Micro I Question Sets

Preference and Utility

2018 Aug Prelim Part 1 Q1

[5 points] Show that $U(\mathbf{x})=U(f(x_1,x_2),x_3)$$U(\mathbf{x})=U\left(f\left(x_1, x_2\right), x_3\right)$ satisfies the conventional definition of (weak) separability; i.e., show that the marginal rate of substitution between $x_1$$x_1$ and $x_2$$x_2$ is invariant to changes in $x_3$$x_3$.

By the chain rule, we can write the MRS as,

$$\begin{array}{}\text{(1)}& MR{S}_{12}=-\frac{\frac{\partial U}{\partial x_1}}{\frac{\partial U}{\partial x_2}}=-\frac{\frac{\partial f}{\partial x_1}}{\frac{\partial f}{\partial x_2}}\end{array}$$

which is invariant to $x_3$$x_3$.

Consumer Theory

2023 June Prelim Part1 Q1 - [IE, SE, Slutsky]

[20 points] Consider a simple quasi-linear utility function of the form $U(x,y)=x+\ln y$$U(x, y)=x+\ln y$​​​. (a) Calculate the income effect for each good and the income elasticity of demand for each good. (b) Calculate the substitution effect for each good. Also calculate the compensated own-price elasticity of demand for each good. (c) Show that the Slutsky equation applies to this function. (d) Show that the elasticity form of the Slutsky equation also applies to this function.

(a)

The Utility max problem is,

$$\begin{array}{}\text{(2)}& \begin{array}{c}maxU(x,y)=x+\ln y\\ \text{s.t. }{p}_{1}x+p_{2}y=w\end{array}\end{array}$$

Now let's consider the interior solution first. ($p_1<w$$p_1 < w$)

Solving the utility max problem, we have the marshalian demand for both goods,

$$\begin{array}{}\text{(3)}& \begin{array}{rl}x(\mathbf{p},w)& =\frac{w}{p_1}-1\\ y(\mathbf{p},w)& =\frac{p_1}{p_2}\end{array}\end{array}$$

The income effect, $x_j\frac{\partial x_i}{\partial w}$$x_j \frac{\partial x_i}{\partial w}$

• For $x$$x$: $y\frac{\partial x}{\partial w}=\frac{y}{p_1}$$y \frac{\partial x}{\partial w} = \frac{y}{p_1}$

• for $y$$y$: $x\frac{\partial y}{\partial w}=0$$x \frac{\partial y}{\partial w} = 0$

Income elasticity of demand: $\frac{\partial x}{\partial w}\frac{w}{x}$$\frac{\partial x}{\partial w} \frac{w}{x}$

• For $x$$x$: $\frac{\partial x}{\partial w}\frac{w}{x}=1$$\frac{\partial x}{\partial w} \frac{w}{x} = 1$

• For $y$$y$: $\frac{\partial y}{\partial w}\frac{w}{y}=0$$\frac{\partial y}{\partial w} \frac{w}{y} = 0$.

Then, consider the corner solution case, the marshalian demands for both goods are,

$$\begin{array}{}\text{(4)}& x(\mathbf{p},w)=0,y(\mathbf{p},w)=\frac{w}{p_2}\end{array}$$

The income effect:

• For $x$$x$: zero

• For $y$$y$: $x\frac{\partial y}{\partial w}=0$$x \frac{\partial y}{\partial w} = 0$

Income elasticity of demand: $\frac{\partial x}{\partial w}\frac{w}{x}$$\frac{\partial x}{\partial w} \frac{w}{x}$

• For $y:1$$y: 1$

(b)

Solve the expenditure min problem, (consider the interior solution first),

$$\begin{array}{}\text{(5)}& \begin{array}{rl}{x}^h(\mathbf{p},u)& =u-\ln p_1+\ln p_2\\ y^h(\mathbf{p},u)& =\frac{p_1}{p_2}\end{array}\end{array}$$

Substitution effect:

• For $x$$x$: $\frac{\partial x}{\partial p_2}=\frac{1}{p_2}$$\frac{\partial x}{\partial p_2} = \frac{1}{p_2}$

• For $y$$y$: $\frac{\partial y}{\partial p_1}=\frac{1}{p_2}$$\frac{\partial y}{\partial p_1} = \frac{1}{p_2}$​​

Then consider the corner solution,

$$\begin{array}{}\text{(6)}& x^h(\mathbf{p},u)=0,y^h(\mathbf{p},u)=e^u\end{array}$$

In this cqase both substitution effects are zero.

(c)

Check the slutsky equation:

$$\begin{array}{}\text{(7)}& \frac{\partial x_i}{\partial p_j}=\frac{\partial h_i}{\partial p_j}-x_j\frac{\partial x_i}{\partial w}\end{array}$$

It holds for interior solution.

For corner solution, it also holds with LHS and RHS are both 0.

(d)

Check the slutsky equation in elasticity form,

$$\begin{array}{}\text{(8)}& {ϵ}_{ij}={ϵ}_{ij}^h-{\alpha }_j{ϵ}_{iw}\end{array}$$

It will hold.

2023 June Prelim Part1 Q3 - [SE]

$[15$$[15$ points] John Maynard gets $\mathrm{\$}3$$\$ 3$ of lunch money per week from his Mom. He only likes peanut butter and jelly sandwiches and this is all he eats for lunch. Peanut butter is sold at $\mathrm{\$}0.05$$\$ 0.05$ per ounce and jelly at $\mathrm{\$}0.10$$\$ 0.10$ per ounce, whereas bread is provided for free from the school cafeteria. J.M. is a particular eater and always makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter. (a) how much peanut butter and jelly will J.M. buy with his lunch money per week in order to maximize his utility?

15 ounce of jelly and 30 ounce of peanut butter. NOte that jelly and peanut butter are perfect complements.

(b) Suppose the price of jelly increases to $\mathrm{\$}0.15$$\$ 0.15$​ per ounce. How much of each commodity will J.M. buy after this price increase?

12 ounce of jelly and 24 ounce of peanut butter.

(c) by how much would his allowance need to increase to compensate him for the price increase?

$0.75.

(d) In what sense does this problem involve only one commodity, namely peanut-butter-and-jelly sandwiches? Write the demand function for peanut-butter-and-jelly sandwiches.

Suppose the income is $w$$w$ and the price of sandwich is $p$$p$,

then the demand for sandwich would be $x=\frac{w}{p}$$x = \frac{w}{p}$.

