<p>I've spent considerable amount of time on this question, in vain. It is from one of the competitive exams for admission to a grad econ program. Help would be tremendously appreciated. Thanks.   </p>

<p>An economy comprises of a consolidated household sector, a firm sector and the
government. The household supplies labour <span class="math-container">$L$</span> to the firm. The firm produces a single good <span class="math-container">$Y$</span> by means of a production function <span class="math-container">$$Y = F(L);\ F'(L) &gt; 0,\ F′′(L) &lt; 0 $$</span> and
maximizes profits <span class="math-container">$$Π = PY −WL $$</span> where <span class="math-container">$P$</span> is the price of <span class="math-container">$Y$</span> and <span class="math-container">$W$</span> is the wage rate.    </p>

<p>The household, besides earning wages, is also entitled to the profits of the firm. The household maximizes utility <span class="math-container">$U$</span>, given by:
<span class="math-container">$$U = \frac12lnC\ +\frac12ln (\frac{M}{P}) \ - d(L)$$</span></p>

<p>where <span class="math-container">$C$</span> is consumption of the good and <span class="math-container">$\frac{M}P$</span> is real balance holding. The term <span class="math-container">$d(L)$</span> denotes the disutility from supplying labour; with <span class="math-container">$ d'(L) &gt; 0,\ d′′(L) &gt; 0 $</span>. The household's budget constraint is given by:
<span class="math-container">$$PC + M = WL + Π + \bar{M} - PT $$</span> ;
where <span class="math-container">$\bar{M}$</span>  is the money holding the household begins with, M is the holding they end up with and <span class="math-container">$T$</span> is the real taxes levied by the government. The government's demand for the good is given by <span class="math-container">$G$</span>.The government's budget constraint is given by:
<span class="math-container">$$M - \bar{M} = PG - PT$$</span>
Goods market clearing implies: <span class="math-container">$Y = C + G$</span>    </p>

<p>(a) Prove that <span class="math-container">$\frac{dY}{dG} ∈ (0,1)$</span>, and that government expenditure crowds out private consumption (i.e., <span class="math-container">$\frac{dC}{dG} &lt; 0 $</span>)  </p>

<p>(b) Show that everything else remaining the same, a rise in <span class="math-container">$\bar{M}$</span> leads to an equiproportionate rise in <span class="math-container">$P$</span></p>

CC BY-SA 4.0
2019-03-04T06:50:17.697Z
Hi. Welcome to Economics.SE. Please read the welcome before posting. We tend to be a little picky about homework questions. Please show your own efforts
emeryville
CC BY-SA 4.0
2019-03-04T09:06:42.620Z
Hi @emeryville to be honest, I've studied introductory macroeconomics, and microeconomics. And this problem seems beyond even intermediate topics. And its not really a homework problem. Its from an old 'entrance' exam paper for an university's grad program. I have essentially calculated Marginal utilities for $ C, \frac{M}P, and \ L$  using the utility function provided and tried to use the equation that marginal utility per money expended be same for all of them. Substituted it back into different equations. And then I am clueless.
DS112
<macroeconomics><utility><production-function><profit-maximization>
Question about an economy with 3 components household, firm and government with functions given
DS112
# Question about an economy with 3 components household, firm and government with functions given

(Asked by: username_0 on 2019-03-03)

I've spent considerable amount of time on this question, in vain. It is from one of the competitive exams for admission to a grad econ program. Help would be tremendously appreciated. Thanks.

An economy comprises of a consolidated household sector, a firm sector and the
government. The household supplies labour $L$ to the firm. The firm produces a single good $Y$ by means of a production function $$Y = F(L);\ F'(L) > 0,\ F′′(L) < 0 $$ and
maximizes profits $$Π = PY −WL $$ where $P$ is the price of $Y$ and $W$ is the wage rate.

The household, besides earning wages, is also entitled to the profits of the firm. The household maximizes utility $U$, given by:
$$U = \frac12lnC\ +\frac12ln (\frac{M}{P}) \ - d(L)$$

where $C$ is consumption of the good and $\frac{M}P$ is real balance holding. The term $d(L)$ denotes the disutility from supplying labour; with $ d'(L) > 0,\ d′′(L) > 0 $. The household's budget constraint is given by:
$$PC + M = WL + Π + \bar{M} - PT $$ ;
where $\bar{M}$ is the money holding the household begins with, M is the holding they end up with and $T$ is the real taxes levied by the government. The government's demand for the good is given by $G$.The government's budget constraint is given by:
$$M - \bar{M} = PG - PT$$
Goods market clearing implies: $Y = C + G$

(a) Prove that $\frac{dY}{dG} ∈ (0,1)$, and that government expenditure crowds out private consumption (i.e., $\frac{dC}{dG} < 0 $)

(b) Show that everything else remaining the same, a rise in $\bar{M}$ leads to an equiproportionate rise in $P$

**Tags:** macroeconomics, utility, production-function, profit-maximization

> **Comments:**
> **username_1**: Hi. Welcome to Economics.SE. Please read the welcome before posting. We tend to be a little picky about homework questions. Please show your own efforts  
> **username_0**: Hi @username_1 to be honest, I've studied introductory macroeconomics, and microeconomics. And this problem seems beyond even intermediate topics. And its not really a homework problem. Its from an old 'entrance' exam paper for an university's grad program. I have essentially calculated Marginal utilities for $ C, \frac{M}P, and \ L$  using the utility function provided and tried to use the equation that marginal utility per money expended be same for all of them. Substituted it back into different equations. And then I am clueless.  

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## 0 answers

