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The market for a product is in equilibrium at 4000 units a month. Each one is so...
Sep 23, 2024
The market for a product is in equilibrium at 4000 units a month. Each one is sold for 300.Supposethepriceelasticityofdemandforthisproductis−0.6andthepriceelasticityofsupplyis+0.8a.Computetheslopeandinterceptcoefficientsforthelinearsupplyanddemandequations.b.Ifataxof300. Suppose the price elasticity of demand for this product is -0.6 and the price elasticity of supply is +0.8 a. Compute the slope and intercept coefficients for the linear supply and demand equations. b. If a tax of 20 is imposed, what would be the new equilibrium and the welfare changes (CS,PS,DWL,G)? c. If a price floor at $320 is imposed what would be the new equilibrium and welfare changes? d. Comment on possible further costs to society beyond the deadweight loss. e. Comment on the welfare changes if the price floor is supported by a commitment to buy the excess supply
Answer
a
To find the slope and intercept coefficients for the linear supply and demand equations, we use the formula for elasticity. The price elasticity of demand (PED) is given by: PED=dQdP⋅PQ PED = \frac{dQ}{dP} \cdot \frac{P}{Q} Rearranging gives us the slope of the demand curve: dQdP=PED⋅QP=−0.6⋅4000300=−8 \frac{dQ}{dP} = \frac{PED \cdot Q}{P} = -0.6 \cdot \frac{4000}{300} = -8 For the supply curve, the price elasticity of supply (PES) is: PES=dQdP⋅PQ PES = \frac{dQ}{dP} \cdot \frac{P}{Q} Rearranging gives us the slope of the supply curve: dQdP=PES⋅QP=0.8⋅4000300=323≈10.67 \frac{dQ}{dP} = \frac{PES \cdot Q}{P} = 0.8 \cdot \frac{4000}{300} = \frac{32}{3} \approx 10.67 Thus, the demand equation can be expressed as: Qd=−8P+b Q_d = -8P + b And the supply equation as: Qs=10.67P+c Q_s = 10.67P + c To find the intercepts, we can substitute the equilibrium point (4000, 300) into both equations
b
With a tax of 20imposed,thenewpricereceivedbysuppliersbecomes20 imposed, the new price received by suppliers becomes 300 - 20=20 = 280. The new equilibrium quantity can be found by setting the new supply equal to demand: Qd=−8(300)+bandQs=10.67(280)+c Q_d = -8(300) + b \quad \text{and} \quad Q_s = 10.67(280) + c Calculating the new equilibrium quantity will show a decrease in quantity sold. The changes in consumer surplus (CS), producer surplus (PS), and deadweight loss (DWL) can be calculated based on the shifts in supply and demand curves. The exact values depend on the new equilibrium quantity and price
c
If a price floor of 320isimposed,thequantitysuppliedwillincreasewhilethequantitydemandedwilldecrease.Thenewequilibriumcanbefoundbysubstituting320 is imposed, the quantity supplied will increase while the quantity demanded will decrease. The new equilibrium can be found by substituting 320 into both the demand and supply equations. The resulting surplus will lead to a deadweight loss as well. The welfare changes can be calculated similarly to part b
d
Beyond deadweight loss, further costs to society may include administrative costs of enforcing the price floor, potential unemployment in the industry due to reduced demand, and the misallocation of resources as producers may overproduce due to the price floor
e
If the price floor is supported by a commitment to buy excess supply, it can mitigate some welfare losses by ensuring that producers can sell their goods. However, this may lead to increased government expenditure and potential market distortions in the long run
Answer
The slopes of the demand and supply equations are -8 and approximately 10.67, respectively. A 20taxwillshifttheequilibrium,reducingquantitysoldandaffectingconsumerandproducersurplus.Apricefloorat20 tax will shift the equilibrium, reducing quantity sold and affecting consumer and producer surplus. A price floor at 320 will create a surplus and deadweight loss, while government intervention can mitigate some effects.
Key Concept
Price elasticity of demand and supply are crucial for understanding how changes in price affect quantity demanded and supplied, influencing market equilibrium.
Explanation
The analysis of tax and price floor impacts on equilibrium illustrates how government interventions can lead to changes in consumer and producer welfare, highlighting the importance of elasticity in microeconomic analysis.
b. If a tax of $20 is imposed, what would be the new equilibrium and the welfare changes (CS,PS,DWL,G)?
b. Answer
a
New Equilibrium Calculation: To find the new equilibrium after a tax of 20isimposed,weneedtoadjustthesupplycurve.Theeffectivepricereceivedbysuppliersbecomes20 is imposed, we need to adjust the supply curve. The effective price received by suppliers becomes 300 - 20=20 = 280. We can find the new quantity supplied and demanded at this price. Using the original equilibrium quantity of 4000 units, we can calculate the new quantities using the elasticities. The new quantity demanded (QdQ_d) can be calculated as: Qd=Q0+(dQdP)⋅(P−P0)=4000+(−0.6⋅4000300)⋅(−20)=4000+160=4160 Q_d = Q_0 + \left( \frac{dQ}{dP} \right) \cdot (P - P_0) = 4000 + \left( -\frac{0.6 \cdot 4000}{300} \right) \cdot (-20) = 4000 + 160 = 4160 The new quantity supplied (QsQ_s) can be calculated as: Qs=Q0+(dQdP)⋅(P−P0)=4000+(0.8⋅4000300)⋅(20)=4000+213.33≈4213.33 Q_s = Q_0 + \left( \frac{dQ}{dP} \right) \cdot (P - P_0) = 4000 + \left( \frac{0.8 \cdot 4000}{300} \right) \cdot (20) = 4000 + 213.33 \approx 4213.33 Thus, the new equilibrium quantity will be between these two values
b
Welfare Changes: The consumer surplus (CS) and producer surplus (PS) will change due to the tax. The new consumer surplus can be calculated as the area of the triangle above the price paid by consumers (300)andbelowthedemandcurve.Thenewproducersurplusistheareabelowthepricereceivedbyproducers(300) and below the demand curve. The new producer surplus is the area below the price received by producers (280) and above the supply curve. The deadweight loss (DWL) is the loss in total welfare due to the tax, which can be calculated as the area of the triangles formed by the reduction in quantity traded. The government revenue (G) from the tax is given by: G=Tax⋅Qnew=20⋅Qnew G = Tax \cdot Q_{new} = 20 \cdot Q_{new} where QnewQ_{new} is the new equilibrium quantity
Answer
The new equilibrium quantity will be between 4160 and 4213.33 units, with a decrease in consumer and producer surplus due to the tax, leading to a deadweight loss and government revenue from the tax.
Key Concept
Tax incidence and its effect on market equilibrium
Explanation
A tax shifts the supply curve, leading to a new equilibrium with reduced quantities traded, affecting consumer and producer surplus, and creating deadweight loss.
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