Microeconomics Cheat Sheet: Definitions, Marginal Utility, Consumer Problems, and More

Posted on May 7, 2024 in Economy

Microeconomics Cheat Sheet

Definitions

Transitivity

  • If the consumer prefers X to Y (X>Y) and Y to Z (Y>Z), then she must also prefer X to Z (X>Z)
  • Consistency of preferences

Strict Monotonicity (S.M)

  • If a consumer strictly prefers one bundle of good over another because it contains more of at least one good and no less of any other good, then their preferences show S.M.
  • MORE IS BETTER! Choosing bundle with more of one good than the other

Monotonicity

  • As more of good increases, the consumer’s utility/satisfaction does not decrease
  • Indifference between bundles that contain the same amount of goods
  • Usually monotonic preferences on a graph are downward sloping
  • More is always preferred

Marginal Utility

  • MRS is how much of good x_2 the consumer is willing to give up to get an additional unit of good x_1
  • +YeTPrTcIAAAAASUVORK5CYII=
  • If a question asks to find the marginal utility of x_1 or x_2, find the derivative of that sole variable only wPrg5LGCm1ZigAAAABJRU5ErkJggg==
  • To calculate the MRS for all the bundles of x_1 and x_2, do MUx_1/MUx_2 and simplify
  • To find value at a given bundle, plug in the bundle into the MRS expression
  • Remember that the given number indicates the amount of x_2 the consumer is willing to give up to get 1 more unit of x_1
  • If asked to write a B.C. as a function of P_1, P_2, I
    • Use the B.C format, do not plug any numbers in yet unless asked
  • Diminishing Marginal Rate of substitution:
    • Refers to the consumer’s willingness to part with less and less quantity of one good in order to get an additional unit of another good
    • wA5HXoKnqOTSQAAAABJRU5ErkJggg==

For Consumer Problems

    • Choosing to allocate goods in order to maximize utility subject to the budget constraint
      • Sarah chooses the allocation of cakes (x_1) and ice cream (x_2) for her graduation party to maximize her utility with the B.C it is subjected to.
      • Mathematically: maxU(x_1,x_2)=x_1x_2 subject to P_1*x_1+P_2*x_2= ?

Solving Lagrange Problems

    • If asked to format the LG equation as a function of x_1, x_2, lambda, P_1, P_2, and I
      • Do not plug in numbers, just write as normal (unless asked otherwise)
      • 0UiXOzgAAAABJRU5ErkJggg==
      • In the first part, you put in the original function
    • Then, take the first order condition (F.O.C, basically the derivative)
      • FSsqLIXAAAALM0iAQkAAACAtjibQwIAAACgDwISAAAAsDgEJAAAAGBxCEgAAADA4hCQAAAAgMUhIAEAAACLQ0ACAAAAFoeABAAAACwOAQkAAABYHAISAAAAsDgEJAAAAGBxCEgAAADAwoj+D3s9AZBfHGLUAAAAAElFTkSuQmCC
    • Then, we move the \lambdax_1 or x_2 over so that we get the number (e.g: 3XF=\lambdax_1, do it for both)
    • Set the lambdas equal to each other
    • Solve for x_2 first
    • Then set up the B.C equation: p_1x_1+p_2x_2=M and plug in x_2 and solve for x_1
    • You will get the optimal values AND the demand functions (x^*_1 and x^*_2)
    • Deriving a demand function for x_1 (x^_1) as function of P_1 and I and vice versa is basically solving the Lagrange problem.
  • Income Elasticity
  • Follow the format gYtsWRszYotDgAAAABJRU5ErkJggg== and the same applies to x_2
  • Solve for both and remember to bring the denominator for the second expression and multiply by the reciprocal to make things easier
  • If it is 1>0, that means that it is a normal good (Income rises, demand increases)
  • If 1
  • If income decreases, person goes from buying expensive steak to cheap ground beef

Computing Quantity demanded

  • The demand functions; just add the p values given for each and you’ll get the quantity demanded

Calculating the Total Effect

Example) Cakes (x_1) and balloons (x_2). The p_1 for cakes is $12 and the p_2 for balloons is $5 and income is $120

  • BC: 12x_1+5x_1=$120
  • Get the original consumption bundle
    • Ex) After calculating for the quantity demanded, you will have the original bundle of (x^*_1,x^*_2)=(3,7) 
  • Find the new consumption bundle given P’_x_1
    • So lets say the cost of cakes(x_1) went up by 20% and originally the price for cakes as $12 
    • Mutiply 20%*$12=2.4 
    • Add: 12+2.4= $14.4 is the new price
  • Reformat the NEW budget constraint 
    • 14.4x_1+5x_1=$120
  • Tangency condition
    • \frac{MU_x_1}{MU_x_2}= p’_x_1/p_x_2  -> Set up the MU’s accordingly and to the new price with the price of the balloons staying the same and solve for x_2
    • Plug x_2 into the new B.C and solve for x_1
  •  we get 2.17 for x_1 
  • The total effect is written as x^*_C-x^*_A= 2.17-3= -.83

Substitution effect 

  • Finding the decomposition bundle
    • From the original bundle (3,7), find the utility
    • Function: x_1x_2^3
      • 3*7^3=1029 utils
    • Then we want the x_2 of the new price 
    • x_2/3x_1=14.4/5=8.64x_1=x_2
  • Reformat
    • x_1x^3_2=1029 utils 
    • x_1(8.64)^3=1029
    • 644.97x_1^4=1029
    • x_1=(1029/644.97)^(1/4)
    • =1.123= x^*_B

The substitution effect is 

1.123-3=-1.8 cakes

The income effect 

IE=x^*_1_C-x^*1_B

=2.17-1.123=1.047 cakes

Sarah increases her consumption for cakes by .877 when her income increases which means that it is normal good