Assume that demand for sugar is a function of income the price of sugar (P_s), a

Question

Assume that demand for sugar is a function of income the price of sugar (P_s), and the price of saccharine (P_c), a sugar substitute, as follows. Find the partial derivatives of this demand function. The elasticity of demand with respect to income is defined as delta Q_d/delta Y middot T/Q_d Find the elasticity of demand with respect to income when Y = 10,000, P_s = 5, and P_c = 7. The own-price elasticity of demand in this example is Find the own-price elasticity of demand when Y = 10,000, P_s = 5, and P_c = 7. The cross-price elasticity of demand refers to the percentage change in the quantity demanded for a good due to a 1% change in the price of another good. In this example, the cross-price elasticity of sugar with respect to saccharine is

Explanation / Answer

Qd = 0.05Y + 10 Pc - 5Ps2

(a) Partial derivative with respect to Income; dQ / dY =0.05

Partial derivative with respect to Pc; dQ / dPc= 10

Partial derivative with respect to Ps; dQ / dPs =10

(b) Y = 10,000 ; Pc = 5 & Ps = 7 Plugging these values in Qd equation

Qd = 0.05(10000) + 10(5) - 5(7)2 => 500 + 50 - 245 => 305

Use partial derivatives calculated in part (A)

Elasticity wrt to income = dQd / dY . Y/Qd

= 0.05 x 10000/305 => 1.63

c) Own Price elasticity of demand =  dQd / d Ps . Ps / Qd

10 x 5 / 305 => o.163

d)  Cross Price elasticity of demand =  dQd / d Pc .  Pc / Qd

10 x 7 / 305 => 0.229

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