(e) Discuss the results of this problem in terms of income and substitution effects involved in the demand for jelly.

if the price ratio changes, the substitution effect will be zero because jelly and peanut butter are complements. so demand for jelly change is only attributed to the income effect.

2022 June Prelim Part1 Q1 - [welfare and duality]

[10 points $]$$]$ A consumer with income $y^0$$y^0$ faces prices ${\mathbf{p}}^{\mathbf{0}}$$\mathbf{p}^{\mathbf{0}}$ and enjoys utility $u^0=v({\mathbf{p}}^{\mathbf{0}},y^0)$$u^0=v\left(\mathbf{p}^{\mathbf{0}}, y^0\right)$. When prices change to ${\mathbf{p}}^{\mathbf{1}}$$\mathbf{p}^{\mathbf{1}}$, the cost of living is affected. To gauge the impact of these price changes, define a cost of living index as the ratio:

(a) Show that $I({\mathbf{p}}^{\mathbf{0}},{\mathbf{p}}^{\mathbf{1}},u^0)$$I\left(\mathbf{p}^{\mathbf{0}}, \mathbf{p}^{\mathbf{1}}, u^0\right)$ is greater (less than) than unity as the outlay necessary to maintain base utility $u^0$$u^0$ rises (falls).

By Sheppard's lemma,

$$\begin{array}{}\text{(10)}& \frac{\partial e(\mathbf{p},u)}{\partial p_i}=x_i^{h\ast }(\mathbf{p},u)>0\end{array}$$

Therefore if some elements of $\mathbf{p}$$\bold p$ increases, i.e. for some $p_i^1>p_i^0$$p_i^1 > p_i^0$ and other $j\ne i$$j\neq i$, $p_j^1=p_j^0$$p_j^1 = p_j^0$, then $e({\mathbf{p}}^1,u^0)>e({\mathbf{p}}^0,u^0)$$e(\bold p^1, u^0) > e(\bold p^0, u^0)$, which implies that $I({\mathbf{p}}^{\mathbf{0}},{\mathbf{p}}^{\mathbf{1}},u^0)\equiv \frac{e({\mathbf{p}}^1,u^0)}{e({\mathbf{p}}^0,u^0)}>1$$I\left(\mathbf{p}^{\mathbf{0}}, \mathbf{p}^{\mathbf{1}}, u^0\right) \equiv \frac{e\left(\mathbf{p}^1, u^0\right)}{e\left(\mathbf{p}^0, u^0\right)} > 1$.

(b) Suppose consumer income also changes from $y^0$$y^0$ to $y^1$$y^1$. Show that the consumer will be better off in the final period whenever $\frac{y^1}{y^0}$$\frac{y^1}{y^0}$ is greater than the cost of living index $I({\mathbf{p}}^{\mathbf{0}},{\mathbf{p}}^{\mathbf{1}},u^0)$$I\left(\mathbf{p}^{\mathbf{0}}, \mathbf{p}^{\mathbf{1}}, u^0\right)$​​.

By the duality,

$$\begin{array}{}\text{(11)}& \begin{array}{c}{y}^0=e({\mathbf{p}}^0,u({\mathbf{p}}^0,y^0))=e({\mathbf{p}}^0,u^0)\\ y^1=e({\mathbf{p}}^1,u({\mathbf{p}}^1,y^1))=e({\mathbf{p}}^1,u^1)\end{array}\end{array}$$

Hence,

$$\begin{array}{}\text{(12)}& \frac{y^1}{y^0}=\frac{e({\mathbf{p}}^1,u^0)}{e({\mathbf{p}}^0,u^0)}\frac{e({\mathbf{p}}^1,u^1)}{e({\mathbf{p}}^1,u^0)}=I({\mathbf{p}}^{\mathbf{0}},{\mathbf{p}}^{\mathbf{1}},u^0)\frac{e({\mathbf{p}}^1,u^1)}{e({\mathbf{p}}^1,u^0)}\end{array}$$

Therefore, devide $I$$I$ for both sides, we obtain,

$$\begin{array}{}\text{(13)}& \frac{y^1}{y^0}/I({\mathbf{p}}^{\mathbf{0}},{\mathbf{p}}^{\mathbf{1}},u^0)=\frac{e({\mathbf{p}}^1,u^1)}{e({\mathbf{p}}^1,u^0)}\end{array}$$

So, suppose $\frac{y^1}{y^0}$$\frac{y^1}{y^0}$ is greater than the cost of living index $I({\mathbf{p}}^{\mathbf{0}},{\mathbf{p}}^{\mathbf{1}},u^0)$$I\left(\mathbf{p}^{\mathbf{0}}, \mathbf{p}^{\mathbf{1}}, u^0\right)$.

$$\begin{array}{}\text{(14)}& \frac{e({\mathbf{p}}^1,u^1)}{e({\mathbf{p}}^1,u^0)}>1\Rightarrow e({\mathbf{p}}^1,u^1)>e({\mathbf{p}}^1,u^0)\end{array}$$

which imlies $u^1>u^0$$u^1> u^0$ (The consumer is better off) under the final period with price ${\mathbf{p}}^1$$\bold p^1$ since $\frac{\partial e}{\partial u}>0$$\frac{\partial e}{\partial u} > 0$.

2022 Aug Prelim Part1 Q1 - [duality, slutsky equation]

[20 points]The utility function for goods $x$$x$ and $y$$y$ is given by:

(a) Calculate the uncompensated (Marshallian) demand functions for $x$$x$ and $y$$y$.

Suppose the price for $x$$x$ is $p_x$$p_x$ and the price for $y$$y$ is $p_y$$p_y$. and the income of the consumer is $w$$w$. By solving the UMP, we have the marshallian demand as follows

$$\begin{array}{}\text{(16)}& x(p,w)=\frac{w}{2p_x},y(p,w)=\frac{w}{2p_y}\end{array}$$

(b) Compute the expenditure function and then use the Shephard's lemma to compute the compensated (Hicksian) demand functions for $x$$x$ and $y$$y$.

