Opportunity Cost

Recall that the graph of the BC is given by x2=−p1p2x1+mp2The opportunity cost of an extra unit of good 1 is p1p2 units of good 2.The opportunity cost of an extra unit of good 2 is p2p1 units of good 1.\text{Recall that the graph of the BC is given by } x_2 = -\frac{p_1}{p_2} x_1 + \frac{m}{p_2} \\ \text{The opportunity cost of an extra unit of good 1 is } \frac{p_1}{p_2} \text{ units of good 2.} \\ \text{The opportunity cost of an extra unit of good 2 is } \frac{p_2}{p_1} \text{ units of good 1.} \\

Comparative Static Analysis

  • Comparative static analysis involves changing one parameter at a time to see how the BC and BS change

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Income and Price Changes

  • Recall that m represents the consumer’s income
    • Increasing m increases the x and y intercepts; thus, the maximum amount of goods for both good 1 and good 2 increase, meaning that budget set grows (represented by change in area) and choice is improved; shifts parallel to the right/outward
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    • Decreasing m decreases the amount of goods available for purchase, thus shrinking the budget set and reducing choice; shifts graph parallel to the left/inwards
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    • Note that the slope does not change in either case because the prices are unaffected
  • Recall that p1 and p2 represent the prices of goods 1 and 2, respectively
    • We will only look at changing the price of one good to keep it simple
    • A decrease in p1 increases the amount of good 1 that the consumer can buy, thus enlarging the budget set and improving choice; makes slope of BC more “shallow” (represented by the black graph)
    • An increase in p1 decreases the amount of good 1 that the consumer can buy, thus shrinking the budget set and reducing choice; makes slope of BC more “steep” (represented by the green graph)
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    • These movements can be thought of as “pivots” around the y-intercept, as the amount of good 2 that can be purchased remains the same

Uniform Ad Valorem Sales Tax

  • Hypothetically, if all prices are changed in the same fashion (such as when the tax rate is increased on all goods), then the BC is moved in a parallel manner to the left
    • This tax policy is known as uniform ad valorem sales tax and is tied to the price of the good
      • “Ad valorem” means that the tax is levied in proportion to the value; i.e. tax is levied at 5% instead of $10
A uniform ad valorem sales tax ar rate t changes the BC fromp1x1+p2x2=m to (1+t)p1x1+(1+t)p2x2=mThis is equivalent to an income decrease.(1+t)p1x1+(1+t)p2x2=m→p1x1+p2x2=m1+t\text{A uniform ad valorem sales tax ar rate t changes the BC from} \\ p_1 x_1 + p_2 x_2 = m \text{ to } (1+t)p_1 x_1 + (1+t)p_2 x_2 = m \\ \text{This is equivalent to an income decrease.} \\ (1+t)p_1 x_1 + (1+t)p_2 x_2 = m \rightarrow p_1 x_1 + p_2 x_2 = \frac{m}{1 + t}
  • A uniform ad valorem sales tax is therefore equivalent to an income tax
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Real World Applications: Food Stamp Program

  • Food stamps are coupons that only allow you to buy one good (food)
  • The question: how does a commodity-specific gift (such as a food stamp) change a family’s budget constraint and affect their decision making?
  • Suppose that m = 100, pF = 1, and pG = 1 where pF represents the price of food and pG represents the price of all other goods
    • The budget constraint is defined as pFxF + pGxG = 100
  • Suppose that 40 food stamps are issued to the family
    • The new budget constraint is pF(xF + 40) + pGxG = 100
The new budget constraint is pF(xF−40)+pGxG=100pFxF−40pF+pGxG=100pGxG=100+40pF−pFxFxG=−pFpGxF+100+40pFpG\text{The new budget constraint is } p_F (x_F - 40) + p_G x_G = 100 \\ p_F x_F - 40 p_F + p_G x_G = 100 \\ p_G x_G = 100 + 40 p_F - p_F x_F \\ x_G = - \frac{p_F}{p_G} x_F + \frac{100 + 40 p_F}{p_G}
  • The budget thus “increases”, but you cannot utilize the entire budget increase image