Question

The utility function of Master Enterprise is

U (T, M) =   T^1/4   M^3/4

Where T and M are the two goods that the Enterprise sells. The owner of the Enterprise is interested in finding out the demand function of its product. Help him:

• Find the demand function for Good T and Good M.
• Use an Income of $8000 and the Price of Good T as $5.00 while the price of Good M as $12 to find the optimum bundles of Good T and M.

Answer 1

$a)\\U(T,M)=T^{(\frac{1}{4})}M^{(\frac{3}{4})}\\MU_T=\frac{1}{4}T^{(−\frac{3}{4})}M^{(\frac{3}{4})}\\MU_M=\frac{3}{4}T^{(\frac{1}{4})}M^{(−\frac{1}{4})}\\MRS_{TM}=\frac{\frac{1}{4}T^{(\frac{−3}{4})}M^{(\frac{3}{4})}}{\frac{3}{4}T^{(\frac{1}{4})}M^{(\frac{−1}{4})}}\\MRS_{TM}=\frac{1}{3}T^{−1}M^{1}\\MRS_{TM}=\frac{M}{3T}\\At \space equilibrium\\MRS=\frac{P_T}{P_M}\\\frac{M}{3T}=\frac{P_T}{P_M}\\M=\frac{P_T}{P_M}\times 3T\\The\space budget\space equition\\P_T\times T+P_M\times M=I\\P_T\times T+P_M\times M\times \frac{P_T}{P_M}\times 3T=I\\P_T\times T+3P_T\times T=I\\4P_T\times T=I\\T=\frac{I}{4P_T}\\M=\frac{P_T}{P_M}\times 3\times \frac{1}{4P_T}=\frac{3I}{4P_M}\\ therefore \space the\space demand \space equition are\\for \space good \space T\\T=\frac{I}{4P_T}\\For \space good\space M\\M=\frac{3I}{4P_M}\\ b)\\I=P_T×T+P_M×M\\8000=5T+12M\\At\space equilibrium\\\frac{MU_T}{MU_M}=\frac{P_T}{P_M}\\\frac{M}{3T}=\frac{5}{12}\\12M=15T\\M=\frac{15T}{12}\\8000=5T+12×\frac{15T}{12}\\8000=20T\\T=400\\M=\frac{15×400}{12}\\M=500$