Given-a-consumer-with-the-utility-function-U-X-1-1-4-X-2-who-faces-a-budget-constraint-of-B-P-1-X-1-P-2-X-2-Show-that-the-expemditure-function-facing-the-consumer-is-B-2P-1-1-2-P-2-1-2

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Question Number 217769 by nECxx2 last updated on 20/Mar/25

$$Givenaconsumerwiththeutility$$$$functionU=X_1^{\frac{1}{4}}+X_{2}whofaces$$$$abudgetconstraintofB=P_1{X}_1{P}_2{X}_2$$$$Showthattheexpemditurefunction$$$$facingtheconsumeris$$$$B=2P_1^{\frac{1}{2}}{P}_2^{\frac{1}{2}}{U}^{\frac{1}{2}}$$

Commented by nECxx2 last updated on 20/Mar/25

$$Pleasehelp$$

Answered by MrGaster last updated on 21/Mar/25

$$\mathcal{L}=X_1^{\frac{1}{4}}+X_2+\lambda (B-P_1{X}_1-P_2{X}_2)$$$$\frac{\partial \mathcal{L}}{\partial X_1}=\frac{1}{4}{X}_1^{-\frac{3}{4}}-\lambda P_1=0\Rightarrow \lambda =\frac{1}{4P_1}{X}_1^{-\frac{3}{4}}$$$$\frac{\partial \mathcal{L}}{\partial X_2}=1-\lambda P_2=0\Rightarrow \lambda =\frac{1}{P_2}$$$$\frac{\partial \mathcal{L}}{\partial \lambda }=B-P_1{X}_1-P_2{X}_2=0$$$$\frac{1}{4P_1}{X}^{-\frac{3}{4}}=\frac{1}{P_2}\Rightarrow X_1^{-\frac{3}{4}}=4\frac{P_1}{P_2}$$$$X_1={\left(4\frac{P_1}{P_2}\right)}^{-\frac{3}{4}}$$$$B=P_1{\left(4\frac{P_1}{P_2}\right)}^{\frac{4}{3}}+P_2{X}_2$$$$X_2=\frac{B-P_1{\left(4\frac{P_1}{P_2}\right)}^{-\frac{4}{3}}}{P_2}$$$$U=X_1^{\frac{1}{4}}+X_2$$$$X_1\stackrel{U=X_1^{\frac{1}{4}}+X_2}{\mathrm{\&}}{X}_2\stackrel{}{\Rightarrow }U={\left({\left(4\frac{P_1}{P_2}\right)}^{-\frac{4}{3}}\right)}^{\frac{1}{4}}+\frac{B-P_1{\left(4\frac{P_1}{P_2}\right)}^{-\frac{4}{3}}}{P_2}$$$$U={\left(4\frac{P_1}{P_2}\right)}^{-\frac{1}{3}}+\frac{B-P_1{\left(4\frac{P_1}{P_2}\right)}^{-\frac{4}{3}}}{P_2}$$$$B=P_{2}U+P_1{\left(4\frac{P_1}{P_2}\right)}^{-\frac{4}{3}}-P_2{\left(4\frac{P_1}{P_2}\right)}^{-\frac{1}{3}}$$$$B=P_{2}U+P_1^{\frac{1}{3}}{P}_2^{\frac{1}{3}}{4}^{-\frac{4}{3}}-P_1^{\frac{1}{3}}{P}_2^{\frac{2}{3}}{4}^{-\frac{1}{3}}$$$$B=P_{2}U+P_1^{\frac{1}{3}}{P}_2^{\frac{1}{3}}(4^{-\frac{4}{3}}-4^{\frac{1}{3}})$$$$B=P_{2}U+P_1^{\frac{1}{3}}{P}_2^{\frac{1}{3}}=(\frac{1}{8}-\frac{1}{2})$$$$B=P_{2}U-\frac{3}{8}{P}_1^{\frac{1}{3}}{P}_2^{\frac{1}{3}}$$$$B=P_{2}U-\frac{3}{8}{\left(P_1{P}_2\right)}^{\frac{1}{3}}$$$$\overline{)\begin{array}{c}B=2P_1^{\frac{1}{2}}{P}_2^{\frac{1}{2}}{U}^{\frac{1}{2}}\end{array}}$$

Commented by nECxx2 last updated on 21/Mar/25

$$Thankyousomuch$$

Commented by nECxx2 last updated on 21/Mar/25

$$Whatwasthereasonfortheequation$$$$atthefirstline?$$

Commented by nECxx2 last updated on 21/Mar/25

$$Alsoyousolvedasthough{there}^{\prime }sa$$$$+signintheequation$$$$B=P_1{X}_1{P}_2{X}_2$$$$nowasB=P_1{X}_1+P_2{X}_2$$$$Thissolvesityet{I}^{\prime }llliketo$$$$understandwhy$$$$$$$$$$$$$$

Commented by nECxx2 last updated on 21/Mar/25

$$AftergoingthroughitagainIhave$$$$evenmorequestions.Hiwcome$$$$line13?$$$$Please,IonlyneedclarityasIam$$$$alearner.Thankyou.$$

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