Price/Income Consumption Curves

demand functions - calculated from budget line and utility function  

• MRS calculated by partial derivatives of utility or given prices
• usually changes w/ respect to price/income of itself or other good
• only depends on own price >> independent good
• remember that demand functions slope downward

Find the demand functions for food and clothing if a consumer's utility function for the 2 was U = C^0.8F^0.2  

• budget constraint >> I = P_CC + P_FF
  • C = (I-P_FF) / P_C
  • F = (I-P_CC) / P_F
  • need to get rid of F to find C demand function
  • need to get rid of C to find F demand function
• MRS = U_C / U_F = P_C / P_F
  • U_C = 0.8C^-0.2F^0.2
  • U_F = 0.2C^0.8F^-0.8
  • MRS = (0.8C^-0.2F^0.2) / (0.2C^0.8F^-0.8) = 4(F/C)
• 4F/C = P_C / P_F
  • 4FP_F = CP_C
• substitute that back into the budget constraint equations
  • C = (I-[CP_C/4])/P_C = I/P_C - C/4
  • 5/4 C = I/P_C >> C = (4/5) (I/P_C)
  • F = (I-[4FP_F]) / P_F = I/P_F - 4F
  • 5F = I/P_F >> F = (1/5) (I/P_F)

price-consumption curve  

• connects points of equal utility on budget lines formed by changing prices

income-consumption curve  

• connects points of equal utility on budget lines formed by changing income