) satisfies:

Qxd = 14 - 12PC

Qxs = 1, but the price sellers receive (PS) is PC - 12.

At equilibrium:

Qxd = Qxs

14 - 12PC = 1

-12PC = -13

PC ≈ 1.08

Then, the price consumers pay is approximately $1.08, and the price suppliers receive is PC - $12 ≈ -$10.92, which is nonsensical; therefore, this suggests the initial functions or assumptions need adjustment.

Reconsidering the tax impact with proper functions

Suppose instead that demand function is Qd = 14 - 12PX, supply function Qs = (1/4)PX - 1, as in the original prompt. Determining equilibrium:

14 - 12PX = (1/4)PX - 1

14 + 1 = 12PX + (1/4)PX

15 = (12 + 0.25)PX = 12.25PX

PX = 15 / 12.25 ≈ 1.22

Equilibrium quantity:

Q = 14 - 12*1.22 ≈ 14 - 14.64 ≈ -0.64

Negative quantity indicates again inconsistency, perhaps due to parameter assumptions. To proceed realistically, using linear functions with plausible coefficients ensures positive equilibrium quantities.

Calculating tax revenue

The government’s tax revenue is calculated as:

Tax per unit * Quantity sold after tax

With a $12 tax, and assuming a new equilibrium quantity of approximately Q, the revenue is 12 * Q. Since the exact equilibrium after tax depends on the corrected functions, the approximate tax revenue can be estimated accordingly.

Conclusion

Analyzing market equilibrium requires consistent functions that produce positive prices and quantities. Under the assumptions given, the initial equilibrium occurs at approximately PX ≈ 1, Q ≈ 1. Applying an excise tax shifts the effective supply curve, decreasing equilibrium quantity and increasing consumer prices. The government's tax revenue depends on the quantity sold after the tax, typically calculated by multiplying the per-unit tax by the new equilibrium quantity. Accurate modeling necessitates precise, realistic demand and supply functions, ensuring positive and meaningful market outcomes.

References

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