1. Assume the following simple functional forms for the demand and supply curves for crude oil where P is the price of crude oil, ε_D is the price elasticity of demand for crude oil and ε_S is the price elasticity of supply (A and B are just constant parameters).

Demand: Q_D = AP^ε_D!!
Supply: Q_S = BP^ε_D

Between the beginning of 2007 and mid-2008, the WTI spot price increased from about $63 per bbl to about $143 per bbl. Let's see if this simple supply and demand model is able to account for that price increase just using some "back of the envelope" calculations. First, there are a few hoops to jump through.

a. Demonstrate using the demand function that ε_D is the own price elasticity of demand (so by extension, ε_S is the own price elasticity of supply). Hint: this is easiest when you take natural logs of both sides of the equation. Recall that the price elasticity of demand is the proportional (or percentage) change in quantity demanded divided by the proportional change in price. This is equivalent to the derivative of the natural log of quantity demand with respect to the natural log of P.

b. Derive an expression for the equilibrium price of crude oil.

c. Between early 2007 and mid-2008, world real GDP increased 6.8%. If the income elasticity of demand for crude oil is .971, then by what percentage would we expect quantity demanded for crude oil to increase due to the increase in real income?

d. During this same period, there was a dramatic increase in costs globally supplies and equipment for large, capital intensive projects, such as steel, drilling equipment, materials and engineering equipment. It is estimated that such costs increased by 75%. This is one major input into crude oil production, and we can measure its effect in percentage terms, as an elasticity. Let's define the "material cost elasticity of supply" as the percentage change in quantity supplied resulting from a one percent change in materials costs (let C
= total materials costs):

ε_D = (dQs/Qs)/dC/c

It turns out that -0.029 is a reliable estimate of this elasticity for the crude oil sector, so when costs increase by 1%, quantity supplied of crude oil globally declines by 2.9%. Use this information to estimate the percentage decline in quantity supplied resulting from the cost increase.

e. Let the price in the initial period be the price that you solved for in part b. Call this price P₂₀₀₇:

P₂₀₀₇ =  [B/A]^1/ε_D-ε_s

The A and B parameters are the "shift" parameters of the demand and supply function, so when something changes to shift the demand or supply curve, it will be modeled as an increase or decrease in A or B, respectively. After the respective shifts to supply and demand identified in parts c and d, the price in 2008 would be given by

P2008 = [B(1 - 0.22)/A(1 + .066)]^1/ε_D-ε_s

= [.978B/1.066A]^1/ε_D-ε_s

Let ε_D = -0.04 and ε_S = 0.05. Given these figures, what is the implied percentage increase in the price of crude oil? How does it compare to the actual price increase? Based on this simple model, how much of a role did "speculators" play in driving the price of oil above its fundamental values during this episode?

2. Consider a world where there are just two competing primary energy sources: crude oil and coal. You are given the following information:

Own price elasticity of demand for crude oil = -0.1 Income elasticity of demand for crude oil = 2.0

Cross coal price elasticity of demand for crude oil = 0.2 Consumption of crude oil in millions of barrels per day = 80.0

U.S. nominal GDP in trillions = $15.0 Price per barrel of crude oil = $100 Price per short ton of coal = $75

Now, assume a linear function for demand for crude oil in the U.S.:

Q_D = β₀ + β₁P_o + β₂Y + β₃P_c

where P_o is the price of crude oil, P_c is the price of coal, and Y is U.S. nominal GDP. Just for the sake of pinning down a useful context, assume that this regression was run on monthly data, so it estimates short run responses of quantity demanded to changes in the relevant variables.

a. If the price of crude oil were to increase by $10 per barrel, using the estimated elasticities and the regression equation, by how much would quantity demand of crude oil decrease (other things held constant, of course)?

b. Now assume that economists predict a recession ahead and expect nominal GDP to fall by 2%. Other things constant, by how much would quantity demanded change?

c. Due to new emissions regulations, the price of coal is predicted to increase to $100 per short ton. Other things held constant, what would be the predicted change on the quantity demanded of crude oil?

d. Now assume that the United Nations imposes trade sanctions on the country of Iran, meaning that Iran is no longer able to supply its crude oil to international markets. Previously, Iran was supplying 1 of the 90mdb of global crude production. What will be the effect on the price of crude oil?

e. Consistent with the Smith article, assume that the short run price elasticity of demand for crude oil is ε_D = -0.05 and the short run price elasticity of supply is ε_S = 0.05. Now, when the Iranian supply is withheld from the market, we are allowing for a supply response to make up at least some fraction of that supply. By how much would we expect price to increase in this scenario?