By duality, the expenditure function is

$$\begin{array}{}\text{(17)}& e(p,u)=e(p,v(p,w))\end{array}$$

Then indirect utility is

$$\begin{array}{}\text{(18)}& v(p,w)=\frac{w}{2p_x^{0.5}{p}_y^{0.5}}\end{array}$$

Then the expenditure function is

$$\begin{array}{}\text{(19)}& e(p,u)=2u{p}_x^{0.5}{p}_y^{0.5}\end{array}$$

By the Shephard's lemma, the Hicksian demand functions are,

$$\begin{array}{}\text{(20)}& \begin{array}{c}{x}^h(p,u)=\frac{\partial e}{\partial p_x}=\frac{u{p}_y^{0.5}}{p_x^{0.5}}\\ y^h(p,u)=\frac{\partial e}{\partial p_y}=\frac{u{p}_x^{0.5}}{p_y^{0.5}}\end{array}\end{array}$$

(c) Compute the total effect of a price change in $p_x$$p_x$ on the Marshallian demand for good $x$$x$​​​ and prove that Slutsky equation holds in this example.

Check for $x$$x$,

$$\begin{array}{}\text{(21)}& \underset{=0}{\underbrace{\frac{\partial x}{\partial p_y}}}=\underset{=\frac{u}{2p_x^{0.5}{p}_y^{0.5}}}{\underbrace{\frac{\partial x^h}{\partial p_y}}}-\underset{\frac{w}{4p_x{p}_y}}{\underbrace{y\frac{\partial x}{\partial w}}}\end{array}$$

The slutsky equation holds since $\frac{u}{2p_x^{0.5}{p}_y^{0.5}}=\frac{w}{4p_x{p}_y}$$\frac{u}{2 p_x^{0.5} p_y^{0.5}} = \frac{w}{4 p_x p_y}$.

2021 June Prelim Part1 Q1 - [Recover direct utility from indirect one]

[15 points] Derive consumer's direct utility function if her indirect utility function has the following form:

and prices are normalized by income, i.e., $p_1=\frac{P_1}{M}$$p_1=\frac{P_1}{M}$ and $p_2=\frac{P_2}{M}$$p_2=\frac{P_2}{M}$ such that the budget constraint is $p_1{x}_1+p_2{x}_2=1$$p_1 x_1+p_2 x_2=1$​​.

By Roy's identity,

$$\begin{array}{}\text{(23)}& \begin{array}{rl}{x}_1& =-\frac{\frac{\partial v}{\partial p_1}}{\frac{\partial v}{\partial y}}=-\frac{\alpha y}{p_1}\\ x_2& =-\frac{\frac{\partial v}{\partial p_2}}{\frac{\partial v}{\partial y}}=-\frac{\beta y}{p_2}\end{array}\end{array}$$

Rearrangement:

$$\begin{array}{}\text{(24)}& \begin{array}{rl}{p}_1& =-\frac{\alpha y}{x_1}\\ p_2& =-\frac{\beta y}{x_2}\end{array}\end{array}$$

The direct utility is then,

$$\begin{array}{}\text{(25)}& u(x_1,x_2)=v(\mathbf{p},1)=A{x}_1^{-\alpha }{x}_2^{-\beta }\end{array}$$

where $A={\alpha }^{\alpha }{\beta }^{\beta }$$A = \alpha^\alpha \beta^\beta$.

2021 Aug Prelim Part1 Q1 - [UMP with multiple goods]

[20 points] The $n$$n$-good Cobb-Douglas utility function is given by:

where $A>0$$A>0$ and $\sum _{i=1}^n{\alpha }_i=1$$\sum_{i=1}^n \alpha_i=1$​. (a) Derive the Marshallian demand functions.

The Marshallian demand for good $i$$i$ is,

$$\begin{array}{}\text{(27)}& x_i^{\ast }(\mathbf{p},w)=\frac{{\alpha }_{i}w}{p_i}\end{array}$$

(b) Derive indirect utility function.

Indirect utility,

$$\begin{array}{}\text{(28)}& v(\mathbf{p},w)=Aw\prod _{i=1}^n{\left(\frac{{\alpha }_i}{p_i}\right)}^{{\alpha }_i}\end{array}$$

(c) Compute the expenditure function.

By duality, $e(\mathbf{p},u)=e(\mathbf{p},v(\mathbf{p},w))$$e(\bold p, u) = e(\bold p, v(\bold p,w))$,

$$\begin{array}{}\text{(29)}& e(\mathbf{p},u)=\frac{u}{A\prod _{i=1}^n{\left(\frac{{\alpha }_i}{p_i}\right)}^{{\alpha }_i}}=\frac{u}{A}\prod _i^n{\left(\frac{p_i}{{\alpha }_i}\right)}^{{\alpha }_i}\end{array}$$

(d) Compute the Hicksian demands.

By Sheppard's lemma,

$$\begin{array}{}\text{(30)}& x^h(\mathbf{p},u)=\frac{\partial e}{\partial p_i}=\frac{u{\alpha }_i}{A}\prod _{j\ne i}^n{\left(\frac{p_j}{{\alpha }_j}\right)}^{{\alpha }_j}\frac{p_i^{{\alpha }_i-1}}{{\alpha }_i^{{\alpha }_i}}\end{array}$$

2020 Oct Prelim Part1 Q1 - [Slutsky equation in elasticity form, engel and cournot aggregation]

[15 points] Barbara consumes 2 goods. We know the following three features of her Marshallian demands: ${\alpha }_1=\frac{1}{3};{\eta }_2=1$$\alpha_1=\frac{1}{3} ; \eta_2=1$ and ${ϵ}_{11}^M=-1$$\epsilon_{11}^M=-1$, where ${\alpha }_1$$\alpha_1$ is the budget share of good 1 , ${\eta }_2$$\eta_2$ is the income elasticity of good 2 and ${ϵ}_{11}^M$$\epsilon_{11}^M$ is the Marshallian own price elasticity of good

Based on this information, please calculate: (a) ${\eta }_1$$\eta_1$​ (income elasticity of good 1 );

By engel aggregation,

$$\begin{array}{}\text{(31)}& {\alpha }_1{\eta }_1+{\alpha }_2{\eta }_2=1\end{array}$$

We solve ${\eta }_1=1$$\eta_1 = 1$.

(b) ${ϵ}_{21}^M$$\epsilon_{21}^M$​ (Marshallian cross price elasticity of good 2 with respect to price of good 1);

By cournot aggregation,

$$\begin{array}{}\text{(32)}& {\alpha }_1{ϵ}_{11}^M+{\alpha }_2{ϵ}_{21}^M+{\alpha }_1=0\end{array}$$

We solve ${ϵ}_{21}^M=0$$\epsilon_{21}^M = 0$. and similarly we solve

(c) ${ϵ}_{12}^h$$\epsilon_{12}^h$ and ${ϵ}_{21}^h$$\epsilon_{21}^h$​ (Hicksian cross-price elasticities).

