Managerial Economics Assignment
Problem I: The demand for a product is given by P = 360 2Q and the supply is P = 30 + 4Q.
a) What will be the market outcome if the price is P = 200? (Shortage of 37.5)
b) What will be the equilibrium price and quantity? (Q=55; P=250)
c) Calculate the price elasticity of demand at the equilibrium point. (εD = 2.27)
d) Calculate the consumer surplus at the equilibrium point. (CS = 3025)
e) Imagine the government imposes a price floor of $200 (we haven't formally discussed this in class but that simply means that the price on the market cannot go below 200 by law). What will be the market outcome? What if the price floor is set at $300? (I'm not going to provide you an answer to this. I want to see how you think and apply basic knowledge to something new. Feel free to discuss amongst yourselves)
f) Calculate the marginal effect of a price change on the consumer surplus at the equilibrium point. This involves writing the CS as a function of P and taking its derivative. Alternatively, you can approximate this by "brute force" if you take the difference between the CS at the equilibrium and the consumer surplus at some close-by price (say at P=251) (ME = 55)
g) If you feel like you need more practice, you can repeat a, b, c, and d for the following supply and demand functions: P = 30 + 2Q, P = 240 Q. You should get a surplus of 45, an equilibrium price of 170 and equilibrium quantity of 70, elasticity of 2.43, and CS of 2450.
Problem II: The demand curve for product X is given by QxD = 220 - PX + 3PY + 0.001I where PY is the price of a related good Y, and I is income. The supply curve for good X is
given by QXS = 10 + 3PX.
a) What is the marginal effect of an increase in PY on the equilibrium price of good X? (3/4)
b) How much do we need to increase income, if we want people to trade 5 more units of product X? (6666.67)
c) Assuming PY = 2 and I = 50, 000, what is the equilibrium price and quantity for good X? (Q=209.5; P=66.5)
d) Assuming the same values for PY and I, find the value of PX that would result in a surplus of 100 units. Also find the value of PX that would result in a shortage of 100 units. (91.5 and 41.5)
Problem III: For each of the following total cost functions, derive the AFC, ATC, AVC, and MC curves.
a) TC = 20Q + 3Q2 + Q3
b) TC = 120 + 2Q2
c) TC = 500 + 2Q2
d) TC = 100Q + 2Q3 + 20
e) TC = 20 + 5Q
Problem IV: A perfectly competitive firm has a total cost function given by T C(Q) = 2Q3 - 20Q2 + 100Q.
a) What will be the optimal quantity produced by the firm if the market price is P = 300? What will be the profit? (Q=10, Profit=2000)
b) How about if the market price is P = 45? What will be the optimal quantity and profit in this case? (Q=0, Profit=0)
c) Find the break even point and the shut down point. (50)
d) Assuming the initial market price of $300, what will be the optimal quantity and profit if a new fixed cost of $1000 has to be incurred by the firm? How about if this FC=$3000?
Problem V: A perfectly competitive firm faces the total cost function T C = 2Q + 2Q2. The firm is part of an industry where the market demand is given by P = 816 2Q and the market supply is given by P = 12 + (Q/100).
a) What is the optimal quantity that this firm will sell on this market? What will be the resulting profit? (Q=3.5, Profit=24.5)
b) Assuming identical firms, how many firms are active on this market (need to be active on this market to satisfy the market equilibrium condition, given their individual production levels)? (N=114.29)
c) What is the shut down point for this firm? (2)
Problem VI: A monopolist with total cost function T C = 30Q + Q2 is facing a market demand given by P = 150 Q.
a) What is the optimal quantity and price the monopolist will set on this market? (Q=30, P=120)
b) What quantity and price would this firm set if it was to behave competitively? (Q=40, P=110)
c) Calculate the price elasticity of demand at the monopoly price and quantity point. Does the Lerner price formula hold? (-4)
Problem VII: The average total cost of a monopolistic firm is ATC = (80/Q) + 20Q. The firm is facing the demand function given by P = 6000 - 20Q.
a) What will be the total profit that this firm will generate if it chooses price and quantity optimally?(Profit=224920)
b) What would be the profit of this firm if it behaved competitively? (Profit=199920)
c) What would be the long run equilibrium price if this market was competitive? (80)
Problem VIII: For each of the following demand and cost functions, find the perfectly competitive and the monopoly outcomes (price, quantity, and profit).
a) P = 200 - Q, T C = 200 + 10Q (QM = 95; PM = 105; QC = 190; PC = 10)
b) P = 160 - 2Q, T C = 50 + Q2 (QM = 26.67; PM = 106.67; QC = 40; PC = 80)
c) P = 1200 - Q, T C = 20Q(QM = 590; PM = 610; QC = 1180; PC = 20)
Problem IX: A perfectly competitive firm has T C = 10 + 10Q - 4Q2 + Q3.
a) What is the optimal production level and the profit, if the market price is P = 26?
(Q = 4; Π = 54)
b) What is the optimal production level and the profit, if the market price is P = 10?
(Q = 2.67; Π = 0.5)
c) What is the minimum market price for which you will choose to produce (P=6)?