Question

The demand function for roses is given as: Q_d = a + bp + fp_c and the supply function is given as: Q_s = c + ep, where: (1) a, b, c, e, and f are constants (with a > 0, b < 0, c > 0, e > 0, and f > 0) and (2) Q_d and Q_s are quantity demanded and supplied, respectively, with p the price of roses, and p_c is the price of chocolates. Based on these equations, write a formula for the equilibrium price of roses, which will depend upon a, b, c, e, f, and p_c. Using this formula, verify that an increase in the price of chocolates will increase the equilibrium price of roses. Of course, without specific values to the constants you cannot draw specific demand and supply curves, but do sketch and explain why an increase in the price of chocolates increases the equilibrium price of roses in this case.

Answer 1

Setting Qd = Qs,

a + bp + fpc = c + ep

p x (b - e) = c - a - fpc

p = (c - a - fpc) / (b - e)

Q = c + [e x (c - a - fpc) / (b - e)] = c + [(ec - ea - efpc) / (b - e)] = (bc - ec + ec - ea - efpc) / (b - e) = (bc - ea - efpc) / (b - e)

Q/pc = [- ef / (b - a)] > 0 [since e > 0, f > 0, (ef) > 0 & (-ef) < 0, and since b < 0, a > 0, (b - a) < 0]

Therefore, as pc (price of chocolate) increases by 1 unit, demand for rose will increase by [ef / (b - a)] units. This is because rose and chocolate are substitutes, as is evident from the positive coefficient in pc in demand function for rose.

In following graph, price (P) and quantity (Q) of roses are shown along vertical and horizontal axes. D0 and S0 are initial demand and supply curves of roses, intersecting at point A with equilibrium price P0 and quantity Q0.

Higher price of a substitute good will increase demand for rose, shifting demand curve rightward, increasing both price and quantity. In above graph, D0 shifts right to D1, intersecting S0 at point B with higher price P1 and higher quantity Q1.