By the Slutsky equation,

$$\begin{array}{}\text{(33)}& {ϵ}_{ij}^M={ϵ}_{ij}^h-{\alpha }_j{\eta }_i\end{array}$$

We solve ${ϵ}_{21}^h=\frac{1}{3}$$\epsilon_{21}^h = \frac13$.

For ${ϵ}_{21}^h$$\epsilon^h_{21}$, since $\frac{\partial x_2^h}{\partial p_1}=\frac{\partial x_1^h}{\partial p_2}$$\frac{\partial x_2^h}{\partial p_1} = \frac{\partial x_1^h}{\partial p_2}$,

$$\begin{array}{}\text{(34)}& \begin{array}{rl}{ϵ}_{12}^h& =\frac{\partial x_1^h}{\partial p_2}\frac{p_2}{x_1}\\ & =\frac{\partial x_2^h}{\partial p_1}\frac{p_2}{x_1}\\ & =\frac{\partial x_2^h}{\partial p_1}\frac{p_1}{x_2}\frac{x_2{p}_2}{x_1{p}_1}\\ & ={ϵ}_{21}^h\frac{{\alpha }_2}{{\alpha }_1}=\frac{2}{3}\end{array}\end{array}$$

2020 Oct Prelim Part1 Q2

[20 points] Guillermo is a first-grader in a public school in Guadalajara. His Mom packs his lunch every morning before he goes to school. The lunch consists of 2 slices of bread and 5 slices of cheese. Guillermo's utility function is given by $U(B,C)=B^{\frac{2}{3}}{C}^{\frac{1}{3}}$$U(B, C)=B^{\frac{2}{3}} C^{\frac{1}{3}}$​ where $B$$B$​ stands for bread and $C$$C$​​​ stands for cheese. One morning his Mom decides that she is tired of packing his lunch and instead would give him 100 pesos to spend freely on bread and cheese at fixed prices. (a) What should be the prices of bread and cheese to induce Guillermo to consume the same daily ration of 2 slices of bread and 5 slices of cheese?

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2020 Aug Prelim Part1 Q3 - [Labor-leisure model]

[15 points] Assume a slightly modified labor-leisure model where a person supplies $L$$L$ hours of labor and consumes two goods: leisure denoted by $Z$$Z$ and composite good denoted by $C$$C$. It takes $t_Z$$t_Z$ and $t_C$$t_C$ time to consume one unit of $\mathrm{Z}$$\mathrm{Z}$ and $\mathrm{C}$$\mathrm{C}$ and the prices of goods $\mathrm{Z}$$\mathrm{Z}$ and $\mathrm{C}$$\mathrm{C}$ are $p_Z$$p_Z$ and $p_C$$p_C$, respectively. The person's utility function is given by $U(Z,C)$$U(Z, C)$, the budget constraint is given by $p_{Z}Z+p_{C}C=M+wL$$p_Z Z+p_C C=M+w L$ where $M$$M$ is non-labor income and $w$$w$ is the wage rate, and the time constraint is given by $t_Z+t_C+L=T$$t_Z+t_C+L=T$, where $T$$T$ denotes total available time. You can think about $(p_Z+w{t}_Z)$$\left(p_Z+w t_Z\right)$ and $(p_C+w{t}_C)$$\left(p_C+w t_C\right)$ as the full prices of $\mathrm{Z}$$\mathrm{Z}$ and $\mathrm{C}$$\mathrm{C}$.

(a) Substituting for $L$$L$, you can merge two constraints into one, and then set-up utility maximization problem with respect to only budget constraint to obtain Marshallian demand functions for $Z$$Z$ and $C$$C$. Write the general form of these two Marshallian functions as $Z^M=$$Z^M=$ $Z^M(\cdot )$$Z^M(\cdot)$ and $C^M=C^M(\cdot )$$C^M=C^M(\cdot)$ making sure that all arguments of these demand functions are correctly included.

Merging two constraints together,

$$\begin{array}{}\text{(35)}& (p_Z+w{t}_Z)Z+(p_C+w{t}_C)C=M+wT\end{array}$$

Then set up the UMP,

$$\begin{array}{}\text{(36)}& \begin{array}{c}maxU(Z,C),\\ \text{s.t. }(p_Z+w{t}_Z)Z+(p_C+w{t}_C)C=M+wT\end{array}\end{array}$$

We can derive the Marshallian demand in general form,

$$\begin{array}{}\text{(37)}& \begin{array}{c}{Z}^M=Z^M(p_Z+w{t}_Z,p_C+w{t}_C,M+wT)\\ C^M=C^M(p_Z+w{t}_Z,p_C+w{t}_C,M+wT)\end{array}\end{array}$$

(b) Formulate the dual of this problem where the agent minimizes the non-labor income $M$$M$ subject to constant utility constraint $U(Z,C)=u^0$$U(Z, C)=u^0$ to obtain the Hicksian demand functions. Write the general form of these two Hicksian functions $Z^h=Z^h(\cdot )$$Z^h=Z^h(\cdot)$ and $C^h=C^h(\cdot )$$C^h=C^h(\cdot)$ and indirect expenditure function $e^{\ast }(\cdot )=M^{\ast }$$e^*(\cdot)=M^*$​ making sure that all arguments of these functions are correctly included.

Hicksian demand:

$$\begin{array}{}\text{(38)}& \begin{array}{c}{Z}^h=Z^h(p_Z+w{t}_Z,p_C+w{t}_C,u^0)\\ C^h=C^h(p_Z+w{t}_Z,p_C+w{t}_C,u^0)\end{array}\end{array}$$

Indirect expenditure function,

$$\begin{array}{}\text{(39)}& e^{\ast }(\mathbf{p},u^0)=(p_Z+w{t}_Z)Z^h(p_Z+w{t}_Z,u^0)+(p_C+w{t}_C)C^h(p_C+w{t}_C,u^0)\end{array}$$

(c) Using the fundamental identity for leisure $Z$$Z$, derive the Slutsky equation for the change in the wage rate $w$$w$​.

