Question: The demand for and supply of labour

The production function for this economy is:

Y = AK^αIn(N)

Where Y= output, A = total factor productivity, K= the capital stock, N= the amount of labour available for work and α = the share of output going to capital. The symbol "In" is the natural log.

Assume that K = 10,000; A = 6.4; α = 0.25; and that the supply of labour is N^s = w , where w = the real wage rate.

a) Derive an algebraic expression for the demand for labour, Based on your results, derive an expression for the elasticity of labour demand with respect to the real wage rate. Use the values for A, K and a shown above to find the demand for labour, the wage rate and the level of output in this economy.

b) Suppose that the supply of labour is now given by In(N^s) = 2ln(w). Find the wage rate and the demand for labour in equilibrium. Explain briefly why the results here are different from what you found in part a). Illustrate how your results differ from part a) using the standard supply and demand diagram.

c) Assume that the supply of labour is once again given by N^s - w. The government wants to raise the level of employment by 25%. To help the government, find the wage rate that would induce a 25% increase in labour supply. Next, using the wage rate you found, find out how many workers would be demanded by firms. By how much does the supply of labour exceed the demand?

d) Looking at the demand for labour, there appear to be two ways to achieve the government's goal: raise either productivity or the capital stock.

a. By how much would "A" have to rise to absorb the increased number of people wishing to work?

b. Suppose that the government introduced policies to encourage firms to invest more. Assuming A remains at its original value of 6.4, what would be the level of the capital stock that would induce firms to increase employment by 25%?

Question 2: The consumption function

For this question, you will need to apply your knowledge of the inter-temporal consumption function as described and particularly in the Appendix to that Chapter which is posted on the course website.

Consumers must decide how much they wish to consume today (c_i), versus in the future (c₂). In making their decisions, they are constrained by the amount of their present income (yi) and future income (y2). They also have an initial endowment of assets (a). The real interest rate in this economy is r and individuals can lend and borrow at this rate.

a) Assume the slope of the inter-temporal utility function is - βc₂/c₁. Based on this, derive a relationship between first period consumption and y₁, y₂, a and r. Show the steps you need to take to get your answer.

b) You are given the following: y₁ = 90; α = 10; y₂ =105; and r = 5%. Find present and future consumption if β = 1. Verify that your results satisfy the inter-temporal budget constraint. What is unique about the values of c₁ and c₂ that you found? How much do consumers save in the present to support consumption in the future?

c) Suppose that the initial endowment of assets increased from 10 to 15. What would be the effect on present and future consumption as well as savings? What effect is operating here?

d) Start again with assets equal to 10 and assume that the interest rate rises to 10')/0. Use your consumption function to find present and future consumption as well as savings. Compare your results with those of part b) above. What has happened to saving?

e) Start with the conditions presented in part b) above. Assume the amount of assets (a) is instead a lump-sum transfer from the government, financed by issuing debt. To balance its budget, the government will raise taxes that will be subtracted from future income. Re-write the consumption function to take account of this and use it to find present and future consumption. Briefly explain what is happening. Explain how your results would be affected if households were income constrained (Le., they can neither borrow or lend at the current rate of interest)?

Question 3: Equilibrium in the goods market

The marginal product of capital for the next period (MPK^f) is related to next period's desired capital stock (K_t+1) in the following way:

MPK^f = 175 - 10K_t+1

a) In this economy, the real rate of interest (r) is 6%; the rate of depreciation (d) is 9%; the marginal rate of taxation on corporate earnings (r) is 25%; and the price of capital (Pk) is 125. Based on this information, find the desired capital stock. To have some understanding of the role of the user cost of capital in determining the desired capital stock, calculate successively the effect of a drop in Pk to 75 and then an increase in r to 50%.

b) Assume that the initial conditions in part a) are in place and that the capital stock in period t is 12 (i.e., K, = 12). Use this information to derive an investment equation showing r as a variable on the right hand side. Based on the information you have, what is the level of gross investment in this economy?

c) The equation below represents real consumption in this economy:

C = 3 + 0.80Y - 100r

Derive an equation for national saving. Then, setting saving equal to investment (using the investment function from part b), derive a relationship for the rate of interest as a function of the other variables that affect saving and investment. If Y= 20 and government spending (G) was 2.92, what would be the real rate of interest?

d) Holding Y constant, what would be the effect of a fall in government spending from 2.92 to 2.0? By how much has investment changed? Verify that saving is equal to the new level of investment. Describe briefly what is happening to both investment and saving?

e) Assume now that the economy is small and open to trade with the rest of the world. As well, assume that G is once again 2.92. Use the saving and investment function you found above to derive the new equilibrium conditions. If the world rate of interest was 7%, what would be the level of net exports (NX)? Use the equilibrium condition to calculate the effect of a fall in G from 2.92 to 2. What effect is operating here?