# Question about an economy with 3 components household, firm and government with functions given

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 $$P$$

• 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 Mar 4 '19 at 6:50
• 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 Mar 4 '19 at 9:06