By duality, $Z^M=Z^M(p_Z+w{t}_Z,p_C+w{t}_C,e^{\ast }(\mathbf{p},u^0))$$Z^M = Z^M\left(p_Z+w t_Z,p_C+w t_C, e^*(\bold p,u^0)\right)$​

Take dervitive w.r.t $w$$w$ for both sides,

$$\begin{array}{}\text{(40)}& \begin{array}{r}\frac{\partial Z^M}{\partial w}=\frac{\partial Z^M}{\partial p_Z}{t}_Z+\frac{\partial Z^M}{\partial p_C}{t}_C+\frac{\partial Z^M}{\partial M}(t_Z{Z}^h+(P_Z+t_{Z}w)\frac{\partial Z^h}{\partial w}+t_C{C}^h+(P_C+t_{C}w)\frac{\partial C^h}{\partial w})\end{array}\end{array}$$

$$\begin{array}{}\text{(41)}& \frac{\partial Z^M}{\partial w}=t_Z\frac{\partial Z^h}{\partial p_Z}+t_C\frac{\partial C^h}{\partial p_C}+\frac{\partial Z^M}{\partial M}((P_Z+t_{Z}w)\frac{\partial Z^h}{\partial w}+(P_C+t_{C}w)\frac{\partial C^h}{\partial w})\end{array}$$

which is the Slutcky equation for the change in the wage rate $w$$w$​

(d) Derive the condition for the substitution effect to be positive, i.e., $\frac{\partial Z^h}{\partial w}>0$$\frac{\partial Z^h}{\partial w}>0$. With the substitution effect being positive, what is the requirement on the income effect in order for the total effect of the change in the wage rate on leisure to be negative, i.e. $\frac{\partial Z^M}{\partial w}<0$$\frac{\partial Z^M}{\partial w}<0$​​​.

2019 June Prelim Part1 Q1

[15 points] Consider a consumer with the so called Stone-Geary utility function:

where $c_i$$c_i$ are interpreted as minimum subsistence levels of $x_i$$x_i$, for $i=1,2$$i=1,2$. (a) Derive the consumer's Hicksian demand functions for $x_i$$x_i$.

$\begin{array}{rl}& min{p}_1+p_2{x}_{2}st\cdots {(x_1-c_1)}^{\alpha }{(x_2-c_2)}^{\beta }-u=0\\ & \mathcal{L}=p_1{x}_1+p_2{x}_2-\lambda ({(x_1-c_1)}^{\alpha }{(x_2-c_2)}^{\beta }-u)\\ & \left[x_1\right]:p_1-\lambda \cdot \alpha {(x_1-c_1)}^{\alpha -1}{(x_2-c_2)}^{\beta }=0\\ & \left[x_2\right]:p_2-\lambda \cdot \beta {(x_1-c_1)}^{\alpha }{(x_2-c_2)}^{\beta -1}=0\end{array}$$\begin{aligned} & \min p_1+p_2 x_2 s t \cdots\left(x_1-c_1\right)^\alpha\left(x_2-c_2\right)^\beta-u=0 \\ & \mathcal L=p_1 x_1+p_2 x_2-\lambda\left(\left(x_1-c_1\right)^\alpha\left(x_2-c_2\right)^\beta-u\right) \\ & {\left[x_1\right]: p_1-\lambda \cdot \alpha\left(x_1-c_1\right)^{\alpha-1}\left(x_2-c_2\right)^\beta=0} \\ & \left[x_2\right]: p_2-\lambda \cdot \beta\left(x_1-c_1\right)^\alpha\left(x_2-c_2\right)^{\beta-1}=0 \\ \end{aligned}$

$$\begin{array}{}\text{(43)}& \Rightarrow \frac{p_1}{p_2}=\frac{\alpha }{\beta }\frac{x_2-c_2}{x_1-c_1}\iff p_1(1-\alpha )(x_1-c_1)=p_2\alpha (x_2-c_2)\end{array}$$

$$\begin{array}{}\text{(44)}& \begin{array}{rl}& x_2^h=c_2+u\left(\frac{p_1}{p_2}\right){\left(\frac{1-\alpha }{\alpha }\right)}^{\alpha }\\ & x_1^h=c_1+u{\left(\frac{p_2}{p_1}\right)}^{\alpha }{\left(\frac{\alpha }{1-\alpha }\right)}^{\alpha }\end{array}\end{array}$$

(b) Derive the expenditure function.

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(c) Compare the expenditure function that you obtained in (b) with the expenditure function you would obtain in case where a consumer's utility function was of a simple Cobb-Douglas form $U=x_1^{\alpha }{x}_2^{\beta },\alpha +\beta =1$$U=x_1^\alpha x_2^\beta, \alpha+\beta=1$​​. Explain the difference between the two expenditure functions?

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2019 June Prelim Part1 Q2 - [Quasilinear utility and welfare]

[15 points] Suppose that the consumer's preferences can be represented by the quasilinear utility function:

(a) Show that the consumer indifference curves are vertically parallel, i.e. their slope depends only on $x_1$$x_1$ and not on $x_2$$x_2$ and confirm that the income elasticity of demand for good 1 is zero.

For each indiffernce curve $x_2=\overline{u}-f(x_1)$$x_2 = \bar u - f(x_1)$, the slope of the IC is $\frac{\partial x_2}{\partial x_1}=-f^{\prime }(x_1)$$\frac{\partial x_2}{\partial x_1} = -f^\prime (x_1)$, which is uncorrelated with $x_2$$x_2$​.

• Cornor solution: $x_1=0$$x_1 = 0$, then income effect would be zero.

• Interior solution:

  From the foc, we know $f^{\prime }(x_1)=\frac{p_1}{p_2}$$f^\prime(x_1) = \frac{p_1}{p_2}$, then take derivitives w.r.s. $w$$w$ to both sides, we have

  $$\begin{array}{}\text{(46)}& f^{\prime \prime }(x_1)\frac{\partial x_1}{\partial w}=0\end{array}$$

  Since $f^{\prime \prime }<0$$f^{\prime \prime}<0$ so we have $\frac{\partial x_1}{\partial w}=0$$\frac{\partial x_1}{\partial w} = 0$

Then we can confirm the income elasticity of demand is,

$$\begin{array}{}\text{(47)}& \frac{\partial x_1}{\partial w}\frac{w}{x_1}=0\end{array}$$

(b) Derive the expressions for CV and EV for changes in $p_1$$p_1$​​ using expenditure function and show that these welfare measures are equal.

$$\begin{array}{}\text{(48)}& \begin{array}{c}CV(p^0,p^1,u^0)=e(p^1,u^0)-e(p^0,u^0)\\ EV(p^0,p^1,u^1)=e(p^1,u^1)-e(p^0,u^1)\end{array}\end{array}$$

So,

$$\begin{array}{}\text{(49)}& \begin{array}{rl}CV-EV& =\underset{p_1^1{x}_1^0+p_2^1{x}_2^0}{\underbrace{e(p^1,u^0)}}-\underset{p_1^1{x}_1^1+p_2^1{x}_2^1}{\underbrace{e(p^1,u^1)}}+\underset{p_1^0{x}_1^1+p_2^0{x}_2^1}{\underbrace{e(p^0,u^1)}}-\underset{p_1^0{x}_1^0+p_2^0{x}_2^0}{\underbrace{e(p^0,u^0)}}\\ & =(p_2^1-p_2^0)(x_2^0-x_2^1)\end{array}\end{array}$$

This is because the MRS is always equal to $f^{\prime }(x_1)$$f^\prime (x_1)$, so $x_1$$x_1$ should be the same under the same price, no matter the utility level. i.e. $p_1^1{x}_1^0=p_1^1{x}_1^1$$p_1^1x_1^0 = p_1^1x_1^1$ and $p_1^0{x}_1^1=p_1^0{x}_1^0$$p_1^0x_1^1 = p_1^0x_1^0$.

And we assume no price change in $p_2$$p_2$ so $p_2^1=p_2^0$$p_2^1 = p_2^0$. Thus we have $CV-EV=0$$CV - EV = 0$​. these welfare measures are equal.

Xiaoyi: 我感觉 $e(p^1,u^0)=p_1^1{x}_1^0+p_2^1{x}_2^0$$e(p^1,u^0)=p_1^1 x_1^0 + p_2^1 x_2^0$ 似乎是有问题的。当price变化的时候，Hicksian demand $x(p,u)$$x(p,u)$ 也需要随price变化才对。这里应该是$e(p^1,u^0)=p_1^1{x}_1^1+p_2^0{x}_2^1$$e(p^1, u^0) =p_1^1x_1^1 + p_2^0x_2^1$​ (因为price for good 2并没有改变，所以不用区分1和0）

There should be four $x_2$$x_2$s (before and after price change on two different utility level). However, the differences between $x_2$$x_2$s under the same price ratio but different utility level should be the same (because $x_1$$x_1$ will not change under the same price and the difference between the utility all comes from $x_2$$x_2$). This property gurantees that $CV=EV$$CV = EV$.

(c) What is the relationship between the change in the Marshallian consumer surplus and the EV and CV measures in this case? Provide a proof of your answer.

Conclusion: $\mathrm{\Delta }CV=EV=CS$$\Delta CV = EV = CS$.

By defination,

$$\begin{array}{}\text{(50)}& \mathrm{\Delta }\mathrm{CS}(p^0,p^1,w)={\int }_{p_1^0}^{p_1^1}{x}_1(p,w)d{p}_1\end{array}$$

$$\begin{array}{}\text{(51)}& EV(p^0,p^1,u^1)={\int }_{p_1^0}^{p_1^1}{x}_1^h(p,u^1)d{p}_1\end{array}$$

$$\begin{array}{}\text{(52)}& CV(p^0,p^1,u^1)={\int }_{p_1^0}^{p_1^1}{x}_1^h(p,u^0)d{p}_1\end{array}$$

To show they are equivalent, we have to show $x_1^h(p,u^1)=x_1^h(p,u^0)=x_1\left(p\right)$$x_1^h\left(p, u^1\right) = x_1^h\left(p, u^0\right) = x_1\left(p\right)$. (We know from (a) that the Marshallian demand for good 1 does not depend on income $w$$w$, so $x_1(p,w)=x_1(p)$$x_1(p,w) = x_1(p)$).

The indirect utility could be write in this way,

$$\begin{array}{}\text{(53)}& v(p,w)=p_{1}f(x_1(p))+p_2{x}_2(p,w)=p_{1}f(x_1(p))+p_2(w-x_1(p))\end{array}$$

By duality, we can derive the expenditure function,

$$\begin{array}{}\text{(54)}& e(p,u)=\frac{u-p_{1}f(x_1(p))+p_2{x}_1(p)}{p_2}\end{array}$$

And obviously we can derive the Hicksian demand for $x_1$$x_1$ by Sheppard's lemma:

$$\begin{array}{}\text{(55)}& x_1^h(p,u)=\frac{\partial e}{\partial p_1}=x_1^h(p)\end{array}$$

which is independent to $u$$u$.

Therefore, $x_1^h(p,u^1)=x_1^h(p,u^0)=x_1(p)$$x_1^h\left(p, u^1\right) = x_1^h\left(p, u^0\right) = x_1(p)$. So we have $\mathrm{\Delta }CV=EV=CS$$\Delta CV = EV = CS$​.

2018 Aug Prelim Part1 Q3 - [duality]

[20 points] Consumer 1 has expenditure function $e_1(p_1,p_2,u_1)=u_1\sqrt{p_1{p}_2}$$e_1\left(p_1, p_2, u_1\right)=u_1 \sqrt{p_1 p_2}$ and consumer 2 has utility function $u_2(x_1,x_2)=43x_1^3{x}_2^{\alpha }$$u_2\left(x_1, x_2\right)=43 x_1^3 x_2^\alpha$. (a) What are the Marshallian demand functions for each of the two goods by each of the consumers? Denote the income of consumer 1 by $M_1$$M_1$ and the income of consumer 2 by $M_2$$M_2$. Notice there will be 4 demand functions in total.

For consumer 🥇

The indirect utility function is,

$$\begin{array}{}\text{(56)}& v(\mathbf{p},M_1)=\frac{M_1}{\sqrt{p_1{p}_2}}\end{array}$$

Then, applying the Roy's identity, we can obtain,

$$\begin{array}{}\text{(57)}& \begin{array}{c}{x}_1^1(\mathbf{p},M_1)=\frac{M_1}{2p_1}\\ x_2^1(\mathbf{p},M_1)=\frac{M_1}{2p_2}\end{array}\end{array}$$

For consumer 🥈

$$\begin{array}{}\text{(58)}& \begin{array}{rl}{x}_1^2(\mathbf{p},M_2)& =\frac{3M_2}{(\alpha +3)p_1}\\ x_2^2(\mathbf{p},M_2)& =\frac{\alpha M_2}{(\alpha +3)p_2}\end{array}\end{array}$$

(b) For what value(s) of the parameter $\alpha$$\alpha$ will there exist aggregate (both consumers) demand functions for $x_1$$x_1$ and $x_2$$x_2$​ that are independent of the distribution of income among consumers.

$\alpha =3$$\alpha = 3$.

2018 June Prelim Part1 Q1 - [Hotelling-Wold Identity]

[10 points] Using Hotelling-Wold identity, derive the consumer's inverse Marshallian demand functions $p_1(x_1,x_2)$$p_1\left(x_1, x_2\right)$ and $p_2(x_1,x_2)$$p_2\left(x_1, x_2\right)$ when the utility functions is of the Cobb-Douglas form $u(x_1,x_2)=A{x}_1^{\alpha }{x}_2^{1-\alpha }$$u\left(x_1, x_2\right)=A x_1^\alpha x_2^{1-\alpha}$, for $0<\alpha <1$$0<\alpha<1$. Prices $p_1=\frac{P_1}{M}$$p_1=\frac{P_1}{M}$ and $p_2=\frac{P_2}{M}$$p_2=\frac{P_2}{M}$ are normalized by income and $x_1$$x_1$ and $x_2$$x_2$​​ denote quantities of goods in question.

`Apply the Hotelling-Wold identity,

$$\begin{array}{}\text{(59)}& p_i(\mathbf{x})=\frac{\partial u(\mathbf{x})/\partial x_i}{\sum _{j=1}^n{x}_j(\partial u(\mathbf{x})/\partial x_j)}.\end{array}$$

2018 June Prelim Part1 Q3 - [quasilinear utility function]

[15 points] An individual receives utility from leisure hours $H$$H$ and from consumption $C$$C$ and has the utility function given by $U(H,C)=C+40\ln (H)$$U(H, C)=C+40 \ln (H)$. The hourly wage rate is given by $w$$w$, the non-labor income is $N$$N$, the price of consumption is $p=1$$p=1$ and there is no saving. There are 16 hours in the day that can be devoted to labor $L$$L$ and leisure $H$$H$; the remaining 8 hours the person has to sleep where, of course, she cannot work while sleeping and also sleeping does not enhance utility. (a) Write the individual's full income constraint and interpret it.

The full income constraint is:

$$\begin{array}{}\text{(60)}& C+wH=16w\end{array}$$

The opportunity cost of having one unit of leisure is $w$$w$ unit of consumption.

(b) Set up the utility maximization problem and solve for the demand equations for $H$$H$ and $C$$C$. Throughout, except in question (e), assume there is an interior solution.

$$\begin{array}{}\text{(61)}& \begin{array}{c}maxU(H,C)=C+40\ln (H)\\ s.t.C+wH=16w\end{array}\end{array}$$

Assuming there is an interior solution, the Marshalian demand for $H$$H$ and $C$$C$ are,

$$\begin{array}{}\text{(62)}& \begin{array}{rl}H& =\frac{40}{w}\\ C& =16w-40\end{array}\end{array}$$

(c) Is the demand for leisure always downward sloping with this utility function?

The demand for leisure is always downward sloping w.r.t. $w$$w$.

(d) Write out the individual supply function for labor. Is it always upward sloping with respect to wage rate?

$$\begin{array}{}\text{(63)}& L=16-H=16-\frac{40}{w}\end{array}$$

Assuming the interior solution, it will be always upward sloping.

(e) Would this person ever choose not to work? If so, with what wages?

This player may choose not to work, in that case we have, $H=16,C=0,L=0$$H = 16, C = 0, L = 0$.

When $w<2.5$$w< 2.5$, there is no labor supply.

(f) Find the slope of an indifference curve with $C$$C$ on the vertical axis and $H$$H$ on the horizontal axis.

$\frac{1}{H}$$\frac{1}{H}$.

(g) Would an incease in non-labor income $N$$N$​​​​​​​ affect the amount of labor supplied? Explain the intuition behind your answer.

No. Because change in $H$$H$ does not contain any income effect because of the quasi-linear utility function. To be more straightforward, consider the new budget constraint: $C+wH=16w+I$$C + wH = 16w + I$. And the new marshallian demand functions are,

$$\begin{array}{}\text{(64)}& \begin{array}{rl}H& =\frac{40}{w}\\ C& =16w+I-\frac{40}{w}\end{array}\end{array}$$

So, non-labor income does not have any impact on leisure $H$$H$​.

The intuition behind: the decision of leisure only depends on its opportunity cost (relative to consumption).

2019 Aug Prelim Part 1 Q1 - [Integrability]

[15 points] Consider a Cobb-Douglas utility function $U(x_1,x_2)=x_1{x}_2$$U\left(x_1, x_2\right)=x_1 x_2$​ and show that the corresponding system of Marshallian demand functions satisfy all conditions required for integrability. Just to make sure you remember what these conditions are, you need to show: (a) budget balanceness (Walras law),

The Marshallian demand function,

$$\begin{array}{}\text{(65)}& \begin{array}{c}{x}_1(\mathbf{p},w)=\frac{w}{2p_1}\\ x_2(\mathbf{p},w)=\frac{w}{2p_2}\end{array}\end{array}$$

Then the Walras law hold because $p_1{x}_1+p_2{x}_2=w$$p_1x_1 + p_2 x_2 = w$.

(b) homogeneity of degree zero in prices and income,

Easy to show.

(c) symmetry of the Slutsky matrix, and

The Slutsky matrix is,

$$\begin{array}{}\text{(66)}& \begin{array}{c}S=\left[\begin{array}{cc}-\frac{w}{4p_1^2}& \frac{w}{4p_1{p}_2}\\ \frac{w}{4p_1{p}_2}& -\frac{w}{4p_2^2}\end{array}\right]\end{array}\end{array}$$

which is obviously symmetric.

(d) negative semidefiniteness of the Slutsky matrix.

First leading principal minor $\le 0(<0)$$\leq0(<0)$

Second leading principal minor $\ge 0$$\geq0$ ($=0$$= 0$).

which is negative semidefinite

Uncertainty

2022 June Prelim Part1 Q3 - [Insurance problem]

[20 points] Consider the following optimal insurance problem: An agent with preferences represented by an expected utility function has a deterministic initial wealth $w>0$$w>0$ and faces a risk of losing $L$$L$ such that $0<L<w$$0<L<w$. The agent can purchase insurance at a premium $\rho \in (0,1)$$\rho \in(0,1)$ per one dolar of coverage to be paid in case of loss. The probability of loss is $\pi \in (0,1)$$\pi \in(0,1)$ and there is nothing the agent can do to affect that probability, i.e. there is no moral hazard. The agent is strictly risk averse with the Bernoulli utility function strictly increasing and twice differentiable. Suppose that $\rho >\pi$$\rho>\pi$, i.e., the premium is not actuarially fair. (a) Show that the optimal insurance amount $a$$a$ is less than the full insurance.

Solve for the optimal insurance amount $a$$a$

The expected utility under optimal insurance could be specified as:

$$\begin{array}{}\text{(67)}& \mathbb{E}U=\underset{a}{max}(1-\pi )U(w-\rho a)+\pi U(w-L+(1-\rho )a)\end{array}$$

The FOC:

$$\begin{array}{}\text{(68)}& \frac{U^{\prime }(w-\rho a)}{U^{\prime }(w-L+(1-\rho )a)}=\frac{\pi }{1-\pi }\frac{1-\rho }{\rho }=\frac{\pi -\rho \pi }{\rho -\rho \pi }<1\end{array}$$

since $\rho >\pi$$\rho > \pi$.

Since the consumer is assumed to be strictly risk averse, $U^{\prime \prime }<0$$U^{\prime \prime} <0$.

Then, $U^{\prime }(w-\rho a)<U^{\prime }(w-L+(1-\rho )a)\iff w-\rho a>w-L+a-\rho a$$U^\prime (w - \rho a) < U^\prime (w - L +(1-\rho) a) \Leftrightarrow w -\rho a > w - L +a- \rho a$, which implies $L>a$$L>a$.

(b) Show that the agent's demand for insurance is an increasing function of the amount of loss $L$$L$​.

From the FOC:

$$\begin{array}{}\text{(69)}& \frac{U^{\prime }(w-\rho a)}{U^{\prime }(w-L+(1-\rho )a)}=\frac{\pi }{1-\pi }\frac{1-\rho }{\rho }\end{array}$$

Let $\frac{\pi }{1-\pi }\frac{1-\rho }{\rho }=A$$\frac{\pi}{1-\pi} \frac{1-\rho}{\rho} = A$, we can write the FOC condition in this way:

$$\begin{array}{}\text{(70)}& U^{\prime }(w-\rho a)=A{U}^{\prime }(w-L+(1-\rho )a)\end{array}$$

Take derivative w.r.t. $L$$L$ for both sides,

$$\begin{array}{}\text{(71)}& -\rho U^{\prime \prime }(w-\rho a)\frac{\partial a}{\partial L}=A{U}^{\prime \prime }(w-L+(1-\rho )a)[-1+(1-\rho )\frac{\partial a}{\partial L}]\end{array}$$

Rearranging, we can derive,

$$\begin{array}{}\text{(72)}& \frac{\partial a}{\partial L}=\frac{U^{\prime \prime }(w-L+(1-\rho )a)}{\rho U^{\prime \prime }(w-\rho a)+(1-\rho )A{U}^{\prime \prime }(w-L+(1-\rho )a)}\end{array}$$

Since the consumer is assumed to be strictly risk averse, $U^{\prime \prime }<0$$U^{\prime \prime} <0$, both the numerator and deminator are negative, so $\frac{\partial a}{\partial L}>0$$\frac{\partial a}{\partial L}>0$, denamd for insurance is an increasing funtction of the loss.

2021 June Prelim Part1 Q2

[15 points] Suppose an individual lives for two periods; his goods are $x_1$$x_1$, the amount consumed in period 1 , and $x_2$$x_2$ the amount consumed in period 2 . His utility function over pairs $(x_1,x_2)$$\left(x_1, x_2\right)$ is:

At the beginning of period 1 , he has an amount of wealth $y>0$$y>0$, which can be allocated to three purposes:

• an amount $0\le x_1\le y$$0 \leq x_1 \leq y$ can be consumed in period 1 .

• an amount $0\le b\le y$$0 \leq b \leq y$ can be stored; this will be available for consumption in period 2 .

• an amount $0\le z\le y$$0 \leq z \leq y$ can be invested into a risky asset; this asset's returns will be available for consumption in period 2. Suppose that with probability $p$$p$, the net return on this risky asset is $Rz$$R z$, where $R>1$$R>1$, and with probability $(1-p)$$(1-p)$ the net return is 0 . Therefore, the amount available for consumption in period 2 will be $b+Rz$$b+R z$ with probability $p$$p$, and $b$$b$ with probability $(1-p)$$(1-p)$. In period 2 , the individual will consume all available resources.

Find the choice of $x_1,b,z$$x_1, b, z$ that maximizes expected utility $E\left[u(x_1,x_2)\right]$$E\left[u\left(x_1, x_2\right)\right]$, subject to the budget constraint $x_1+b+z\le y$$x_1+b+z \leq y$

The marginal expected utiltity is positive, so we write the budget contraint as an equation.

Set up the lagrange function:

$$\begin{array}{}\text{(74)}& \mathcal{L}=p[\ln x_1+\ln (b+Rz)]+(1-p)[\ln x_1+\ln b]-\lambda [x_1+b+z-y]\end{array}$$

The FOCs:

$$\begin{array}{}\text{(75)}& \begin{array}{rl}& \left[x_1\right]:\frac{p}{x_1}+\frac{1-p}{x_1}-\lambda =0\\ & [b]:\frac{p}{b+Rz}+\frac{1-p}{b}-\lambda =0\\ & [z]:\frac{pR}{b+Rz}+-\lambda =0\end{array}\end{array}$$

Plus the budget constraints,

Rearranging:

• $b=\frac{(1-p)R}{R-1}{x}_1$$b=\frac{(1-p) R}{R - 1}x_1$​

• $z=\frac{pR-1}{R-1}{x}_1$$z=\frac{p R-1}{R-1} x_1$​

And plus it into the budget constraint, we have $x_1=\frac{2}{y}$$x_1 = \frac{2}{y}$.

Then, $b=\frac{(1-p)R}{R-1}\frac{y}{2}$$b = \frac{(1-p)R}{R-1}\frac{y}{2}$, $z=\frac{pR-1}{R-1}\frac{y}{2}$$z = \frac{pR-1}{R-1}\frac{y}{2}$.

2018 Aug Prelim Part1 Q2

[5 points] Consider a utility function in income and prices:

and relying on the concept (coefficient) of relative risk aversion, determine the type of risk preferences that this utility function describes.

Producer Theory

2023 June Prelim Part1 Q2 - [Cost function]

[15 points] For a general strictly quasi-concave production function $q=f(x_1,x_2,\dots ,x_n)$$q=f\left(x_1, x_2, \ldots, x_n\right)$, assume that in the short-run $x_1$$x_1$ is fixed at some ${\overline{x}}_1$$\bar{x}_1$ level and show that: (a) the long-run average cost curve (LAC) is the lower envelope of the entire family of short-run average cost curves (SACs) obtained by varying ${\overline{x}}_1$$\bar{x}_1$​.